Citation
Inverse kinematic analysis of robot manipulators

Material Information

Title:
Inverse kinematic analysis of robot manipulators
Creator:
Manseur, Rachid, 1954- ( Dissertant )
Doty, Keith L. ( Thesis advisor )
Lam, Herman ( Reviewer )
Principe, Jose C. ( Reviewer )
Ritter, Gerhard X. ( Reviewer )
Selfridge, Ralph G. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1988
Language:
English
Physical Description:
viii, 139 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Geometric angles ( jstor )
Geometry ( jstor )
Inner products ( jstor )
Inverse kinematics ( jstor )
Iterative methods ( jstor )
Kinematics ( jstor )
Mathematical variables ( jstor )
Matrices ( jstor )
Robotics ( jstor )
Robots ( jstor )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis Ph. D
Kinematics ( lcsh )
Manipulators (Mechanism) ( lcsh )
Robotics ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
Computer controlled robot manipulators are becoming an important part of automated manufacturing plants thereby creating a need for reliable and fast control algorithms that can improve the performance of robot manipulators in industrial applications. An important part of such control algorithms is the inverse kinematics portion which consists of computing the values of the robotic joint variables corresponding to a desired and effector position and orientation. This work is based on a new approach that uses orthogonality of rotation matrices to reduce the problem to a simpler form. The reduction techniques are first used to analyze to the kinematics of four-degree-of-freedom (DOF) robots. The results obtained are then applied to the study of five- and six-degree-of-freedom manipulators. Fast one-and two-dimensional numerical techniques for solving five- and six-DOF arms of arbitrary geometry are developed. These new methods provide a large reduction in computational complexity and can be easily implemented in real-time applications. another contribution of this work is a classification of robot geometries in terms of inverse kinematic complexity. Some new sufficient structural conditions for the possibility of closed-form solutions for five- and six-DOF robot manipulators are described. In the case of six-DOF arms, structural conditions for the applicability of a one-dimensional iterative technique are also provided. Finally, in the example applications of the techniques presented here, we describe a six-degree-of freedom manipulator capable of achieving a particular end-effector pose in sixteen distinct configurations.
Thesis:
Thesis (Ph. D.)--University of Florida, 1988.
Bibliography:
Includes bibliographical references.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Rachid Manseur.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Rachid Manseur. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
001113629 ( ALEPH )
19866092 ( OCLC )
AFL0266 ( NOTIS )

Downloads

This item has the following downloads:


Full Text















INVERSE KINEMATIC ANALYSIS
OF ROBOT MANIPULATORS












By


RACHID MANSEUR


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1988

..























To those who fought for the freedom and education of all Algerians.

..













ACKNOWLEDGMENTS


This work would not have been possible without the

teachings, the constant guidance, and the expert advice I received from my advisor, Professor Keith L. Doty. I am

especially grateful for his encouragements, his enthusiasm,

his great sense of humor, and his friendliness which made this work so enjoyable.

I would like to thank Dr. Herman Lam, Dr. Gerhard X. Ritter, Dr. Ralph G. Selfridge, and Dr. Jose C. Principe for their excellent service on my final committee. Thanks are also due to Dr. Giuseppe Basile and Dr. Eginhard J. Muth for their previous service on my committee.

I also wish to express my appreciation to Dr. Carl D. Crane III, for his help with the computer graphics simulations and the photographs of Figure 9.4 of this text,

and to Dr. Joseph Duffy for taking the time to answer my questions.

The financial support I received, throughout my

doctoral studies, from the Mathematics Department and from the Electrical Engineering Department of the University of Florida, in the form of teaching assistantships, is gracefully acknowleged. Special thanks are due to


iii

..









Dr. Peyton Z. Peebles and to Dr. Robert L. Sullivan for their timely help and understanding.

I would like to take this opportunity to thank Ms. Greta Sbroco for being such a great graduate sectretary, for caring, and for her constant encouragements.

All the students of the Machine Intelligence Laboratory also deserve recognition for creating such a friendly and productive working environment. In particular, I would like to thank Dr. Subbian Govindaraj and Mr. David Alderman for their help in the laboratory.

I would also like to express my gratitude and my love to my family, specially my parents, Omar Amokrane Manseur and Dhia Yamina Si-Ahmed, for their support, their understanding and their patience throughout the long years of my education.

My gratitude also goes to my parents-in-law, Tahar Zbiri and Faiza Houasnia for their help and support, and for entrusting me with their daughter.

A very special and sweet thought goes to my wife,Zohra, for her support, her patience throughout our common student life, and for my beautiful children, Mehdi and MayaNibal, for being there and making it all worthwhile.

Finally, I wish to thank all our friends in Gainesville for making it such a great city to live in.

..















TABLE OF CONTENT



Page

ACKNOWLEDGEMENTS. iii ABSTRACT. vii CHAPTERS

1 INTRODUCTION. .1

2 THE INVERSE KINEMATICS PROBLEM. 5

Notation and Mathematical Preliminaries. 5 Problem Definition. 10

3 EXISTING SOLUTIONS. 13

Closed-Form Architectures. 13 Record and Playback. 14 Numerical Techniques. 15

4 NEW APPROACH. 18

Link-Frames Assignment. 18 The Reduced System of Equations. 19 Additional Inverse Kinematic Equations. 24 Solving Inverse Kinematic Equations .25 Exchanging Base and End-Effector Frames. 27,

5 SOLVING 4-DOF MANIPULATORS. 29

Reduced System of Equations. 29 Special 4-DOF Manipulator Geometries 35

6 SOLVING 5-DOF MANIPULATORS. 51

One-Dimensional Iterative Method .51
Five-DOF Robots with Closed-Form*
Solution. 63.

..















7 SOLVING 6-DOF MANIPULATORS. 70

Reduction to a 4-DOF Problem. 70 Two-Dimensional Iterative Technique .72 one-Dimensional Method. 82 Closed-Form Solution. 83

8 ORTHOGONAL MANIPULATORS. 87 9 APPLICATION EXAMPLES. 92

Example l:.The PUMA 560 .92 Example 2: The GP66. 98 Example 3: The 0M25 Manipulator. 108 Example 4: 0M37 Manipulator. 122
Example 5: A General Geometry 6-DOF
Manipulator. 124

10 CONCLUSION AND FUTURE WORK. 126

APPENDIX SOLVING FOR.13 REFERENCES. 136 BIOGRAPHICAL SKETCH. 139


vi,

..














Abstract of Dissertation Presented
to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy



INVERSE KINEMATIC ANALYSIS OF
ROBOT MANIPULATORS

By

RACHID MANSEUR

August 1988


Chairman: Dr. Keith L. Doty
Major Department: Electrical Engineering


Computer-controlled robot manipulators are becoming an important part of automated manufacturing plants thereby creating a need for reliable and fast control algorithms that can improve the performance of robot manipulators in industrial applications. An important part of such control algorithms is the inverse kinematics portion which consists of computing the values of the robotic joint variables corresponding to a desired end-effector position and orientation. This work is based on a new approach that uses orthogonality of rotation matrices to reduce the problem to a simpler form. The reduction techniques are first used to analyze the kinematics of four-degree-of-freedom (DOF) robots. The results obtained are then applied to the study


vii

..








of five- and six-degree-of-freedom manipulators. Fast oneand two-dimensional numerical techniques for solving fiveand six-DOF arms of arbitrary geometry are developed. These new methods provide a large reduction in computational complexity and can be easily implemented in real-time applications. Another contribution of this work is a classification of robot geometries in terms of inverse kinematic complexity. Some new sufficient structural conditions for the possibility of closed-form solutions for five- and six-DOF robot manipulators are described. In the case of six-DOF arms, structural conditions for the applicability of a one-dimensional iterative technique are also provided. Finally, in the example applications of the techniques presented here, we describe a six-degree-of freedom manipulator capable of achieving a particular endeffector pose in sixteen distinct configurations.


viii

..














CHAPTER 1
INTRODUCTION



An important part of computer control algorithms for open serial kinematic chains is the inverse kinematics section. In any robotic application, the' hand or endeffector of the robot may move along a trajectory specified as a sequence of points at which the end-effector pose (orientation and position) is known. While this trajectory

is specified in Cartesian coordinates, the motion of the robot is controlled through individual joint actuators that

produce the necessary rotation in revolute joints, or the translation in prismatic joints. The robot controller must,

therefore, be supplied the values of the joint variables corresponding to the end-effector pose, i.e., the

coordinates in joint space of the robot hand for each point along the trajectory must be computed. The conversion oftrajectory locations from Cartesian coordinates to joint coordinates is referred to as the inverse kinematics problem.

A desirable inverse kinematic algorithm is one capable of producing the joint coordinates in real-time. While the

robot hand is at, or approaching, one location along the trajectory, the algorithm must be able to produce the joint

..









coordinates for the next pose. In tasks where speed and precision are important, the real-time requirement puts heavy constraints on the computation time of the inverse kinematic algorithm.

The forward kinematics problem, the conversion from joint space to Cartesian space, is a much simpler problem that has a unique closed-form solution. In most cases a robot manipulator can achieve a desired end-effector pose in more than one configuration. The question of just how many distinct solutions there are to the inverse kinematics problem of general six-degree-of-freedom (DOF) robot manipulators has interested a few researchers. Roth, Rastegar, and Scheinman (1973) put an upper bound of 32 on the degree of a polynomial equation (in one joint variable) that can be derived from the inverse kinematics problem of six-DOF manipulators. A similar result was obtained by Duffy and Crane (1980), using the equivalence between an open 6-revolute-DOF kinematic chain and the 7-revolute

single-loop spatial mechanism. Therefore, the number of inverse kinematic solutions for 6-revolute-DOF manipulators could be at most 32. More recently, Lee and Liang (in

press), using Duffy's method, were able to reduce the degree of the inverse kinematic polynomial equation to 16, thereby reducing the upper bound on the number of inverse kinematic solutions to 16. Tsai and Morgan (1984), illustrating a new inverse kinematic method capable-of producing all soldtions,

..









found a robot manipulator and an end-effector pose with 12 possible solutions. Manseur and Doty (in press) described a simple manipulator geometry capable of achieving a

particular end-effector pose in 16 distinct configurations, thereby closing the proof that 16 is indeed the maximum achievable number of inverse kinematic solutions for six-DOF robot manipulators. The manipulator and the pose for which

the sixteen solutions were found and the inverse kinematic solution search algorithm used are discussed in Chapter 9, Example 3 of this dissertation.

Another desirable property of an inverse kinematic algorithm is the capability of computing more than one solution, so that a solution can be chosen according to some optimality or collision avoidance criteria. Although f or

manipulators with simple geometries, such as the PUMA 560 industrial robot, several possible solutions can be obtained

in closed-form, this multiple solution property conflicts with the real-time requirement discussed earlier for many other robots that must rely on iterative techniques.

After introducing the notation and some mathematical preliminaries and a brief discussion of existing inverse kinematic methods, we present a new-approach to, the -in verse kinematic problem based on a reduced set of nonlinear equations. This new approach is then used to analyze the kinematics of four-, five-, and .six-degree-of -freedom manipulators. -.-Some simple and efficie nt iterative

..






4

techniques are described and sufficient manipulator structural conditions for the applicability of these methods are determined. All the methods developed in this

dissertation are illustrated by examples in Chapter 9. Chapter 10 summarizes the final results, discusses the

contributions of this work to the f ield of robotics, and presents related topics and areas of future research.

..













CHAPTER 2
THE INVERSE KINEMATICS PROBLEM



Notation and Mathematical Preliminaries The DH Parameters

A robot manipulator is modelled as an open kinematic chain of rigid bodies (links) connected by joints. A

reference frame is assigned to each link along the chain starting with the base frame F0, assigned to the fixed link, up to the end effector frame Fn, for a manipulator with n degrees of freedom (DOF). The position and orientation of frame Fi=(xi, Yi, zi), with respect to the preceding frame Fi_, are entirely described by the four DH-parameters di, 8Gi, ai and ai (Denavit and Hartenberg 1955). These parameters are illustrated in Figure 2.1 and defined as: di = distance between the common normal to axes zi-1 and z

and the common normal to zi and zi+1 measured along

axis zi.

i = the angle of rotation about zi so that xi becomes

parallel to xi-1 when ei = 0.

a = the length of the common normal to axes zi-1 and zi. ai = the angle of rotation about xi so that zi becomes

parallel to zi-1 when ai = 0.

..













I/





1*
di I





F x F
Cj I




X
iI i










link (i- i)ink i











Figure 2.1. The DH-parameters.

..









When joint i is revolute, parameter ei is the joint variable and, if joint i is prismatic, the joint variable is di. When applicable, di measures the translation along axis zi_.


Homogeneous Matrices

If a vector iu = [iux, iuy, iuz]T is expressed in frame Fi, its expression with respect to frame Fi_, i-lu, satisfies


i-lu Ci -Siri Sici aiCi iux
-x i.11 1 i ii Ux


= (2 .1)
i-1a 1
i z 0 i i di
z i z

1 0 0 0 1 1



or in a compact notation,


i-1u u
= Ai (2.2)
1 1



where Ti=cos(ai), ci=sin(ai), Ci=cos(8i), and Si=sin(ei) and Ai is the homogeneous frame-transform matrix (Paul 1981). The leading superscript indicates the frame of expression.

The homogeneous matrix transform merely expresses the fact that frame Fi can be obtained from frame Fi-1 by the following sequence of basic transforms:

1. Rotation about zi_1 of angle ei whose homogeneous matrix is

..











Ci Si

-Si Ci
i i


Rz (ei) =


2. Translation the matrix

1 0
Trz(di) =


0 0 0 0


0 0 1 0 0 0 0 1


of di units along axis zi-1 described by


0 0 0 1 0 0


0 0 1 d 0 0 0 1


3. Translation of homogeneous transform


Trx (ai) =


ai units along axis xi, with


1 0 0 ai

0 1 0 0 0 0 1 0 0 0 0 1


4. Rotation about xi of angle ai,


Rx(a i) =


1 0 0 0


0 7i -Ci
0 ai i 0 a1 r


0 0 0 1


The matrix A1 of Eq. (2.1) is obtained by the product (Paul, 1981),


(2.3)


(2.4)


(2.5)


(2.6)

..








Ai = Rz(9i) Trz(di) Trx(ai) Rx(ai). A useful decomposition of matrix Ai is


Ai = Ai Bi (2.7)


with the definitions


Ai = Rz(ei) (2.8)

and

Bi = Trz(di) Trx(ai) Rx(ai). (2.9)


Explicitly, matrix Bi is



1 0 0 ai

0 Ti -ai 0 G ki
Bi (2.10)
0 ai Ti di 0 0 0 1

0 0 0 1


where Gi is the upper left 3 x 3 in Bi and ki is the upper right 3 x 1 vector of Bi. The upper left 3 x 3 matrix in Ai is the rotation matrix Ri necessary to align the unit vectors of Fi with their counterparts in Fi_1, while vector



aiCi

i = aiSi

di


positions the origin of Fi with respect to Fi_.1

..









A compact and useful expression for Ai is


(2.11)


Rotation matrices are orthogonal, so Ri-l= RT, where the superscript T denotes the transpose operation, and the inverse of matrix Ai can be expressed as


Ci



Sci
-s0
0


Si

CiT i

-Cii
-c0 a
0


-ai

-oidi

-ridi
1


RiT (-RiTli)
= (2.12)
0 0 0 1


Problem Definition


If the orientation of the end-effector is specified by the rotation matrix R, necessary to align the unit vectors of the end-effector frame Fn with the corresponding vectors of base frame F0, and the position of the origin of the endeffector frame is given as a vector p with respect to the base frame F0, then the end-effector pose is adequately described by the 4 x 4 matrix


nx bx tx n b ty nz b tz 0 0 0


n b t p R p

0 0 0 1 0 0 0 1


Ai-1


(2.13)

..










where

nx b tx Px

n = ny b = by t= t p = p

nz bz tz Pz


and

n b t

R = n b ty

nz bz tz



The inverse kinematics problem for a n-degree-offreedom manipulator consists of finding a set of joint variables values, called a solution set, that will satisfy the equation


A1 A2 A3 A4 A5.An = P. (2.14)


This matrix equation gives rise to a system of nonlinear equations whose complexity depends on the manipulator geometry, as described by the DH-parameters.

At least six degrees of freedom are required to arbitrarily position and orient a rigid body in space. Therefore, when n is larger than six, the manipulator is redundant and the system of equations implied by Eq. (2.14) is underconstrained. If n is less than 6," the system becomes overconstrained and when n is equal to 6, the inverse kinematic problem is exactly specified. In this

..









research we will address the inverse kinematics problem of non-redundant robot manipulators.

Most existing industrial manipulators are 5- or 6degree-of-freedom robots, hence, it is of practical importance to solve Eq. (2.14) for n=5 and n=6. The

numerical techniques developed in this text are based on a complete inverse kinematic analysis of four-degree-offreedom manipulators. Therefore, this research will aim at solving Eq. (2.14) for robots with four, five, and six joint axes.

Although the techniques described in this text can be applied to manipulators having prismatic joints (Manseur and Doty 1988), we concentrate on all-revolute six-DOF manipulators; n is six and all joints are assumed revolute.

..













CHAPTER 3
EXISTING SOLUTIONS



Closed-Form Architectures


The ability to compute the coordinates in joint space of an end-effector pose given in Cartesian space is an important criterion in the design of computer-controlled manipulators. A desirable property for an industrial manipulator is the possibility of computing the joint

variables necessary to position and orient the end-effector as specified in Cartesian space, in closed-form. Pieper

(1968) has shown that a closed-form solution is possible when the manipulator has three adjacent joint axes intersecting at a common point. The inverse kinematic

problem reduces then to a quartic polynomial equation in one

of the joint variables. Manipulators with the last three joint axes intersecting are said to be "wrist-partitioned". Computationally efficient methods for computing the

position, velocity, and acceleration inverse kinematics for this type of manipulators have been presented by

Featherstone (1983), Hollerbach and Sahar (1983), Paul and Zhang (1986), and Low and Dubey (1986). Several industrial six- and five-DOF manipulators such as the PUMA series

robots are of the wrist-partitioned type. If, on top of

..








having a wrist, the manipulator has some added structural feature such as two parallel or intersecting joint axes then closed-form solutions may be obtained in a simpler form than a quartic polynomial equation. This is the case of the PUMA 560 robot whose inverse kinematics are discussed in Example 1 of Chapter 9. An algebraic method for solving the inverse kinematics of the PUMA 560 can be found in Craig (1986) and a geometric approach is described in Fu, Gonzalez, & Lee (1987). Another sufficient condition for closed-form solutions is that three adjacent joint axes be parallel (Duffy 1980, Fu, Gonzalez, & Lee 1987).



Record and Plavback


An industrial robot manipulator is usually equipped with sensors that can measure information such as joint variable values and rates of change of those values. A

method that avoids the computational complexity of the inverse kinematic problem altogether- consists of remotely guiding a robot end-effector trajectory by activating each joint separately while storing joint space coordinates and information -from -the sensors at -selectedpoints alongC the trajectory. The robot can then indefinitely repeat the recorded motion. Should the robot be needed for a different task or should a change in the workcell occur that requires different end-effector -trajectories the motion of the Irobot will have to be recorded again.

..










Numerical Techniques


Many six- and five-DOF kinematic structures lack the necessary architectural simplicity for closed-form inverse kinematic solutions. Solving such manipulators requires the use of numerical iterative techniques. For six-DOF robots, equation (2.14) can be expressed as a system of six nonlinear equations in the six joint variables of the form


fl(e1, e2, e3, e4, e5, e6) = Px f2(91' e2' e3' G4' e5' e6) = Py f3(81' e2, e3, e4, 5, 6) = Pz f4(el, e2, 83, e4, e5, e6) = a f5(e1, e2, e3, e4, S5, e6) = E f6(811 e2, e3, e4, e5, e6) =

where px, Py, and pz are the coordinates of the origin of the end-effector frame and a, e, and are either the Euler angles or the roll-pitch-yaw angles derived- from the orientation matrix R of the end-effector frame (Paul 1981).

The six-dimensional equation is then solved by use of a direct or modified Newton-Raphson ot- similar technique. Multidimensional iterative techniques for solving the inverse kinematics problem of manipulators of arbitrary architecture are described by Angeles (1985-, -1986), Goldenberg- Benhabib', & Fentorn (1985)."- Goldenberg:and Lawrence (1985). The computational efficiency of these

..









methods is hindered by the need to compute the inverse of the manipulator Jacobian at several points.

Linares & Page (1984) and Kazerounian (1987) describe techniques that solve the inverse kinematic problem by varying one joint variable at a time so as to minimize the difference, measured by a defined norm, between the endeffector pose as computed from the current joint variables values and the desired pose. This technique has the

advantages of guaranteed convergence and reliability even at a singular position. This method requires computation of the forward kinematics at each iteration and it has a computational complexity comparable to that of a Modifiedmultidimensional Newton-Raphson.

After reducing the problem to a polynomial system of four equations in only four of the joint variables, Tsai and Morgan (1984) used a homotopy map method, for solving systems of polynomial equations in several variables, to find the solutions of the inverse kinematics problem of revolute five- and six- degree-of-freedom manipulators of* arbitrary architecture. The method finds all solutions but its computational complexity renders it impractical for many applications.

Lumelsky (1984) presented an iterative algorithm that finds estimates for three of the joint variables and solves in closed form for the remaining three variables at each iteration. The method applies to a particular type of arm

..






17

geometry (that of the GP66 robot discussed in Example 2, Chapter 9, of this dissertation) and converges to an accurate end-effector position, but to a less accurate approximation of the end-effector orientation.

..













CHAPTER 4
NEW APPROACH



Link-Frames Assignment


Some simplification in the mathematical description of the inverse kinematics problem can be obtained if certain simple rules for assigning the link-frames are applied.

In selecting frame Fi, the direction of vector zi is always chosen so that twist angle ai is in the interval [O,r). If ai = 0, then vectors zi_1 and zi are parallel and the common normal can be arbitrarily located along both axes. In this case the position of Frame Fi should be chosen so that di is equal to zero.

For an n-DOF robot, frame Fn, attached to the endeffector, can be chosen so that it differs from link frame Fn_1 only by a rotation of angle en about Zn_.l In other words, Fn can be selected so that dn = an = an = 0, without loss of generality. We prove this point for n = 6, but it is valid for any relevant value of n. Let us assume that Eq. (2.14) is to be solved with a 6-DOF manipulator for which d6, a6, or a6 is not equal to 0, then the homogeneous matrix transform A6 decomposes into


A6 = A6 B6

..









as given in Eq. (2.7). Equation (2.14) is then equivalent to


A1 A2 A3 A4 A5 A6 =P B6-1


where the right hand side of this last equation is seen to be a constant pose matrix for a manipulator described by the left hand side (i.e. one for which d6=a6=a6=0 so that A6 =A6)"

When joint 1 is not prismatic, dI is constant and the origin of the base frame F0 can be positioned so that dI is equal to 0.



The Reduced System of Equations


For a 6-DOF arm, Eq. (2.14) becomes


A1 A2 A3 A4 A5 A6 = P (4.1)


and it yields twelve non trivial scalar equations in the six unknown variables. It is desirable to reduce-this system to a minimal number of equations involving as few of the joint variables as possible. For all-revolute, 6-DOF

manipulators, Tsai and Morgan (1984) have establishe that with respect to frame F3, the z-component of the position vector 3p and that of vector 3t along with the inner products (3t.3p) and (3 p. 3p) provide 4 equations in only 4 of the unknowns, thereby reducing the complexity of the problem. The process of obtaining these four equations

..








involved multiplying the A-matrices and simplifying the expressions obtained for the elements of 3t and 3p. Besides being lengthy, this method does not allow insight into the mechanisms that make the simplifications possible. The

approach presented here provides the same results with much less effort and greater insight by taking advantage of the properties of rotation transformations.

By writing the product of two A matrices-in the form



RiRj (Rilj + Ii)
0 0 0 1



we divide Eq. (4.1) into a position equation


p = RI(R2(R3(R4(R516+15)+14)+13)+12)+1I (4.2)


and an orientation equation


R = R1 R2 R3 R4 R5 R6. (4.3)


With the frame assignment conventions discussed, 16=0 whenjoint 6 is revolute. Equation (4.2) then simplifies to

.p =-RI(R2(R3(R415+l4)+3)+l2)+11. (4.4) .


Three independent scalar equations for Px, Py, and Pz can be obtained from Eq. (4.4) and more equations can be selected out of the 9 scalar equations implied by Eq. (4.3).

..









Since rotations are orthogonal transformations, they leave inner products invariant, hence


Ru Rv = u v (4.5)


for any rotation matrix R and any vectors u and v. A

special case of (4.5) that is very useful is


Ru v = u R-1v. (4.6)


These properties are extremely efficient in eliminating algebraic terms and unnecessary joint variables when applied to Eqs. (4.3) and (4.4) if it is further recognized that


Ri-lli = [ai,diai,dii]T (4.7)

and

Ri-l = [0,oi,ri], (4.8)


where z = [ 0, 0, 1]T, are always independent of 8ei. Also, due to the frame assignments discussed above,

R6 z = R6-1 z =

in all cases since frame F6 is chosen to force a6 =0.

By repeated use of Eqs. (4.5) and (4.6), we obtain four reduced equations from Eqs. (4.3) and (4.4).

tz equation.

tz = t z = (R z) z

t = (R1 R2 R3 R4 R5 R6 z) z

tz = (R1 R2 R3 R4 R5 z) z

t = z (R5-1 R4-1 R3-1 R2- R1-1 z) (4.9)

..









pz equation

p = R1 R2 R3 R4 q

with

q = 15 + R4 1(14 + R3-1(13 + R2-1(12 + RI-I1I))), so that

Pz = p z = q (R4- 1 R3-1 R2-1 RI-I) z. (4.10)


p.t equation.

p t = R5-1q z (4.11)


p.p equation.
p.p = p2 = q.q = q2. (4.12)


Since Rl-llI and Rl-lz are independent of ei (Eqs. (4.7) and (4.8)), vector q and Eqs. (4.9)-(4.12) are easily seen to be independent of the first and last joint variables and therefore form a system of 4 equations in 4 unknowns. Figure 4.1 illustrates this discussion. With 16=0, vector t, which coincides with z5, and the position vector p of the origin of frame F6 are invariant in the rotation R6

(rotation about z5 which can only affect the end-effector orientation). Rotation about z has no effect on the zcomponent of any vector expressed in frame F0. Hence, pz and t are independent of I as well. Finally, since rotation about z0 moves all the robotic structure as a bloc, it does not affect the length of vector p or the inner product of t and p. ,

..







I

I 6


Z


I
e n


Ib




I P


A
Zo


Yo
F
Xo O







Figure 4.1. Rotations about z0 or z5 do not affect
tz, Pz, t.p, and p.p.

..









The reduced system of equations (4.9)-(4.12) determines

candidate solutions for joint variables 2, 3, 4, and 5. Once this system of equations is solved, the remaining two variables can be found by using more equations from (4.1) and then tested for consistency. The power of this approach

will become apparent for specific manipulators as further simplification using Eqs. (4.5)-(4.8) becomes obvious. Furthermore, simplification by use of rotation inner-product invariance is computationally economical and provides

greater insight into the structure and properties of the inverse kinematic equations.



Additional Inverse Kinematics Equations


Equations (4.9)-(4.12) are necessary, but not sufficient. Although they are satisfied by all solution sets of Eq. (4.1), they are also, in general, satisfied by extraneous solutions. This problem was reported by Tsai and

Morgan (1984) as

Another problem with considering Eqs. (4.9)-(4.12) alone is the presence of sign ambiguities. In many

practical situations, one of the equations will allow a closed-form solution for either the sine or the cosine function of a resolute variable 8. The other function needs

to be computed using the Pythagorean identity, which offers two values opposite in sign. 'Although both. signs can be tried in the search for a solution, in some cases the number

..









of sign ambiguities can be reduced by considering more constraints from Eqs. (4.3) and (4.4). Additional equations will also help filter out extraneous solutions and in some cases will ease the solution-finding process rather than complicate it. The x- and y-components of vectors t and p provide convenient additional constraints at the cost of introducing the variable e1. Equations


tx = R1 R2 R3 R4 R5 z x, (4.13)

t = R1 R2 R3 R4 R5 z y, (4.14)

Px = (R1(R2(R3(R415+14)+13)+12)+11) x, (4.15)
and

py = (R1(R2(R3(R415+14)+13)+12)+11) y, (4.16)

where


1 0

x = 0 and y= 1

0 0


are the usual canonical unit vectors, are still independent of 96.



Solving Inverse Kinematic Equations


Once the reduced set of equations (4.9)-(4.12) and the additional equations (4.13)-(4.16) have been expanded, the problem becomes that of extracting the values of the joint angles from the equations which are in terms of the sines

..









and cosines of the angles. In this section, we describe

some of the techniques that can be used for this task.

Certain simple arm geometries allow a closed form

solution. For such arms, one of the equations will have the form

aS+bC=d

where S and C are the sine and cosine, respectively, of some angle a. If the constants a, b, and d are known, then there are two possible solutions when a2 + b2 d2


e = atan2[d,1(a2+b2-d2)] atan2(b,a)


where atan2(v,w) returns the angle arctan(v/w) adjusted to the proper quadrant according to the sign of the real numbers v and w.

A special case occurs when a = 0 or b = 0. The

equation can then be solved for S or C separately. The

other variable can be obtained from the Pythagorean identity

S2 + C2. .
S +C= 1 (4. 17)


with a sign ambiguity. Again, this leads to two possible values for the angle 8,


e = atan2(S, 1(1 S2)) if S is computed or

8 = atan2( /(l C2), C) if C is the known

variable.

A value of 0 can be directly and uniquely obtained when two linear equations in the sine and cosines of one angle

..









are obtained. In this case the values of S and C are computed and the angle 9 is then given by

8 = atan2(S, C).



Exchanginq Base and End-Effector Frames


The inverse kinematics problem consists of finding joint variables that realize a given relationship between two frames, the base frame F0 and the end-effector frame FnThe roles of these two frames are in fact interchangeable as we illustrate in Figure 4.2. This means that the problem can be viewed as finding the joint variables necessary for the robot to achieve the base frame as viewed from the endeffector frame. This problem reversal requires that the DHparameters be rearranged and intermediate frames be reassigned as illustrated in Figure 4.1 but it can be useful in many ways. For example, several computationally

efficient inverse kinematic techniques have been developed for robots with the last three joint axes intersecting at a common point (Featherstone 1983, Hollerbach and sahar 1983, Paul and Zhang 1986, Low and Dubey 1986). The same techniques can be used for a robot whose first three axes intersect by reversing the roles of end--effector and base frame. In the next chapters, we will use this problem reversal technique to avoid repetitious developments.

..







end-e ffector


base


base


end-effector

X4
Z5+ t6 *


Figure 4.2.' Interchanging base andend-effector
frames.

..













CHAPTER 5
SOLVING 4-DOF MANIPULATORS Reduced System of Equations


For 4-DOF robot arms, the inverse kinematic problem is solved when 4 joint values are found that satisfy the equation


A1 A2 A3 A4 = P. (5.1)


Equation (5.1) decouples into a position equation


R1 (R2 (R3 14 + 13) + 12) + 11 = p, (5.2)


and an orientation equation given by


R1 R2 R3 R4 = R. (5.3)


When the fourth joint .is revolute 14=0 is obtained by proper choice of frame F4 and Eq. (5.2) simplifies to


R1 (R2 13 + 12) + 11 = p. (5.4)


A reduced system of four equations in the sines and cosines of joint angles e1 and e3 can be derived by considering the quantities tz, pz and the inner products t.p and p.p expressed in frame F1

..









Vector t is given by

t = R z = R1 R2 R3 R4 z


where z is the third canonical unit vector z = [ 0, 0, 1]T Since twist angle a4 is equal to 0,


R4 z = [04 S4, -U4 C4, T4]T = [0, 0, 1]T = z, and the expression for t simplifies to


t = R1 R2 R3 z. (5.5)


Multiplying by R1-1 yields


R1 1 t = R2 R3 z

and the inner product of each side of this equality with vector z provides


z (RI-1 t) = z (R2 R3 z).


Eq. (4.5) applied to both sides of this last equation gives


R1 z t =R2-1 z R3 z or

(R1 z).t (R2-1 z).(R3 z) = 0. (5.6)

Since R2-1 = [0, 02, 2] T does not depend on 829 this last equation is independent of joint variables 2 and 4.

Subtracting vector 1- from both sides of Eq. (5.4) and multiplying by R-Q yields

..








R2 13 + 12 = R1-1 (p 11) (5.7)


and taking the inner-product with vector z provides


(z R2 13) + (z 12) = (z R1-1 p) (z RI-1 11). Applying (4.5) to the first term of both sides of this equation gives (after rearranging terms) R1 p R2-1 z 13 = z RI- 11 + z 12. (5.8) The right hand side of Eq. (5.8) is constant since


ai

Ri-1 i = dici and z.1i=di are independent of ei.

diri


Multiplying Eq. (5.7) by R2-1 gives

13 + R2-1 12 = R2-1 RI-1 (p 11) (5.9)


and multiplication of Eq. (5.5) by R2-1 R1-1 yields


R3 z = R2-1 R1-1 t. (5.10)


The inner product of corresponding sides of equations (5.9) and (5.10) produces


(13 + R2-1 12).(R3 z) = [R2- R1-1 (p 11)].[R2-1 RI-1 t]. Repeated use of properties (4.4) and (4.5) and reordering simplifies this last equation to

..









11.t [R2-1 12 R3 z] = t.p [R3-1 13 z]. (5.11) Equation (5.11) is also independent of 92 and 84.

Using Eq. (5.4), the inner-product p.p satisfies p.p = [R1 (R2 13 + 12) + 11].[R1 (R2 13 + 12) + 11']. Expanding the left hand side, using inner-product invariance of rotations where needed, and rearranging terms yield 13.R2- 1 12 + p.11 = [p.p + 11.11 12.12 13.13]/2. (5.12)


Equations (5.11), (5.2), (5.18), and (5.12) form a linear system in the variables S1, C1, S3 and C3. The four equations obtained are


alty S1 + altx C1 + a2C3 S3 02c3d2 C3 = r1 (5.13)

01tx S1 Olty C1 + 0203 C3 = r2 (5.14)

0lpx S1 alPy C1 a2a3 S3 = r3 (5.15)

alPy S1 + alpx C1 + a2a3d2 S3 + a2a3 C3 = r4 (5.16)


with

rI = t.p T3d3 dltz T23d2 (5.17)

r2 = 273 -rtz (5.18)

r3 = T(d tz) + d2 + T2d3 (5.19)

r4 (p.p + al2+d 2-a22-d22-a3 2-d32)/2

-dlPz r2d2d3. (5.20)


The linear system of equations formed by Eqs. (5.13)-(5".16) will be _referred to as the reduced system for a four-DOF

..








manipulator. Although di can be assumed zero without loss of generality for a 4-DOF manipulator, this system of equations will be used for 4-DOF sections of larger manipulators (next chapters) for which the parameter corresponding to dI will, in general, not be zero. Hence, dI is assumed not equal to zero at this point.

A unique solution to the reduced system is given by



S1 r1

C1 = H-1 r2 (5.21)
H1 2 (5.21)
S3 r3

C3 r4


where


alty altx a2G3 -a2a3d2

altx -alty 0 a3
H = y, (5.22)
0alPx -lPy -a2a3 0
alpy alPx a2a3d2 a2a3



when matrix H is nonsingular. Unique values of 81 and 83 are thus obtained from the values of Sl, Cl, S3, and C3. The case where H is not invertible is discussed in the next sections because of its interesting implications.

With 81 and 83 known, Eq. (5.7) provides away to solve for e2. Indeed, when expanded, the first 2 components yield


(o2d3-'2a3S3) S2 + (a2+a3C3) C2 = ClPx + SlPy al (5.23)

..








and

(a2+a3C3) S2 (a2d3-r2a3S3) C2 =

-1lSlPx + T1C1Py + al(pz-dl). (5.24) When the determinant of this linear system of equations in S2 and C2 is not 0, a unique value of e2 can be computed. Otherwise, we can obtain e2 uniquely from another linear system of equations in S2 and C2,


(r2a3C3+a2r3) S2 + a3S3 C2 = Cltx + Slty (5.25)

and

C3S3 S2 (T263C3+a2T3) C2

= 1Sltx + iClty + aitz, (5.26)


derived from Eq. (5.5). Note that 92 can also be computed using a system of two equations formed by Eq. (5.23) or Eq. (5.24) and one of Eqs. (5.25) and (5.26). The Appendix

shows that the determinants of the two systems of equations above are simultaneously zero only when joint axis 2 aligns with another joint axis which puts the arm in a degenerate configuration.

To complete the 4-DOF solution set, we use Eq. (5.2) which can be rewritten as


R4 = RI1 R21 RI-1 R.


The first column vector of R4, obtained by multiplying both sides by the first canonical unit vector x = [1,0,0] T

..










C4 nx

S4R2 1 R x = R31 R2 R1 ny

0 n



can be used to compute the last variable e4. This shows

that a 4-DOF inverse kinematic problem will, in general (general in the sense that matrix H is nonsingular), yield a unique solution set. However, for some manipulator

geometries and/or some particular end-effector poses, the problem may have more than one solution.



Special 4-DOF Manipulator Geometries


Equation (5.21) is valid only when matrix H is invertible. The determinant of matrix H, computed from Eq. (5.22), is given by


dH = 61 62 al[a2 (a32 wI + a32 w2) + 2 02 03 a3 d2 w3]

+0 a~ 2 2 +02 d2 22 2
+ C3 a3 12 (a2 '+ o2 2 ) + c2 a1.] w4 (5.27)


where the quantities wI, w2, w3, and w4 are defined, in terms of the components of pose vectors t and p, as


wI= t x2 + ty2 w2= Px2 + py2

t
~w3.Pxt y tx
W4 -x tx

..









Analyzing dH

Equation (5.27) shows that the value of dH depends on the seven robot parameters 0l, U2, 03, al, a2, a3, and d2 as well as the pose quantities wl, w2, w3, and w4. However,

for certain robot structures dH is equal to zero no matter what the end-effector pose is. The expression of dH above

provides us with a way to find all such 4-DOF robot

geometries. Due to our link frames assignment, the only robot parameter in the expression of dh that can be negative is d2. By expanding Eq. (5.27) we get dH = ao2 a3 2 wI + C 1 032 a1 a2 w2

+ 2 0I 2 23 a1 a3 d2 w3 + a12 C3 a22 a3 w4
+20223 a3d2 w4+022
1 a 2 2d22 4 22 3 a1 a3 w4 (5.28)


where only the quantities d2, w3, and w4 can be negative. If an arm structure is such that dH is zero for every possible end-effector pose, then dH will be zero even for a pose with positive w3 and w4.

If we assume w3, and w4 non negative, then with d2negative, dH can be zero if the equality


-2 01 02 03 a1 a3 d2 w3 =

01 02 al a2 a32 wI + 01 02 032 al a2 w2

+U12 3 a22 a3 w4 + 0l2 022 03 a3 d22 w4 + 022 03 aI2 a3 w4 holds. However, such an equality is actually a condition on pose quantities wl, w2, w3, and w4. We conclude that robot

..









structures for which dH is always zero (independent of the end-effector pose) have DH--parameters 01, 02, 03, a,, a2, a3, and d2 for which each of the six terms in Eq. (5.28) is individually zero. These terms, in turn, are zero when particular structure parameters are equal to zero. For

example, al=a02=0 will make the determinant zero. In order to enumerate the minimum number of distinct combinations of zero parameters that make d11 = 0, we examine all possible cases when a particular parameter is zero. We get seven

simpler expressions of dH, listed in Table 5-1., by

separately assuming each relevant parameter to be equal to zero.

Table 5-1 provides a simple mean for finding all (poseindependent) 4-DOF robot geometries for which matrix H will be singular. In the next section, we show that the inverse

kinematics problem for such robots can still be solved by use of the reduced system of equations (5.13)-(5.16). Special 4-DOF Arm Structures

A trivial condition occurs when two consecutive joint axes coincide somewhere along the arm. Such a degenerate

condition is detected by Iai = ai= 0 for some joint i. In

this case, the manipulator loses one degree of freedom and becomes a redundant 3-DOF arm. If a solution set exists for such an arm, there will be an infinite number of solution sets. A careful analysis of Table 5-1 shows that there are only ten minimal, non-trivial, conditions on the arm

..









Table 5-1. Special expressions for dH. Condition dH

1 01 = 0 22 3 a12 a3 w4

2 02 = 0 012 o3 a22 a3 w4

3 03 = 0 010 2 al a2 a32 w1

4 al = 0 012 03 a3 (a2 + 22 d22) w4

5 a2 = 0 a22 03 a3[2 01 al d2 w3 + (a12 + 012 d22)w41

6 a3 = 0 010 2 032 a a2 w2

7 d2 = 0 01 02 a1 a2 (32 W2 + a32 w1) + 03 a3 (022 a12 + 012 a22) w4


geometry (pose-independent) for dH = 0. All ten conditions are listed and described in Table 5-2 and illustrated in Figure 5.1.

The first three conditions in Table 5-2 follow from the first entry of Table 5-1. Conditions 4 and 5 are derived from the second entry in Table 5-1 after dropping duplicate conditions already established. Continuing in this fashion, all of Table 5-2 can be completed. Observe that entry 7 in Table 5-1 does not add any new conditions into Table 5-2 since all minimal sets of zero parameters implied by dH=0 in entry 7 have already been accounted for.

SIt must be noted that dH can still be zero for 4-DOF
H!
arm geometries not listed in the preceding. Table. However from the discussion above, we see that such a situation can only happen at particular end-effector poses whereas dH will

..









Table 5-2. Special structures of 4-DOF manipulators Condition Description


1 al=a2=0 First 3 joint axes (1,2,3) are parallel


2 0l=03=0 First 2 axes (1,2) are parallel and
Last 2 axes (3,4) are parallel

3 al=a3=0 First 2 axes (1,2) are parallel and

last 2 axes (3,4) intersect


4 C2=a3=0 Last three axes (2,3,4) are parallel


5 a2=a3=0 Middle axes (2,3) are parallel and
last 2 axes (3,4) intersect


6 C3=al=0 First 2 axes (1,2) intersect & last 2
axes (3,4) are parallel


7 a3=a2=0 Middle axes (2,3) intersect & last 2
axes (3,4) are parallel


8 al=a3=0 First 2 axes (1,2) intersect & last 2
axes (3,4) intersect


9 al=a2=d2=0 First 3 axes (1,2,3) intersect 10 a2=a3=0 Middle 2 axes (2,3),intersect and last
2 axes (3,4) intersect



be zero for the geometries described in Table 5-2 at any pose. We now examine in detail the inverse kinematics of each of the special 4-DOF robot architectures described in Table 5-2 and illustrated in Figure 5.1.

..




















1. First three joint axes
are parallel.


3.Axes 1 and 2 are parallel,
axes 3 and 4 intersect.


5.Axes 2 and 3 are parallel,
3 and 4 intersect.


2.Axes 1 and 2 are parallel,
3 and 4 are parallel.


4.Axes 2, 3, and 4 are parallel.


6.Axes 1 and 2 intersect
2 and 3 are parallel.


Figure 5.1. Special 4-DOF structures.

..




























7.Axes 2 and 3 intersect,
3 and 4 are parallel.


9.Axes 1 and 2 intersect,
3 and 4 intersect.


8.Axes 1, 2, and 3 intersect.


10.Axes 2 and 3 intersect,
3 and 4 intersect.


Figure 5.1--Continued.-

..








The reduced system of equations (5.13)-(5.16) can still be efficiently used to find all the solution sets of Eq. (5.1) when matrix H is singular.

Case 1: al=a2=0. First three joint axes are parallel (Entry 1 in Table 5-2).The reduced system of equations becomes


alty S1 + altx C1 + a2a3 S3 = rI (5.29)

0 = r2 (5.30)

0 = r3 (5.31)

alPy S1 + alpx C1 + a2a3 C3 = r4. (5.32)


Equations (5.30) and (5.31) are constraints on pose parameters tz and pz respectively. Only end-effector poses that satisfy pz = dl + d2 + d3 and tz = 73 (Eqs. (5.18) and (5.19)) are solvable with this arm geometry.

Equations (5.29) and (5.32) still allow a solution in the style of Pieper (1968) by first eliminating S3 and C3 from the equations. This can be done by solving for S3 and C3 and substituting in the Pythagorean identity (4.17) to get


{[rl (alty S1 + altx Cl)]/a2a3)2 +

{[rl (alPy S1 + alPx Cl1)]/a2a3) 2 1. (5.33) With the trigonometric identities


Sl = 2 tl/(l + t12) and C1 = (1 t12)/(l + tl2)

..








where tI = tan(e1/2), Eq. (5.33) yields a quartic polynomial equation in tI. With t1 computed, a value of e1 is obtained and e3 can be computed uniquely from Eqs. (5.29) and (5.32). The remaining angles (e2 and 84) can be computed as indicated earlier.

We propose a method that allows better insight without the complexity of a quartic polynomial equation. For

simplicity, the sine and cosine of a sum of angles will be represented according to Cijk=cos(ei+ej) and Sij=sin(ei+ej).

As described in chapter 4, a set of inverse kinematic equations can be obtained by expressing the components of vectors t and p and the inner products t.p and p.p in terms of the joint variables ei, i=l, . 3. The equations obtained are

t = a3 S123 (5.34)

ty = -a3 C123 (5.35),

Px = a3 C123 + a2 C12 + al C1 (5.36)

py = a3 S123 + a2 S12 + a1 Sl (5.37)
t.p= al C23 + a2a3 S3 + T3(dl+d2) + d3 (5.38)
p.p = 2(ala3 C23 + a2a3 C3 + ala2 C2) + ct (5.39) where

ct = a12 + a22 + a32 + d12 + d22 + d32

+ 2(d1d2 + d1d3 + d2d3).


Equations (5.34) and (5.35) yield S123 and C123 directly, soa unique value of 8123 81+82+83 is obtained.

..









With e123 known, Eqs. (5.36) and (5.37) become (elbow

equations)

Px a3 C123 = a2 C12 + al C1 (5.36')

py a3 S123 = a2 S12 + a1 S1 (5.37')


and can be solved for C2 by


C2 = [(Px a3 C123)2 + (py a3 S123) 2

a2 a12] / (2 a1 a2)


which is obtained by applying the cosine law to the triangle having links 1 and 2 as its sides. Two values of 92 follow from e2=atan2( /(1-C22), C2)


and a unique value of 81 can then be computed from Eqs. (5.36') and (5.37') which yield a linear system in S1 and C1 when S12 and C12 are expanded using sum of angles trigonometric identities. Joint variable 3, 83 is given by


63 = 6123 81 82


and the solution set is completed when the last angle 04 is. computed as shown earlier. This development proves that there can be at most 2 solution sets for a 4-DOF arm with this particular geometry.

Case 2: ai=a3=0. The first two joint axes are parallel and the last two joint axes are parallel. The reduced

system is

..








alty S1 + altx Cl = rI (5.40)

0 = r2 (5.41)

a2a3 S3 = r3 (5.42)

alPy S1 + alPx C1 + a2a3d2 S3 + a2a3 C3 = r4. (5.43) Equation (5.41) imposes a constraint on pose parameter tz, tz = 2T3/T1. When this constraint is satisfied, Eq. (5.40) can be solved and yields two distinct values for e1. Then Eqs. (5.42) and (5.43) form a linear system in S3 and C3 which can be solved uniquely for e3. With E1 and e3

computed, e2 and e4 can be uniquely obtained as shown earlier. Here again we find at most two solution sets.

Case 3: al1=a3=0. First two joint axes are parallel and last two joint axes intersect. The reduced system becomes alty S1 + altx Cl + a2a3 S3 02o3d2 C3 = r1 (5.44)

a2a3 C3 = r2 (5.45)

0 = r3 (5.46)

alPy S1 + alPx C1 = r4. (5.47)


From Eq. (5.19), the pose constraint r3 = 0 translates to


SPz = di + d2 + T2d3"


For a pose matrix that satisfies this constraint, two possible values of 93 can be obtained from Eq. (5.45). For each of those e3 values, a unique value of el is computed from the linear system in S1 and C1 formed by Eqs. (5.44)

..








and (5.47). The two solution sets are then completed as shown previously.

Case 4: a2=a3=0. The last three joint axes are

parallel. The reduced system simplifies to


alty S1 + altx C1 = r1 (5.48)

altx S1 alty C1 = r2 (5.49)

1px S1 alPy C1 = r3 (5.50)
alPy S1 + alPx C1 + a2a3 C3 = r4. (5.51)


Two out of the first three equations (Eqs. (5.48)-(5.50)) can be used to solve uniquely for el. The third (unused

equation) becomes a realizability constraint on the pose. With 81 known, Eq. (5.51) yields a value for C3 which in turn gives two possible values for e3. Two solution sets can be obtained after computing e2 and e4.

Case 5: a2=a3=0. The intermediate joint axes are

parallel and the last two axes intersect. The reduced sytem becomes


alty S1 + altx C1 + a2a3 S3 = rl (5.52)

altx S1 7lty C1 = r2 (5.53)

0lpx S1 alPy C1 = r3 (5.54)

aly S1 + alPx C1 r4. (5.55)

Two out of the last three equations (5.53)-(5.55) can be solved uniquely for el. The third equation becomes a

realizability constraint on pose element tz or pz depending

..








on the chosen equation. With el known, two values of 83 can be computed from the value of S3 derived from Eq. (5.52).

Case 6: a3=al=0. The last two joint axes are parallel and the first two axes intersect. The reduced system is 0 = r1 (5.56)

altx S1 at y C1 = r2 (5.57)

alpx S1 alPy C1 a2a3 S3 = r3 (5.58)

a2a3d2 S3 + a2a3 C3 = r4. (5.59)

Equation (5.56) is a realizability constraint on pose

parameter tz (Eq. (5.17)). For an end-effector pose that satisfies this constraint, Eq. (5.57) yields two values for e1. Equations. (5.58) and (5.59) can then be solved for 83 uniquely.

Case 7: a3=a2=0. The last two joint axes are parallel and the intermediate two axes intersect. The reduced system is


alty S1 + altx C1 = rI (5.60)

altx S1 1ty C1 = r2 (5.61)

aPpx S1 alPy Ci a2a3 S3 = r3 (5.62)

alPy S1 + alPx C1 + a2a3d2 S3 = r4. (5.63)


Here, Eqs. (5.60) and (5.61) yield a unique value for e1, then one of Eqs. (5.62) or (5.63)-can be used to solve. for

..








S3 thereby providing two values for e3, the remaining equation is a pose constraint.

Case 8: al=a2=d2=0. The first three joint axes

intersect and the reduced system becomes


0 = rI (5.64)

aitx S1 aity C1 + a2a3 C3 = r2 (5.65)

oipx S1 alPy C1 C2a3 S3 = r3 (5.66)

0 = r4. (5.67)


Equations (5.64) and (5.67) impose constraints on pose parameters tz and pz. Here again a solution can be obtained in form of a quartic polynomial equation in tl=tan(e1/2) by eliminating e3 from Eqs. (5.65) and (5.66) as we did earlier in case 1.

With e1 known, e3 can be uniquely obtained from Eqs. (5.65) and (5.66) and the solution set completed as usual. This method puts an upper bound of 4 on the number of solution sets since at most four distinct values of 01 can be obtained from the quartic polynomial equation in tI.

An easier inverse kinematic analysis of this structure can be obtained if the roles of end-effector frame and base frame are reversed and the intermediate link-frames are reassigned accordingly. This will put the three

intersecting axes at the end-effector position instead of at the base. The 4-DOF structure is seen to be equivalent to one that has a2 = a3 = 0 which is discussed in case 10.

..








With this analysis, we find that there can be at most two solution sets.

Case 9: al=a3=0. The first two joint axes intersect and the last two joint axes intersect. The reduced system is

a2a3 S3 a2a3d2 C3 = r1 (5.68)

aitx S1 alty C1 + a2a3 C3 = r2 (5.69)

alpx S1l alPy C1 = r3 (5.70)

0 = r4. (5.71)


Here, r4 = 0 poses a constraint on pose parameter pz. Equation (5.70) yields two distinct values of el, then Eqs. (5.68) and (5.69) will form a linear system in S3 and C3 which can be solved for a unique value of e3 for each value of e1. Two solution sets are thus obtained.

Case 10: a2=a3=0. The intermediate two joint axes intersect and the last two joint axes intersect. The reduced system becomes


alty S1 + altx C1 a263d2 C3 = r1 (5.72)

itx S1 lty C1 + 0203 C3 = r2 (5.73)

alpx S1l 0lPy C1 = r3 (5.74)

alPy S1 + alPx C1 = r4. (5.75)


Equations (5.74) and (5.75) yield a unique solution for el. The value of el obtained-can be substituted in Eq. (5.72) or (5.73) to solve for C3 -which provides twop possible- values

..









for e3. The unused equation is a constraint on the endeffector pose parameter tz.

When a3 = 0 or d2 = 0, the conditions for dH=O obtained have all been already discussed, hence there are only ten minimal pose- independent arm geometry conditions for which the reduced system is singular. In all ten cases, we found at most two distinct inverse kinematic solution sets.

To summarize the above cases, we find that a four-DOF robot manipulator will in general have a unique inverse kinematic solution set. At most two solution sets can be found when the arm has one of the following special structures:

1. Three consecutive joint axes that are parallel.

2. Three consecutive joint axes intersect.

3. Two consecutive pairs of parallel or intersecting joint axes.

4. Three consecutive joint axes such that two intersect and two are parallel.

5. Three consecutive joint axes such that the firsttwo intersect and the last two intersect.

..














CHAPTER 6
SOLVING FIVE-DOF MANIPULATORS



One-Dimensional Iterative Technique


With five degrees of freedom, Eq. (2.14) takes the form


A1 A2 A3 A4 A5 = P. (6.1)


and after multiplying both sides of this equation by A1- 1 we obtain


A2 A3 A4 A5 = Q (6.2)

with

Q = A1 P. (6.3)


When G1 is known, matrix Q is fully determined and can be viewed as a pose matrix for a 4-DOF arm whose structure is described by the left hand side of Eq. (6.2) which merely expresses a 4-DOF problem. In Chapter 5, we have seen that a 4-DOF problem can always be solved in closed-form, hence the remaining joint variables can be computed as shown earlier.

Since we only need to know one of the joint variables to solve for the whole solution set, the inverse kinematics problem of five-DOF manipulators reduces to finding the

..









value of the first joint variable only, and getting closedform values for the remaining variables.

Let the column vectors of pose matrix Q of Eq. (6.3) be given by m, c, u, and q, in order, so that


mx cx ux my cy Uy mz cz uz 0 0 0


m c u q 0 0 0 1


u = R1-1 R z = RI-1 t


q = R1 p p-


R-1 11 = R1-l(p 11).


From the left hand side of Eq. (6.2), two vectors corresponding to vectors u and q, are given by


(6.6)


UL and qL'


uL = R2 R3 R4 z


qL = R2(R3 14 + 13) + 12.


A nonlinear function of el can be defined as a difference between corresponding quantities from the left and the right side of Eq. (6.2). For example, the difference between the inner-products (uL.qL) and (u.q) yields the function


f(e1) = uL q- u.q.


then


(6.4)


and


(6.5)


and


(6.7) (6.8)


(6.9)

..









If the value of e1 used to compute pose matrix Q in Eq. (6.3) does correspond to a solution set, then Eq. (6.2) will hold, vectors uL and qL will be exactly equal to u and q, respectively, and function f will equal zero. In other

words solution sets of Eq. (6.1) correspond to zeros of function f defined in Eq. (6.9). Hence, the inverse

kinematics problem of 5-DOF robot manipulators reduces to solving the one-dimensional equation


f(e1) = 0.


The zeros of f can be found by use of any suitable onedimensional technique such as Newton-Raphson or the secant method. Once 91 is known, the solution set can be completed by solving Eq. (6.2) in closed form as we showed in Chapter 5. The solution set can then be checked for consistency with Eq. (6.1) to determine whether the one found is extraneous or not because the zeros of f are not always part of a solution set of the manipulator.

Computinq f(el_. Using Eqs. (6.7) and (6.8), the inner product uL-qL is given by


UL qL = (R2 R3 R4 z) (R2(R3 14 + 13) + 12).


If we apply properties (4.5) and (4.6) repeatedly, this last equation becomes


uL-qL = z.(R4-114) + z.(R4-1R3-113) + z.(R4-1R3-1R2-112)

..








or after computing the z-components of the terms in parentheses,


uL.qL = T4d4 + a3a4 S4 a3d304 C4 + r4d3

+ U4 S4 (a2 C3 + a2d2 S3)

04 C4 (-a2r3 S3 + C2d2T3 C3 + r2d203)

+ T4 (a2a3 S3 o2d2a3 C3 + T2d2'2). (6.10)


This last equation shows that 93 and 94 must be known before we can compute uL.qL. With el known, 93 and 84 can be obtained by solving Eq. (6.2) as described in Chapter 5. The coordinates of vectors u and q and the inner-products u.q, and q.q are necessary for the 4-DOF inverse kinematic method of Chapter 5. Equation (6.5) yields


t C1 + ty S1

u = R- t = -71 tx S1 + Tity C1 + ltz (6.11)

t1 x S1 lty C1 + Tltz and from Eq. (6.6),


Px C1 + Py Sl al

q = -rlpx S1 + Py C1 + lPz (6.12)

alPx S1 alPy C1 + rlPz


where we have assumed (R1-1 11) = [al, 0, 0]T since dl=0 by proper positioning of frame Fo.

..









The inner-products u.q and q.q can then be easily computed by


u.q = uxqx + uyqy + Uzqz

and

q.q = qx2 + qy2 + qz2


when the numeric values of the components of u and q have been obtained. These inner products can be obtained from Eqs. (6.5) and (6.6) as well,


u.q = (R1-1 t) (R-1 (p 11)) = t (p 11), u.q = tx (Px al Cl) + ty (py al Sl) (6.13)

and

q.q = R1-(p 1i) R1-1(p 11) = (p 11).(p 11), q.q = p.p + 11.11 2(p.11)

q.q = p.p + a12 2 (pxal C1 + pyal Sl). (6.14)


Equations (6.11)-(6.14) clearly show that all components of u and q, and the inner-products u.q and q.q are linear functions of S1 and C1, a result that will prove useful in the next section.

To summarize, f(e1) can be computed for a given value of e1 according to the following steps:

Step 1. From the current estimate of e1, Compute the components of u and q and the inner products u.q and q.q as shown in Eqs. (6.11)-(6.14).

..









Step 2. Compute e2 and 83 from the reduced system of equations


a2uy S2 + a2ux C2 + a3o4 S4 o3a4d3 C4 = r1 (6.15)

a2ux S2 a2uy C2 + 0304 C4 = r2 (6.16)

o2qx S2 92qy C2 o3a4 S4 = r3 (6.17)
a2qy S2 + a2qx C2 + a3a4d3 S4 + a3a4 C4 = r4 (6.18)

with

rl = q.u 74d4 d2uz 73T4d3 (6.19)

r2 = TT4 2u (6.20)

r3= T2(d2 qz) + d3 + T3d4 (6.21)

r4 = (q.q+a22+d22-a32-d32-a42-d42)/2

d2qz r3d3d4, (6.22)


derived from Eqs. (5.13)-(5.20) by proper index substitution (the indexes are incremented to fit the 4-DOF problem of Eq. (6.2)). Vectors u and q play the roles of vectors t and p respectively. The last system of equations gives the values of e2 and 84. Equations (5.23) and (5-.24), with the proper index changes,


(o3d4- T3a4S4) S3 + (a3+a4C4) C3

= C2qx +S2qy a2 (6.23)

and

(a3+a4C4) S3 (a3d4-T3a4S4) C3

= -72S2qx + 2C2qy + a2(qz-d2), (6.24)

..









can be solved for 83. Another way to obtain 83 is by using the equations


(T3u4C4+a3 T4) S3 + 64S4 C3 = C2ux + S2uy (6.25) and

04S4 S3 (r364C4+63T4) C3 =

-T 2S2ux + T2C2Uy + U2Uz, (6.26)


derived from Eqs. (5.25) and (5.26) by incrementing the indexes. With 81, 82, and 83 known, uL-qL can be computed as in Eq. (6.10) and f(81) is then given by Eq. (6.8).

The ability to compute f(81) when e is given is

sufficient to implement a practical Newton-Raphson algorithm for finding the zeros of function f. The algorithm can be programmed according to the following steps:

Step 1. Obtain an initial estimate for 01. As for all iterative methods, the closer the initial estimate of e is to a true solution, the faster the convergence will be. If the end-effector of the robot is tracking a trajectory given as a finite set of end-effector poses, a good estimate for finding the solution set for a pose along the trajectory is the value of 81 corresponding to the preceding pose on the trajectory.

Step 2. Compute 83 and 84 and then f(81) as described earlier.

..









Step 3. Compute the derivative df/d61 of f with respect to 81. A numeric approximation of this derivative is given by


df/de1 = (f(E1+68El) -f(E1)]/681, (6.27)


where S8 is a small increment of 81. Note that this

approximation requires another function evaluation at

(E1+6E1).

Step 4. Obtain a new estimate for 01 by the onedimensional Newton-Raphson method, i.e.


e1(new) 81 f(81)/(df/d01). (6.28)


Steps 2 to 4 must be repeated until 1el is obtained to the desired accuracy. The solution set can then be completed by using the values of 82, e3 and 04 as computed at the last iteration and by computing 05 uniquely from


C5

R5x = S5 = R4-1 R3-1 R2-1 RI- m.

0

and

85 = atan2(S5,C5).


The one-dimensional method just described is flexible in terms of the choice of function f to be used. A

different function can be implemented. The only

requirements are that f(91) be computable for any value of

..









G1 and some known relationship between the zeros of f and the solution sets of Eq. (6.1). For example, another choice of f may be


f(E1) = qL.qL q.q


(6.29)


or any difference between corresponding quantities from the left and right side of Eq. (6.2). The function choice is important in terms of minimal computation complexity and filtering of extraneous solutions which are discussed next. In all practical experiments the function defined in Eq. (6.8) has given good results.

Extraneous solutions. An extraneous solution set is one that the iterative method converges to, i.e. it satisfies the reduced system of equations (6.15)-(6.18!)- and f(91) = 0 but yet it is not a solution for Eq. (6.1). The iterative method just described may converge to such a, set. This problem was also reported by Tsai and Morgan (1984) who developed a different inverse kinematic method that makes use of a similar reduced system of equations.

In deriving the reduced system of equations (5.13)(5.16) in chapter 5, vectors u and q,and the inner-products u.q and q.q are the only pose related quantities that were involved. This means that -a solution set obtained by convergence of the method just described does not necessarily satisfy-other pose requirements from Eq.- (6.1). Extraneous solutions can be filtered out by- a choice of

..









function f that constrains more of the end-effector pose elements at the expense of computation time or by checking all solutions found for consistency with one or more endeffector pose elements.

Iteratinq on e5. An equivalent one-dimensional

iterative technique can be implemented based on a function of 85 instead of 91. Recall from Chapter 2 that the

homogeneous matrix A4 decomposes into


A4 =.A4 B4


where B4 is a homogeneous matrix fully determined by

parameters a4, d4, and a4 and independent of 94. Rightmultiplication of Eq. (6.1) by (A5-1B4-1) yields


A1 A2 A3 A4 Q (6.30)

with

Q = P A51B4 -I (6.31)


When 85 is given, matrix Q becomes a known pose matrix for the 4-DOF problem expressed by equation (6.30). Vectors u and q are given by


u = R 1 z (6. 32)

and

q R R5- 1(-G41k4) + p, (6.33)


where G4 is the rotation part of homogeneous matrix B4,

..










1 0 0 74 0 04
0 0


0

-04 74

0


a4

, k4 = 0 is the position vector

d4


of B4, and R and p are the usual rotation matrix and position vector of end-effector pose P. Explicitly, we get



nx4 S5 + bxU4 C5 + txT 4
u = n y4 S5 + b y4 C5 + tyT4 (6.34)

nzU4 S5 + bza 4 C5 + tzT4


and


(-nxa4d4 + bxa4) S5

(nxa4 + b a4d4) (-n ya4d4 + b ya4) S5

(n ya4 + by c4d4) (-nza4d4 + bza4) S5

(nza4 + bz"4d4)


C5 tx 4d4 + Px C5 tyT4d4 + Py C5 tz74d4 + Pz


Expressions of inner-products u.q and q.q in terms of S5 and C5 can be obtained from Eqs. (6.32) and (6.33),


u.q =[R R51G4 z] R (R5 (-G4 k4) + p]-


and


B4 =


(6.35)

..









q.q = [R (R5- 1(-G4-1k4) + p] [R (R5-1(-G4-1k4) + p].' With the use of properties (4.5) and (4.6) as necessary and rearranging terms, the equations yield u.q = 04 [(n.p) S5 + (b.p) C5] + 74(t.p) d4 (6.36) and

q.q = -2[a4d4 (n.p) a4 (b.p)] S5

2[a4 (n.p) + C4d4 (b.p)] C5

274d4 (t.p) + a42 + d42 + p.p, (6.37) where we used the fact that



n.p nxPx + nypy + nzpz

R- p= b.p bxPx + bypy + bzpz

t.p txPx + typy + tzpz



Here again, we note that uz, Pz, u.q, and q.q are linear functions of S5 and C5.

With the components of u and q and the inner products u.q and q.q computed, a one-dimensional iterative method can be implemented as described earlier with a function f(85) given by


f(85) = uLqL u.q (6.38)-


which will converge to a value of e5.

..









5-DOF Robots with Closed-Form Solution


Certain Five-degree-of-freedom robots with simple geometries may yield inverse kinematic equations that can be solved directly and without need for a numeric technique such as Newton-Raphson. In Chapter 5, some particular 4-DOF robot structures were found for which the reduced system of equations (5.13)-(5.16) was overspecified i.e. the matrix H of the linear system was singular. The analysis of these special geometries showed that one or two of the four equations of the reduced system became constraint equations on pose elements, particularly, elements tz, Pz, t.p, and p.p.

In the case of 5-DOF robots, the quantities uz, qz, u.q, and q.q (u playing the role of t and q that of p) are either linear functions of S1 and C1 as shown in Eqs. (6.1l)-(6.14) or linear functions of S5 and C5 as shown in Eqs. (6.34)-(6.37). Either way, the constraint equations described in the ten cases of chapter 5 can be used to solve for the correct value of el or e5 directly without need for an iterative technique. This result means that if a 5-DOF robot manipulator has a 4-DOF section with one of the special geometries discussed in Chapter 5, then the arm can be solved in closed form. We now proceed to prove this point.

The 5-DOF inverse kinematics problem of Eq. (6.1) can be reduced to the 4-DOF one of Eq. (6.2). In this case, the

..









reduced system of equations (6.15)-(6.18) must be solved. By substituting the expressions of uz, qz, u.q, and q.q from Eqs. (6.11)-(6.14) into Eqs. (6.19)-(6.22) and rearranging, we get


rI = (-alt y aid2tx) S1 + (-altx + d21t y) C1

+ t Px typy rld2tz T3d3r4 T4d4, (6.39)'


r2 = -a1r2tx S1 + 1 T2ty C1 Tr12tz + T3T4, (6.40)


r3 -a1r2Px S1 + 1 T2Py C1

r2Pz + r 2d2 + d3 + T3d4, (6.41)

and

r4= (-alpy aid2Px) S1 + (-alPx + d2alPy) C1

-Tld2Pz + 73d3d4

+ (p.p +a12+a22+d22-a32-d32-a42-d42). (6.42)


These last four equations prove that the terms rl, r2, r3, and r4 are all of the form


ri = ril S1 + ri2 C1 + ri3, i=l, . 4,


where the constants rij are fully determined by the arm parameters and the end-effector pose elements.

Another choice is to use Eq. (6.30). The reduced

system of equations is given by Eqs. (5.13)-(5.16) with all elements of pose matrix P replaced by corresponding elements of matrix Q of Eq. (6.31). The ri quantities become linear expressions in S5 and C5 and have the form

..









ri = ril S5 ri2 C5 + ri3, i=l, . 4.


Indeed, if we replace uz, qz, u.q, and q.q by their expressions in terms of S5 and C5, as given by Eqs. (6.34)(6.38) and substitute in Eqs. (5.17)-(5.20) (substitute for tz, pz, t.p, and p.p and let dl=0 ), we obtain rI = a4(n.p) S5 + a4(b.p) C5 + r4(t.p)

r3d3 7'2d2r3 d4, (6.43)


r2 = -rla4nz S5 -Tla4bz C5 TT4tz + 23, (6.44)


r3 = (Tla4d4nz -Tla4bz) S5 + (rla4nz + r1a4d4bz) C5

+ (Tlr 4d4tz -Tlpz) + d2 + T2d3, (6.45)

and

r4 =-2[a4d4 (n.p) a4 (b.p)] S5

2[a4 (n.p) + a4d4 (b.p)] C5 2r4d4 (t.p) r2d2d3
+ (p.p + al2 a -d22 a3 2

d32 + a42 + d42)/2. (6.46)


In the analysis of special four-DOF geometries in Chapter 5, we found cases where the reduced system of equations included a constraint of the form ri = 0. Such a constraint applied to one of Eqs. (6.39)-(6.46) will usually yield a value of 81 or e5 which in turn makes the 5-DOF inverse kinematics problem solvable in closed form.

Case 1: Three joint axes are parallel. When the

parallel axes are the first three (i.e. axes 1, 2 and 3), Eq. (6.30) can be used. Case 1 of Chapter 5 shows that

..









r2=0 and r3=0. These two constraints and Eqs. (6.44) and (6.45) yield a system of equations in S5 and C5,


r21 $5 + r22 C5 = -r23 r31 S5 + r32 C5 = -r33,


which can be solved for 05 when the determinant given by (r21r32 r31r22) is not zero, otherwise there is no solution. With 85 known, the remaining angles can be obtained in closed-form.

If the last three axes are parallel, a similar result is obtained by exchanging the roles of base and end-effector frames. When the intermediate axes are parallel, Eq. (6.2) should be used. The constraints r2 = r3 = 0 then yield a value of 01 and the inverse kinematic problem can be solved in closed form as well.

Case 2: two consecutive sets of two parallel axes. If axes 1 and 2 are parallel and axes 3 and 4 are parallel, Eq. (6.30) and Chapter 5, case 2 yield r2 = 0 which can be used to solve for 95 from Eq. (6.44). If axes 2 and 3 are

parallel and axes 4 and 5 are parallel, then using Eq. (6.2) and Eq. (6.40) will yield a value of 91.

Case 3: Two parallel axes followed by two intersecting axes. When this special geometry concerns the first four joint axes of the 5-DOF arm, using Eq. (6.30) and Chapter 5, case 3 yields r3 0. This constraint applied to Eq. (6.45) gives a value of 95. If the upper part of the 5-DOF robot

..









has the special structure, Eq. (6.2) can be used and the constraint r3 = 0 applies to Eq. (6.41). Angle aI can be directly computed.

Case 4: Two intersecting axes followed by two parallel axes. This structure corresponds to Chapter 5, case 6. Here, the constraint is rI = 0 and, as in the preceding cases, 81 or 85 can be directly computed from Eq. (6.39) or (6.43), respectively.

Case 5: Three intersecting axes. Pieper (1968) has

shown that a six-DOF manipulator with three intersecting axes can always be solved in closed form. This result

applies to the simpler case of five-DOF robots. This

structure corresponds to case 8 of Chapter 5 where the constraints are rl=0 and r4=0. If the three intersecting axes are the first three, Equation (6.30) should be used. With Eqs. (15) and (6.46), a value of a5 can be obtained directly. This same method can be used when the last three axes intersect by first exchanging the roles of end-effector and base frames. When the intermediate three axes are intersecting, use of Eqs. (6.2), (6.39) and (6.42) will yield a value of al.

Case 6: Two consecutive sets of two intersecting axes. This structure is analyzed in case 9 of Chapter 5. The

constraint equation is r4 = 0. Depending on where this

special structure is located along the five-DOF arm, 01 or

..









e5 can be directly computed by use of Eqs. (6.42) or (6.46) respectively.

In the special geometries described in Chapter 5, cases 5, 7, and 10, we did not find a constraint of the form ri=0, yet a five-DOF arm having one of these particular geometries can still be solved in closed form. We now study these

special cases as they apply to five-degree-of-freedom robots.

Case 7: Three joint axes are such that they either intersect or they are parallel two at a time. This type of structure is studied in cases 5, 7, and 10 of Chapter 5. Assuming this geometry concerns axes 3, 4, and 5 of the five-DOF arm, Eq. (6.2) should be used. From Chapter 5,

case 5 and case 10, we see that the last two equations of the reduced system, Eqs. (6.17) and (6.18) have the form


o2(qx S2 qy C2) = r3 a2(qy S2 + qx C2) =r4


where qx, qy, r3, and r4 are all linear expressions in S( and C1. A quartic polynomial equation in t1 = tan(91/2) is readily obtained by squaring and adding the last two equations,


qx2 + qy2 = (r3/o2)2 + (r4/a2)2

and substituting S1 = 2t1/(l+tl2) and C1 = (1-t12)/(l+t12). This polynomial can be solved for 81 and the solution set

..









completed as described earlier. Similarly, from case 7 of Chapter 5, we get the equations


a2(uy S2 + ux C2) = r, C2(ux S2 uy C2) = r2


corresponding to Eqs.(6.15) and (6.16) of the reduced system of equations. Here again a quartic polynomial equation in tI is obtained by eliminating S2 and C2.

When the three axes with the special geometry are located elsewhere along the five-DOF structure, a similar result can be obtained by using equation (6.30) or by

exchanging the roles of base and end-effector frames.

To summarize the above cases, we find that a 5-DOF robot manipulator will allow closed-form solutions if any of the following conditions is satisfied:

1. Three consecutive joint axes are parallel.

2. Three consecutive joint axes intersect.

3. There are two consecutive sets of two joint axes that are either parallel or intersecting.

4. Three consecutive joint axes are such that two intersect and two are parallel.

5. Three consecutive joint axes are such that the first two intersect and the last two intersect.

Note that these conditions are not exclusive of one another.- For example, arms that satisfy condition 5 include those that satisfy condition 2.

..














CHAPTER 7
SOLVING SIX-DOF MANIPULATORS



Reduction to.a 4-DOF Problem


At least six degrees of freedom are required for a robot manipulator to be able to arbitrarily position and orient its end-effector within its workspace. Equation

(2.14), with n equal to six, yields


A1 A2 A3 A4 A5 A6 = P. (7.1)


If both sides of this equality are multiplied by (A1 A2)-, the equation becomes


A3 A4 A5 A6 = Q (7.2)

with
Q a2-1 A1- P. (7.3)


When 81 and 82 are known, matrix Q is fully determined and can be viewed as a pose matrix for a 4-DOF arm whose structure is described by the left hand side of Eq. (7.2) which merely expresses a 4-DOF problem. In Chapter 5, we have seen that a 4-DOF problem can always be solved in closed-form, hence the remaining joint variables can be computed from Eq. (7.2).

..









First we show that a similar result can be obtained if 85 and 86 are known or if 81 and 06 are known instead of the first two joint variables.

In the development of the 4-DOF inverse kinematics solution, we have used the simplifying assumption that the last frame had DH-parameters d, a, and a all equal to zero. Although this assumption is obviously correct in the case of Eq. (7.2), we must show that it can be obtained in other cases. As shown in Eq. (2.7), a homogeneous matrix Ai decomposes into Ai = Ai Bi where Ai and Bi are given by Eqs. (2.8) and (2.9) and Ai is a homogeneous matrix for which DHparameters a, d, and a are zero. If the values of 81 and 86 are known, Equation (7.1) now reduces to the 4-DOF problem


A2 A3 A4 A5 = Q (7.4)

where

Q = A P- P A6- 1 B5 1 (7.5)


Similarly, If 85 and 86 are known, the inverse kinematic task becomes that of solving the 4-DOF case


Al A2 A3 A4 = Q/ (7.6)

with

Q = P A6-1 A5-I B4-I. (7.7)


In the following section, we will show how a twodimensional iterative technique can be implemented to solve the inverse kinematics problem of six-DOF robot

..









manipulators. Although this technique can equally be developed using Eqs. (7.4) or (7.6), it will be based on Eq. (7.2) for convenience.



Two-Dimensional Iterative Technique


Since we only need to know 2 of the joint variables to solve for the whole solution set, the inverse kinematics problem of six-DOF manipulators can be reduced to finding the values of the first two joint variables only, and getting closed-form values for the remaining variables. A numerical technique aimed at finding the values of 81 and E2 can be implemented by defining a system of two nonlinear equations in 81 and e2,


f(e1,82) = 0 (7.8)

g(ElE2) = 0, (7.9)


that can be solved using an iterative method such as a twodimensional Newton-Raphson.

From the left hand side of Eq. (7.2), two vectors uL and qL, corresponding to vectors u and q, (vectors u and q relate to pose Q as shown in Eq. (6.4)), are given by


UL = R3 R4 R5 z (7.10)

and


(7.11):


qL = R3(R4 15 + 14) + 13-

..









We define two nonlinear functions of 91 and E2 as differences between the inner-products uL qL, qL.qL and the inner-products u.q and q.q, respectively;


f(81,82) = uLqL u.q, (7.12)

g(81,82) qL-qL q.q. (7.13)


If the values of 81 and E2 used to compute pose matrix Q in Eq. (7.3) do correspond to a solution set, then Eq. (7.2) will hold and vectors UL and qL will be exactly equal to u and q forcing both functions f and g to be equal to zero. In other words solution sets of Eq. (7.1) correspond to zeros of the functions f and g defined in Eqs. (7.12) and (7.13).


Computing f(812) and q(91,82).

In order to compute the values of f and g for a given pair (81,82), the components of vectors u, q and the inner products u.q and q.q are needed to solve the 4-DOF equation (7.2) which in turn allows computation of inner products ULqL and qL.qL and finally the values of f and g.

Vectors u and q, computed from Eq. (7.3), are

u = R2-1 R -1 R z = R -1 R t (7.14)
u=2 1. 2 1

and

q = R2-1 [R1-1 (p 11) 12]. (7.15)


If we consider the components of vector t as expressed with respect to frame Fl,

..










Itx Cl tx + S1 ty

it = it = R -1 t = -r1S1 x+ TiCI ty + 01 tz
y 1.1 x 1 y 1
itz aS1 tx o1C1 ty + T1 tz


then vector u is given by

C2 1tx + S2 it

u = R2-1 it = -72S2 itx + T2C2 it + 02 It (7.16)

a2S2 itx 2C2 ity + T2 itz

To obtain the components of vector q, first we rewrite Eq. (7.15) as

q = R2-1(RI- 1 p R-1 11) R2-1 12 and we define

1ipx Cl Px + S1 py

1p 1p = R1 = 1SI Px + Cl Py + 01 Pz

1Pz _iSi Px 0Cl Py + irl Pz


vector q is then given by


C2 ( px al) + S2 py a2

q = -r2S2 1px + r2C2 1py + 02 pz 2d2 (7.17)

2S2 Px a2C2 py + 2 Pz T2d2

The inner product u.q can be derived from Eqs. (7.14) and (7.15),


u.q = (R2 RI1 t). R2-R1 [(p 11) -R1 12].

..









Using (4.4) and (4.5) as needed and expanding yields


u.q = t.(p 1i) R1-1t 12, or

u.q = t.p t.1It.2 which gives


u.q = t.p altx C1 alt y S1 a2 itx C2

2 it S2 d2 Itz. (7.18)


Similarly, the square of the length of vector q, q.q, is given by


q.q = R2-1R1-1 (P 1I)-R1 12] R2-1R1-1 [(p 1)-R1 12 where we factored out R2-1R1-1 in the expression of q from Eq. (7.15). Using (4.5) and (4.6) and expanding again leads to

q.q = p.p + 11.11 + 12.12 2(p.11 + R1- 1p.12 + R1 11.12) or

q.q = -2a2 [(a1 + 1px) C2 + 1py S2] 2d2 1Pz

2a 1px + p.p + a12 + a22 + d22 (7.19)


Equation (7.2) gives rise to a reduced system similar to that of Eqs. ((5.13)-(5.16) with the required shift in indexes,


a3uy S3 + a3ux C3 + a4o5 S5 4o5d4 C5 = r1 (7.20)

U3ux S3 U3uy C3 + G4a5 C5 = r2 (7.21)

..









o3qx S3 a3qy C3 a4a5 S5 = r3 (7.22)

a3qy S3 + a3qx C3 + a4a5d4 S5 + a4a5 C5 = r4 (7.23)


with


rI = q.u 75d5 d3uz T4T5d4 (7.24)

r2 475 3uz (7.25)

r3 T3(d3 qz) + d4 + '4d5 (7.26)

r4= (q.q + a32 + d32 a42 d42 a52 d52 )/2

d3qz T4d4d5. (7.27)


Solving this system of equations will yield the values of 83 and 85. The value of e4 can then be computed from the two equations


(a4d5- T4a5S5) S4 + (a4+a5C5) C4 = C3qx + S3qy a3 (7.28)

and

(a4+a5C5) S4 (o4d5-r4a5S5) C4 =

-T3S3qx + r3C3qy + C3(qz-d3) (7.29) (7.29)


derived from Eqs. (5.23) and (5.24), or from the equations (T4u5C5+"4-5) S4 + c5S5 C4 = C3u + S3uy (7.30)

and

a5S5 S4 (4a5C5+475) C4 =

-T3S3ux + r3C3uy + C3uz, (7.31)


corresponding to Eqs. (5.25) and (5.26).

..








We can now compute the inner products uL*qL and qL.qL. By incrementing the indexes in Eq. (6.10), we derive uLqL = T5d5 + a4o5 S5 o4d4a5 C5 + r5d4

+ 05 S5 (a3 C4 + a3d3 S4)

05 C5 (-a3 4 S4 + a3d3r4 C4 + r3d3o4)

+ T5 (a3a4 S4 03d304 C4 + T3d3 3). (7.32)

Vector qL, obtained from the left hand side of Eq. (7.2), is


qL = R3(R4 15 + 14) + 13 (7.33)


and the square of its length is given by


qL'qL = (R3(R4 15 + 14) + 13) (R3(R4 15 + 14) + 13) or

qL'qL = (15+R4-114+R4-1R3-113) (15+R4-114+R4-1R3-113), after factoring out (R3 R4) and using inner product invariance of rotations. Multiplying out the terms in parentheses and using (4.5) and (4.6) where necessary, the last equation yields


qL L = 2 [a5 C5(a4 + a3 C4 + a3d3 S4)

+ a5 S5(-a3r4S4 + a3d3T4 C4 + r3d3C4 + C4d4)

+ d5 (T4d4 +a3C4 S4 o3d3u4 C4 + d373 4)

+ a3a4 C4 + C3d3a4 S4 + 'T3d3d4]

+ a32 + d32 + a42 + d42 + a52 +d52. (7.34)

..









Given a pair (81, e2), the corresponding values of f(el,e2) and g(81,82) are obtained by the following steps:

Step 1. For initial values of l81 and 82, compute the coordinates of vectors u and q as given by Eqs. (7.16) and (7.17). The inner products u.q and q.q can be computed using the regular inner product formula,


u.q = Uxqx + uy qy + Uzqz

and

q.q = qx2 + q2 + qz2


Step 2. Solve the reduced system of Eqs. (7.20)-(7.23) for 03 and 85.

Step 3. Compute the value of 84 from Eqs. (7.28) and (7.29) or Eqs. (7.30) and (7.31).

Step 4. Compute the inner products uL*qL and qL.qL, given by Eqs. (7.32) and (7.34), respectively, and compute the values of f and g as given by Eqs. (7.12) and (7.13). Two-Dimensional Newton-Raphson

The zeros of f and g can be iteratively computed and checked for consistency with Eq. (7.1). If a computer

program for evaluating the two functions is available, the partial derivatives of f and g with respect to el and 02, denoted fl, f2 and gl, g2 respectively, can be numerically approximated by


fl(el,e2)= 1f/ 1 =1f( +681',e2)f(e1,2)]/661, (7.35)

..









f2 ('1'2)= Bf/E2 = [f(E1,82+E682) -f(1,82)]/692, (7.36) and

g91(e1,e2)= 2g/G1 = (g(91+681e2)-g(E112)]/e681, (7.37) g2('lE2)= ag/aG2 = [g(G1E,2+E2)-g(E112)]/6892, (7.40) where 681 and 682 are small increments of 8 and 82

respectively.

The two-dimensional Newton-Raphson technique for solving the inverse kinematics problem for a six-revoluteDOF robot arm of arbitrary architecture proceeds according to the following steps:

Step 1. Assume an initial estimate of 81 and E2 and compute 83, E4, and 95

Step 2. From the values of e1, 82, 83, 84, and E5 compute f(911,2) and g(911e2) as in Eqs. (7.12) and (7.13).

Step 3. Compute the partial derivatives of f and g with respect to 1 and 92 by numeric approximations as shown earlier.

Step 4. Obtain a new estimate for 8 and 82 by the two-dimensional Newton-Raphson method, i.e.
_-1i
ei 81 fl f2 f(G1182)

E2 2 91 92 g(9 1' 2)
new

Step 5. Repeat steps 2 to 5 until desired accuracy is attained.

..









Step 6. Complete the solution set by uniquely

computing G6 from



C6 nx

R6x X 6 R5_ 14_ 13_ 12_ 1Rin y (7.41)

0 n



Step 7. Check the solution set for consistency with Eq. (7.1).

Choice of functions f and g. The functions f and g

defined by Eqs. (7.12) and (7.13) are computationally economical since they do not require computation of the forward kinematics or the inverse jacobian of ,the manipulator. In fact, even the value of e6 is not required to compute f and g since Eq. (7.41) is considered only after convergence. Unfortunately, a pair (01, 82) for which both f and g are zero is not guaranteed to correspond to a solution set of Eq. (7.1). Extraneous solution sets can be

converged to as well. These are joint values that will*

yield an end-effector pose that is not exactly the desired one.

Other functions that fully constraint the end-effector pose can be defined at the cost of greater computational complexity. Wu and Paul (1982) have shown that the

difference between actual and desired end-effector poses can be expressed as a 6 x 1 vector xe given by

..











xI nL (P PL)

x2 bL (P P L)

x3 bL (P PL)
xe = = (7.42)
x4 (tL.b t.bL)/2

x5 (nL-t n.tL)/2

x6 (bL.n b.nL)/2


where n, b, t, and p are the vectors that describe the desired end-effector pose P as defined in (2.12) and vectors nL, bL, tL, and PL are their corresponding vectors from the left hand side of equation (7.1). Two functions can be

defined as


f(81,2) = x12 + x22 + x32 (7.43)
222 ( .4
g(e1,2) = x42 + x52 + x6 (7.44)


A pair (81e,2) that is a zero of both f and g is guaranteed to correspond to a solution set of Eq. (7.1) so that the iterative method described above will only converge to a true solution. However, now, the forward kinematics must be computed at each iteration since the components of vectors nL, bL, tL, and PL are all needed to evaluate functions f and g as defined by Eqs. (7.42) and (7.43).

..








One-Dimensional Method


The inverse kinematic problem for six-DOF manipulators reduces to a five-DOF one when the first or the last joint variable is known. Equation (7.1) becomes


A2 A3 A4 A5 A6 =Q, (7.45)

with

Q = A1-1 P (7.46)


when G1 is known, and


A1 A2 A3 A4 A5 = Q, (7.47)

with

Q = P 61B5 (7.48)


if 06 is known. In both cases, a five-DOF problem is

obtained. When the resulting five-DOF problem is solvable in closed form, knowledge of 81 or E6 is then sufficient to yield a complete solution set. The inverse kinematic

problem then reduces to finding one joint angle which can be accomplished by a one-dimensional iterative technique. In chapter 6, we found that a sufficient condition for closed form solutions of 5-DOF manipulators is that they have one of the special structures listed at the end of Chapter 6. Six-DOF arms that include a five-DOF segment with this type of geometry can then be solved using a onedimensional iterative method. This iterative technique can be implemented in much the same way as described in Chapter

..









6 for five-degree-of-freedom arms. Assuming Eq. (7.45) is to be solved, we define a function f of 9i by


f(91) = uL-qL u.q (7.49)


where vectors UL, qL, u, and q are defined as earlier. Given a value of 81, vectors u and q are computed from Eq. (7.46), the values of the remaining joint variables are computed in closed form from Eq. (7.45) as indicated in Chapter 6 and Appendix B, the inner product uL.qL can then be obtained as in Eq. (6.10) with the proper index adjustments, and the value of f is then given by Eq. (7.49). As we have seen before, the ability to compute the function f for a given value of E1 allows the implementation of a practical one-dimensional Newton-Raphson algorithm. Therefore, we can conclude that a six-degree-of-freedom manipulator with two consecutive pairs of intersecting or parallel joint axes or three consecutive joint axes that are parallel and/or intersecting two at a time can be solved by use of a one-dimensional iterative technique instead of the two-dimensional method required for an arm of arbitrary architecture.



Closed-Form Solution


Some six-degree-of-freedom manipulators with simple geometries do not require any iterative method since- they can be solved in closed-form. Pieper (1968) has shown that

..









a sufficient condition for closed-form solutions is that three consecutive axes be intersecting. The inverse kinematics problem then reduces to finding the zeros of a quartic polynomial. In the literature, It seems to be common knowledge that three consecutive joint axes that are parallel is another sufficient geometric condition for

closed form solutions.

The analysis of Chapter 5 and Appendix A showed that under certain conditions, the reduced system of equations (7.20)-(7.23) included constraint equations of the form


ri = 0. (7.50)


The quantities ri, i=l, ,4, are functions of 81 and 92, as we have seen earlier. By looking for conditions under which a joint variable value can be directly obtained from an equation having the form of Eq. (7.50), we find two more sufficient six-DOF robot structure conditions for closed-form solutions (excluding the already known conditions of three parallel or three intersecting axes).

When the first two joint axes of a manipulator are parallel so that ai=0, then ci=0, ri=l, and the z-components of vectors u and q, given by Eqs. (7.16) and (7.17), become


uz= 2 (-ty C12 + tx S12) + r2tz (7.51)

and


(7-.52)-


Sqz = -a2' (Px S12 Py C12) + T2 (Pz-d2)':

..









This shows that r2 and r3, as given in (7.25) and (7.26), become linear functions of S12 and C12.

When joint axes 3 and 4 are parallel and joint axes 5 and 6 are parallel, the reduced system oe equations (7.20)(7.23) becomes similar to that of case 2 of Chapter 5,


a3uy S3 + a3ux C3 = rI (7.53)

0 = r2 (7.54)

c4a5 S5 = r3 (7.55)

a3qy S3 + a3qx C3 + a4a5d4 S5 + a4a5 C5 = r4. (7.56)


Equation (7.54) yields two possible values for G12, each of which will provide two possible values of 05 from Eq. (7.55) when substituted in the expression of r3. The remaining

joint values can then be computed in closed form. A similar development occurs when axes 3 and 4 are parallel and axes 5 and 6 intersect.

To summarize, a six-DOF manipulator has a closed-form solution if one of the following conditions is satisfied:

1. Three consecutive joint axes are parallel.

2. Three consecutive joint axes intersect at one point.

3. The arm is formed of three sets of two parallel axes. This structure is illustrated in Figure 7.1(a).

4. The arm has two sets of two parallel joint axes followed or preceded by two intersecting axes. These

structures are illustrated in Figure 7.1(b) and 7.1(c).

..




















a. 3 pairs of parallel joint axes.


b. 2 pairs of parallel joint axes followed by 2
intersecting joint axes.


C. 2 pairs of parallel joint axes preceded by 2
intersecting joint axes.



Figure 7.1. 6-DOF manipulators with closed-form
inverse kinematics.

..













CHAPTER 8
ORTHOGONAL MANIPULATORS



Definition: An n-axes, serial kinematic chain of revolute or prismatic joints is orthogonal if all twist angles ai, i=l, . n, along the chain are 0 or v/2. An open orthogonal kinematic chain will be called an orthogonal manipulator (Doty 1986).

Six-DOF orthogonal manipulators can be classified in terms of the values of their twist angles ai, i=l, . 5. Twist angle a6 is always assumed to be zero in this text. In fact, the value of a6 can be chosen arbitrarily since z6 is not a joint axis. Therefore, there are only 25 = 32 distinct classes of orthogonal manipulators, as listed in Table 8-1, 8 of which have 4 or more adjacent parallel joint axes which reduces their capability to less than six degrees of freedom. As a result, there are only 24 types of. six-joint orthogonal manipulators with full spatial position and orientation capability.

A convenient notation for this classification of orthogonal manipulators is obtained by assigning a 5-bit binary number to each of these 24 types in which bit i is 0 if ai= 0 and bit i is 1 if ai=7r/2. For example, a

manipulator with twist angles'

..









a5 r/2, a4=7r/2, a3=0, a2=0, and al =7r/2


belongs to the class 11-001 of orthogonal manipulators.

Since most industrial robot arms are orthogonal, it is worthwhile to consider the inverse kinematics problem with respect to these manipulators. The A-matrices associated with orthogonal arms have one of the two following forms


Ai (a=0) -


-Si Ci

0

0


aiCi siSi di

1


(8.1)


Ai(a==7r/2) =


Si aiCi

-Ci asii 0 di

0 1


Further computational simplification is obtained in the inverse kinematic equations with orthogonal manipulators since

RiZ Rilz = Z if ai= 0 and

Ri1z y if ai= v/2.


Doty (1986) has shown that, of the 24 classes of nontrivial orthogonal manipulators, those with 2 non-zero


(8.2)

..









twist angles (classes 01-001, 01-010, 01-100, 10-100 and 10010) have closed-form solutions. The inverse kinematic analysis of Chapter 7 shows that the most complex six-DOF robot structure can be solved by use of a two-dimensional iterative technique. Simpler structures only require a onedimensional numerical technique and some even simpler structures can be solved in closed-form.

In Table 8-1, we provide a list of 'all thirty-two orthogonal manipulator classes in which we indicate the degenerate geometries and, for the twenty four nondegenerate classes, we indicate a suitable inverse kinematic

method necessary for solving the most complex arm structure within that class. It must be understood that intersecting axes cannot be considered according to a classification based on the values of the twist angles alone. The choice of inverse kinematic method indicated in Table 8-1 is based solely on the presence of parallel axes within a given class. Simpler inverse kinematic methods can be used if any of the special structures discussed in chapters 5, 6, an d 7are present.

In Chapter 9, the inverse kinematics of four orthogonal manipulators are described -in more detail.

..









Table 8-1. Inverse kinematics of orthogonal manipulators

Class Method Justification


1 00-000. D All six axes are parallel

2 00-001 D Five consecutive parallel axes

3 00-010 D Four consecutive parallel axes

4 00-011 D Four consecutive parallel axes

5 00-100 CF Three consecutive parallel axes

6 00-101 CF Three consecutive parallel axes

7 00-110 CF Three consecutive parallel axes

8 00-111 CF Three consecutive parallel axes

9 01-000 D Four consecutive parallel axes

10 01-001 CF Three consecutive parallel axes

11 01-010 CF -Three pairs of parallel axes

12 01-011 1-D Two pairs of parallel axes

13 01-100 CF Three consecutive parallel axes

14 01-101 2-D

15 01-110 2-D

16 01-111 2-D

..










Table 8-1. Class


--Continued Method


Justification


Five consecutive parallel axes Four consecutive parallel axes Three consecutive parallel axes Three consecutive parallel axes Three consecutive parallel axes Two pairs of parallel axes





Four consecutive parallel axes Three consecutive parallel axes Two pairs of parallel axes



Three consecutive parallel axes


Notation: D

CF

1-D 2-D


= Degenerate geometry = Closed-Form

= One Dimensional iterative method = Two-Dimensional iterative method


10-000 10-001 10-010 10-011 10-100 10-101 10-110 10-111 11-000 11-001 11-010 11-011 11-100 11-101 11-110 11-111


D

D

CF CF CF 1-D

2-D 2 -D

D

CF 1-D

2-D CF 2-D 2-D 2-D

..















CHAPTER 9
APPLICATION EXAMPLES



Example 1: The PUMA 560


A popular orthogonal manipulator geometry, the PUMA 560, is described by the kinematic parameters given in Table 9-1 and illustrated in Figure 9.1. This manipulator has a spherical wrist and therefore allows closed-form solutions (Pieper 1968). Inverse kinematic solutions have been

proposed by numerous authors for this type of arm (Lee and Ziegler 1984; Craig 1986; Paul and Zhang 1986).


Table 9-1. PUMA 560 kinematic parameters

Joint d G) a a (rd)


e3

94


85.


7r/ 2

0

fr/ 2 7r/ 2 7r/72

0


This example is utility of the approach


included here to demonstrate the already outlined and to contrast it

..


Full Text
xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0008224200001datestamp 2009-01-28setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title Inverse kinematic analysis of robot manipulators dc:creator Manseur, Rachiddc:publisher Rachid Manseurdc:date 1988dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00082242&v=0000119866092 (oclc)001113629 (alephbibnum)dc:source University of Florida



PAGE 1

,19(56( .,1(0$7,& $1$/<6,6 2) 52%27 0$1,38/$7256 %\ 5$&+,' 0$16(85 $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$

PAGE 2

7R WKRVH ZKR IRXJKW IRU WKH IUHHGRP DQG HGXFDWLRQ RI DOO $OJHULDQV

PAGE 3

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

PAGE 4

'U 3H\WRQ = 3HHEOHV DQG WR 'U 5REHUW / 6XOOLYDQ IRU WKHLU WLPHO\ KHOS DQG XQGHUVWDQGLQJ ZRXOG OLNH WR WDNH WKLV RSSRUWXQLW\ WR WKDQN 0V *UHWD 6EURFR IRU EHLQJ VXFK D JUHDW JUDGXDWH VHFWUHWDU\ IRU FDULQJ DQG IRU KHU FRQVWDQW HQFRXUDJHPHQWV $OO WKH VWXGHQWV RI WKH 0DFKLQH ,QWHOOLJHQFH /DERUDWRU\ DOVR GHVHUYH UHFRJQLWLRQ IRU FUHDWLQJ VXFK D IULHQGO\ DQG SURGXFWLYH ZRUNLQJ HQYLURQPHQW ,Q SDUWLFXODU ZRXOG OLNH WR WKDQN 'U 6XEELDQ *RYLQGDUDM DQG 0U 'DYLG $OGHUPDQ IRU WKHLU KHOS LQ WKH ODERUDWRU\ ZRXOG DOVR OLNH WR H[SUHVV P\ JUDWLWXGH DQG P\ ORYH WR P\ IDPLO\ VSHFLDOO\ P\ SDUHQWV 2PDU $PRNUDQH 0DQVHXU DQG 'KLD
PAGE 5

7$%/( 2) &217(17 3DJH $&.12:/('*(0(176 LLL $%675$&7 9 &+$37(56 ,1752'8&7,21 7+( ,19(56( .,1(0$7,&6n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n 6ROXWLRQ Y

PAGE 6

62/9,1* '2) 0$1,38/$7256 5HGXFWLRQ WR D '2) 3UREOHP 7ZR'LPHQVLRQDO ,WHUDWLYH 7HFKQLTXH 2QH'LPHQVLRQDO 0HWKRG &ORVHG)RUP 6ROXWLRQ 257+2*21$/ 0$1,38/$7256 $33/,&$7,21 (;$03/(6 9 ([DPSOH O7KH 380$ n ([DPSOH 7KH *3 ([DPSOH 7KH 20 0DQLSXODWRU ([DPSOH 20 0DQLSXODWRU ([DPSOH $ *HQHUDO *HRPHWU\ '2) 0DQLSXODWRU &21&/86,21 $1' )8785( :25. $33(1',; 62/9,1* )25 5()(5(1&(6 %,2*5$3+,&$/ 6.(7&+ YL

PAGE 7

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f URERWV 7KH UHVXOWV REWDLQHG DUH WKHQ DSSOLHG WR WKH VWXG\ YLL

PAGE 8

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

PAGE 9

&+$37(5 ,1752'8&7,21 $Q LPSRUWDQW SDUW RI FRPSXWHU FRQWURO DOJRULWKPV IRU RSHQ VHULDO NLQHPDWLF FKDLQV LV WKH LQYHUVH NLQHPDWLFV VHFWLRQ ,Q DQ\ URERWLF DSSOLFDWLRQ WKHn KDQG RU HQG HIIHFWRU RI WKH URERW PD\ PRYH DORQJ D WUDMHFWRU\ VSHFLILHG DV D VHTXHQFH RI SRLQWV DW ZKLFK WKH HQGHIIHFWRU SRVH RULHQWDWLRQ DQG SRVLWLRQf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

PAGE 10

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f URERW PDQLSXODWRUV KDV LQWHUHVWHG D IHZ UHVHDUFKHUV 5RWK 5DVWHJDU DQG 6FKHLQPDQ f SXW DQ XSSHU ERXQG RI RQ WKH GHJUHH RI D SRO\QRPLDO HTXDWLRQ LQ RQH MRLQW YDULDEOHf WKDW FDQ EH GHULYHG IURP WKH LQYHUVH NLQHPDWLFV SUREOHP RI VL['2) PDQLSXODWRUV $ VLPLODU UHVXOW ZDV REWDLQHG E\ 'XII\ DQG &UDQH f XVLQJ WKH HTXLYDOHQFH EHWZHHQ DQ RSHQ UHYROXWH'2) NLQHPDWLF FKDLQ DQG WKH UHYROXWH VLQJOHORRS VSDWLDO PHFKDQLVP 7KHUHIRUH WKH QXPEHU RI LQYHUVH NLQHPDWLF VROXWLRQV IRU UHYROXWH'2) PDQLSXODWRUV FRXOG EH DW PRVW 0RUH UHFHQWO\ /HH DQG /LDQJ LQ SUHVVf XVLQJ 'XII\nV PHWKRG ZHUH DEOH WR UHGXFH WKH GHJUHH RI WKH LQYHUVH NLQHPDWLF SRO\QRPLDO HTXDWLRQ WR WKHUHE\ UHGXFLQJ WKH XSSHU ERXQG RQ WKH QXPEHU RI LQYHUVH NLQHPDWLF VROXWLRQV WR 7VDL DQG 0RUJDQ f LOOXVWUDWLQJ £ QHZ LQYHUVH NLQHPDWLF PHWKRG FDSDEOHRI SURGXFLQJDOO VROXWLRQVn

PAGE 11

IRXQG D URERW PDQLSXODWRU DQG DQ HQGHIIHFWRU SRVH ZLWK SRVVLEOH VROXWLRQV 0DQVHXU DQG 'RW\ LQ SUHVVf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

PAGE 12

WHFKQLTXHV DUH GHVFULEHG DQG VXIILFLHQW PDQLSXODWRU VWUXFWXUDO FRQGLWLRQV IRU WKH DSSOLFDELOLW\ RI WKHVH PHWKRGV DUH GHWHUPLQHG $OO WKH PHWKRGV GHYHORSHG LQ WKLV GLVVHUWDWLRQ DUH LOOXVWUDWHG E\ H[DPSOHV LQ &KDSWHU &KDSWHU VXPPDUL]HV WKH ILQDO UHVXOWV GLVFXVVHV WKH FRQWULEXWLRQV RI WKLV ZRUN WR WKH ILHOG RI URERWLFV DQG SUHVHQWV UHODWHG WRSLFV DQG DUHDV RI IXWXUH UHVHDUFK

PAGE 13

&+$37(5 7+( ,19(56( .,1(0$7,&6 352%/(0 1RWDWLRQ DQG 0DWKHPDWLFDO 3UHOLPLQDULHV 7KH '+ 3DUDPHWHUV $ URERW PDQLSXODWRU LV PRGHOOHG DV DQ RSHQ NLQHPDWLF FKDLQ RI ULJLG ERGLHV OLQNVf FRQQHFWHG E\ MRLQWV $ UHIHUHQFH IUDPH LV DVVLJQHG WR HDFK OLQN DORQJ WKH FKDLQ VWDUWLQJ ZLWK WKH EDVH IUDPH ) DVVLJQHG WR WKH IL[HG OLQN XS WR WKH HQG HIIHFWRU IUDPH )Q IRU D PDQLSXODWRU ZLWK Q GHJUHHV RI IUHHGRP '2)f 7KH SRVLWLRQ DQG RULHQWDWLRQ RI IUDPH ) ^[A \MB ]A` ZLWK UHVSHFW WR WKH SUHFHGLQJ IUDPH )IBL DUH HQWLUHO\ GHVFULEHG E\ WKH IRXU '+SDUDPHWHUV GMB k DA DQG 'HQDYLW DQG +DUWHQEHUJ f 7KHVH SDUDPHWHUVDUH LOOXVWUDWHG LQ )LJXUH DQG GHILQHG DV GMB GLVWDQFH EHWZHHQ WKH FRPPRQ QRUPDO WR D[HV =MA DQG ]A DQG WKH FRPPRQ QRUPDO WR ]A DQG ]A PHDVXUHG DORQJ D[LV ]A e WKH DQJOH RI URWDWLRQ DERXW =MB VR WKDW [A EHFRPHV SDUDOOHO WR ZKHQ A DMB WKH OHQJWK RI WKH FRPPRQ QRUPDO WR D[HV ]AA DQG ]A WKH DQJOH RI URWDWLRQ DERXW [A VR WKDW ]A EHFRPHV SDUDOOHO WR ]AA ZKHQ

PAGE 14

)LJXUH 7KH '+SDUDPHWHUV

PAGE 15

:KHQ MRLQW L LV UHYROXWH SDUDPHWHU A LV WKH MRLQW YDULDEOH DQG LI MRLQW L LV SULVPDWLF WKH MRLQW YDULDEOH LV GA :KHQ DSSOLFDEOH GA PHDVXUHV WKH WUDQVODWLRQ DORQJ D[LV +RPRJHQHRXV 0DWULFHV ,I D YHFWRU X >X[ [X\ LV H[SUHVVHG LQ IUDPH )A LWV H[SUHVVLRQ ZLWK UHVSHFW WR IUDPH )ABO[ X VDWLVILHV LLX ; &L 6LUL 6LDL DL&L OX; LBOX\ 6L &L7L &LDL DL6L r+ Lf] Ds 7/ Gs [X] O RU LQ D FRPSDFW QRWDWLRQ L X L X f ZKHUH U A FRV DAf DMB VLQDef &A FRV Af DQG 6A VLQkAf DQG $A LV WKH KRPRJHQHRXV IUDPHWUDQVIRUP PDWUL[ 3DXO f 7KH OHDGLQJ VXSHUVFULSW LQGLFDWHV WKH IUDPH RI H[SUHVVLRQ 7KH KRPRJHQHRXV PDWUL[ WUDQVIRUP PHUHO\ H[SUHVVHV WKH IDFW WKDW IUDPH )A FDQ EH REWDLQHG IURP IUDPH )AA E\ WKH IROORZLQJ VHTXHQFH RI EDVLF WUDQVIRUPV 5RWDWLRQ DERXW RI DQJOH M ZKRVH KRPRJHQHRXV PDWUL[ LV

PAGE 16

5]Lf &L 6L 6MB &L f 7UDQVODWLRQ RI F/ XQLWV DORQJ D[LV GHVFULEHG E\ WKH PDWUL[ 7U]GLf R R O GM f 7UDQVODWLRQ RI DA XQLWV DORQJ D[LV [A ZLWK KRPRJHQHRXV WUDQVIRUP 7U[DLf D f 5RWDWLRQ DERXW ;MB RI DQJOH 5[DLf UL DL R &7L I/ f 7KH PDWUL[ RI (T f LV REWDLQHG E\ WKH SURGXFW 3DXO f f n

PAGE 17

$L 5]kLf 7U] GLf 7U[DA 5[8Lf $ XVHIXO GHFRPSRVLWLRQ RI PDWUL[ $L LV $L $L %L f ZLWK WKH GHILQLWLRQV $L 5]Lf DQG f %L 7U]GLf 7U[DLf 5ARAf f ([SOLFLWO\ PDWUL[ %L LV %L 7L &7L DL UL GL f ZKHUH *L LV WKH XSSHU OHIW [ LQ %L DQG NL LV WKH XSSHU ULJKW [ YHFWRU RI %Lr 7KH XSSHU OHIW [ PDWUL[ LQ $L LV WKH URWDWLRQ PDWUL[ 5L QHFHVVDU\ WR DOLJQ WKH XQLW YHFWRUV RI )L ZLWK WKHLU FRXQWHUSDUWV LQ )LB ZKLOH YHFWRU L DLFL DLVL SRVLWLRQV WKH RULJLQ RI )L ZLWK UHVSHFW WR )LBL

PAGE 18

$ FRPSDFW DQG XVHIXO H[SUHVVLRQ IRU $A LV f 5RWDWLRQ PDWULFHV DUH RUWKRJRQDO VR 5A 5A7 ZKHUH WKH VXSHUVFULSW 7 GHQRWHV WKH WUDQVSRVH RSHUDWLRQ DQG WKH LQYHUVH RI PDWUL[ $A FDQ EH H[SUHVVHG DV FL 6L DL VLUL &LUL DL fDLGL VLDL &L&7L 7L ULGL 5L7 5L7OLf f 3UREOHP 'HILQLWLRQ ,I WKH RULHQWDWLRQ RI WKH HQGHIIHFWRU LV VSHFLILHG E\ WKH URWDWLRQ PDWUL[ 5 QHFHVVDU\ WR DOLJQ WKH XQLW YHFWRUV RI WKH HQGHIIHFWRU IUDPH )Q ZLWK WKH FRUUHVSRQGLQJ YHFWRUV RI EDVH IUDPH )J DQG WKH SRVLWLRQ RI WKH RULJLQ RI WKH HQG HIIHFWRU IUDPH LV JLYHQ DV D YHFWRU S ZLWK UHVSHFW WR WKH EDVH IUDPH )4 WKHQ WKH HQGHIIHFWRU SRVH LV DGHTXDWHO\ GHVFULEHG E\ WKH [ PDWUL[ Q[ E[ IF[ Q\ E\ E\ 3[ 3\ Q E W S 5 S Q] 3]

PAGE 19

ZKHUH Q[ E[ IF[ 3[ Q Q\ E E\ W IF\ I S 3\ B Q] E] E] 3= DQG 5 Q[ E[ Q\ E\ Q] E] W W W ; \ ] 7KH LQYHUVH NLQHPDWLFV SUREOHP IRU D QGHJUHHRI IUHHGRP PDQLSXODWRU FRQVLVWV RI ILQGLQJ D VHW RI MRLQW YDULDEOHV YDOXHV FDOOHG D VROXWLRQ VHW WKDW ZLOO VDWLVI\ WKH HTXDWLRQ $ $ $ $ $rrr$Q 3 f 7KLV PDWUL[ HTXDWLRQ JLYHV ULVH WR D V\VWHP RI QRQOLQHDU HTXDWLRQV ZKRVH FRPSOH[LW\ GHSHQGV RQ WKH PDQLSXODWRU JHRPHWU\ DV GHVFULEHG E\ WKH '+SDUDPHWHUV $W OHDVW VL[ GHJUHHV RI IUHHGRP DUH UHTXLUHG WR DUELWUDULO\ SRVLWLRQ DQG RULHQW D ULJLG ERG\ LQ VSDFH 7KHUHIRUH ZKHQ Q LV ODUJHU WKDQ VL[ WKH PDQLSXODWRU LV UHGXQGDQW DQG WKH V\VWHP RI HTXDWLRQV LPSOLHG E\ (T f LV XQGHUFRQVWUDLQHG ,I Q LV OHVV WKDQ WKH V\VWHP EHFRPHV RYHUFRQVWUDLQHG DQG ZKHQ Q LV HTXDO WR WKH LQYHUVH NLQHPDWLF SUREOHP LV H[DFWO\ VSHFLILHG ,Q WKLV

PAGE 20

UHVHDUFK ZH ZLOO DGGUHVV WKH LQYHUVH NLQHPDWLFV SUREOHP RI QRQUHGXQGDQW URERW PDQLSXODWRUV 0RVW H[LVWLQJ LQGXVWULDO PDQLSXODWRUV DUH RU GHJUHHRIIUHHGRP URERWV KHQFH LW LV RI SUDFWLFDO LPSRUWDQFH WR VROYH (T f IRU Q DQG Q 7KH QXPHULFDO WHFKQLTXHV GHYHORSHG LQ WKLV WH[W DUH EDVHG RQ D FRPSOHWH LQYHUVH NLQHPDWLF DQDO\VLV RI IRXUGHJUHHRI IUHHGRP PDQLSXODWRUV 7KHUHIRUH WKLV UHVHDUFK ZLOO DLP DW VROYLQJ (T f IRU URERWV ZLWK IRXU ILYH DQG VL[ MRLQW D[HV $OWKRXJK WKH WHFKQLTXHV GHVFULEHG LQ WKLV WH[W FDQ EH DSSOLHG WR PDQLSXODWRUV KDYLQJ SULVPDWLF MRLQWV 0DQVHXU DQG 'RW\ f ZH FRQFHQWUDWH RQ DOOUHYROXWH VL['2) PDQLSXODWRUV Q LV VL[ DQG DOO MRLQWV DUH DVVXPHG UHYROXWH

PAGE 21

&+$37(5 (;,67,1* 62/87,216 &ORVHG)RUP $UFKLWHFWXUHV 7KH DELOLW\ WR FRPSXWH WKH FRRUGLQDWHV LQ MRLQW VSDFH RI DQ HQGHIIHFWRU SRVH JLYHQ LQ &DUWHVLDQ VSDFH LV DQ LPSRUWDQW FULWHULRQ LQ WKH GHVLJQ RI FRPSXWHUFRQWUROOHG PDQLSXODWRUV $ GHVLUDEOH SURSHUW\ IRU DQ LQGXVWULDO PDQLSXODWRU LV WKH SRVVLELOLW\ RI FRPSXWLQJ WKH MRLQW YDULDEOHV QHFHVVDU\ WR SRVLWLRQ DQG RULHQW WKH HQGHIIHFWRU DV VSHFLILHG LQ &DUWHVLDQ VSDFH LQ FORVHGIRUP 3LHSHU f KDV VKRZQ WKDW D FORVHGIRUP VROXWLRQ LV SRVVLEOH ZKHQ WKH PDQLSXODWRU KDV WKUHH DGMDFHQW MRLQW D[HV LQWHUVHFWLQJ DW D FRPPRQ SRLQW 7KH LQYHUVH NLQHPDWLF SUREOHP UHGXFHV WKHQ WR D TXDUWLF SRO\QRPLDO HTXDWLRQ LQ RQH RI WKH MRLQW YDULDEOHV 0DQLSXODWRUV ZLWK WKH ODVW WKUHH MRLQW D[HV LQWHUVHFWLQJ DUH VDLG WR EH ZULVWSDUWLWLRQHG &RPSXWDWLRQDOO\ HIILFLHQW PHWKRGV IRU FRPSXWLQJ WKH SRVLWLRQ YHORFLW\ DQG DFFHOHUDWLRQ LQYHUVH NLQHPDWLFV IRU WKLV W\SH RI PDQLSXODWRUV KDYH EHHQ SUHVHQWHG E\ )HDWKHUVWRQH f +ROOHUEDFK DQG 6DKDU f 3DXO DQG =KDQJ f DQG /RZ DQG 'XEH\ f 6HYHUDO LQGXVWULDO VL[ DQG ILYH'2) PDQLSXODWRUV VXFK DV WKH 380$ VHULHV URERWV DUH RI WKH ZULVWSDUWLWLRQHG W\SH ,I RQ WRS RI

PAGE 22

KDYLQJ D ZULVW WKH PDQLSXODWRU KDV VRPH DGGHG VWUXFWXUDO IHDWXUH VXFK DV WZR SDUDOOHO RU LQWHUVHFWLQJ MRLQW D[HV WKHQ FORVHGIRUP VROXWLRQV PD\ EH REWDLQHG LQ D VLPSOHU IRUP WKDQ D TXDUWLF SRO\QRPLDO HTXDWLRQ 7KLV LV WKH FDVH RI WKH 380$ URERW ZKRVH LQYHUVH NLQHPDWLFV DUH GLVFXVVHG LQ ([DPSOH RI &KDSWHU $Q DOJHEUDLF PHWKRG IRU VROYLQJ WKH LQYHUVH NLQHPDWLFV RI WKH 380$ FDQ EH IRXQG LQ &UDLJ f DQG D JHRPHWULF DSSURDFK LV GHVFULEHG LQ )X *RQ]DOH] t /HH f $QRWKHU VXIILFLHQW FRQGLWLRQ IRU FORVHGIRUP VROXWLRQV LV WKDW WKUHH DGMDFHQW MRLQW D[HV EH SDUDOOHO 'XII\ )X *RQ]DOH] t /HH f 5HFRUG DQG 3OD\EDFN $Q LQGXVWULDO URERW PDQLSXODWRU LV XVXDOO\ HTXLSSHG ZLWK VHQVRUV WKDW FDQ PHDVXUH LQIRUPDWLRQ VXFK DV MRLQW YDULDEOH YDOXHV DQG UDWHV RI FKDQJH RI WKRVH YDOXHV $ PHWKRG WKDW DYRLGV WKH FRPSXWDWLRQDO FRPSOH[LW\ RI WKH LQYHUVH NLQHPDWLF SUREOHP DOWRJHWKHU FRQVLVWV RI UHPRWHO\ JXLGLQJ D URERW HQGHIIHFWRU WUDMHFWRU\ E\ DFWLYDWLQJ HDFK MRLQW VHSDUDWHO\ ZKLOH VWRULQJ MRLQW VSDFH FRRUGLQDWHV DQG LQIRUPDWLRQ IURP WKH VHQVRUV DW VHOHFWHGnSRLQWV DORQJn WKHn WUDMHFWRU\ 7KH URERW FDQ WKHQ LQGHILQLWHO\ UHSHDW WKH UHFRUGHG PRWLRQ 6KRXOG WKH URERW EH QHHGHG IRU D GLIIHUHQW WDVN RU VKRXOG D FKDQJH LQ WKH ZRUNFHOO RFFXU WKDW UHTXLUHV GLIIHUHQW HQGHIIHFWRU WUDMHFWRULHVnnWKHfPRWLRQ RI WKHURERW ZLOO KDYH WR EH UHFRUGHG DJDLQ

PAGE 23

1XPHULFDO 7HFKQLTXHV 0DQ\ VL[ DQG ILYH'2) NLQHPDWLF VWUXFWXUHV ODFN WKH QHFHVVDU\ DUFKLWHFWXUDO VLPSOLFLW\ IRU FORVHGIRUP LQYHUVH NLQHPDWLF VROXWLRQV 6ROYLQJ VXFK PDQLSXODWRUV UHTXLUHV WKH XVH RI QXPHULFDO LWHUDWLYH WHFKQLTXHV )RU VL['2) URERWV HTXDWLRQ f FDQ EH H[SUHVVHG DV D V\VWHP RI VL[ QRQOLQHDU HTXDWLRQV LQ WKH VL[ MRLQW YDULDEOHV RI WKH IRUP A k # H n f 3 I n k k n n f 3 I p@B k k p n n f 3 Ipn k k H n f D I L k k p n n f k I p &' WR k p n n f ZKHUH S[ 3\ DQG S] DUH WKH FRRUGLQDWHV RI WKH RULJLQ RI WKH HQGHIIHFWRU IUDPH DQG D DQG S DUH HLWKHU WKH (XOHU DQJOHV RU WKH UROOSLWFK\DZ DQJOHV GHULYHGn IURP WKH RULHQWDWLRQ PDWUL[ 5 RI WKH HQGHIIHFWRU IUDPH 3DXO f 7KH VL[GLPHQVLRQDO HTXDWLRQ LV WKHQ VROYHG E\ XVH RI D GLUHFW RU PRGLILHG 1HZWRQ5DSKVRQ RU VLPLODU WHFKQLTXH 0XOWLGLPHQVLRQDO LWHUDWLYH WHFKQLTXHV IRU VROYLQJ WKH LQYHUVH NLQHPDWLFV SUREOHP RI PDQLSXODWRUV RI DUELWUDU\ DUFKLWHFWXUH DUH GHVFULEHG E\ $QJHOHV f *ROGHKEHIJ\ %HKKDELE )HQWRUU fnA *•OGQEIJADKG /DZUHQFH f 7KH FRPSXWDWLRQDO HIILFLHQF\ RI WKHVH

PAGE 24

PHWKRGV LV KLQGHUHG E\ WKH QHHG WR FRPSXWH WKH LQYHUVH RI WKH PDQLSXODWRU -DFRELDQ DW VHYHUDO SRLQWV /LQDUHV t 3DJH f DQG .D]HURXQLDQ f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f XVHG D KRPRWRS\ PDS PHWKRG IRU VROYLQJ V\VWHPV RI SRO\QRPLDO HTXDWLRQV LQ VHYHUDO YDULDEOHV WR ILQG WKH VROXWLRQV RI WKH LQYHUVH NLQHPDWLFV SUREOHP RI UHYROXWH ILYH DQG VL[ GHJUHHRIIUHHGRP PDQLSXODWRUV RI DUELWUDU\ DUFKLWHFWXUH 7KH PHWKRG ILQGV DOO VROXWLRQV EXW LWV FRPSXWDWLRQDO FRPSOH[LW\ UHQGHUV LW LPSUDFWLFDO IRU PDQ\ DSSOLFDWLRQV /XPHOVN\ f SUHVHQWHG DQ LWHUDWLYH DOJRULWKP WKDW ILQGV HVWLPDWHV IRU WKUHH RI WKH MRLQW YDULDEOV DQG VROYHV LQ FORVHG IRUP IRU WKH UHPDLQLQJ WKUHH YDULDEOHV DW HDFK LWHUDWLRQ 7KH PHWKRG DSSOLHV WR D SDUWLFXODU W\SH RI DUP

PAGE 25

JHRPHWU\ WKDW RI WKH *3 URERW GLVFXVVHG LQ ([DPSOH &KDSWHU RI WKLV GLVVHUWDWLRQf DQG FRQYHUJHV WR DQ DFFXUDWH HQGHIIHFWRU SRVLWLRQ EXW WR D OHVV DFFXUDWH DSSUR[LPDWLRQ RI WKH HQGHIIHFWRU RULHQWDWLRQ

PAGE 26

&+$37(5 1(: $3352$&+ /LQN)UDPHV $VVLJQPHQW 6RPH VLPSOLILFDWLRQ LQ WKH PDWKHPDWLFDO GHVFULSWLRQ RI WKH LQYHUVH NLQHPDWLFV SUREOHP FDQ EH REWDLQHG LI FHUWDLQ VLPSOH UXOHV IRU DVVLJQLQJ WKH OLQNIUDPHV DUH DSSOLHG ,Q VHOHFWLQJ IUDPH )A WKH GLUHFWLRQ RI YHFWRU LV DOZD\V FKRVHQ VR WKDW WZLVW DQJOH LV LQ WKH LQWHUYDO >7f ,I WKHQ YHFWRUV DQG ] DUH SDUDOOHO DQG WKH FRPPRQ QRUPDO FDQ EH DUELWUDULO\ ORFDWHG DORQJ ERWK D[HV ,Q WKLV FDVH WKH SRVLWLRQ RI )UDPH )MB VKRXOG EH FKRVHQ VR WKDW GA LV HTXDO WR ]HUR )RU DQ Q'2) URERW IUDPH )Q DWWDFKHG WR WKH HQG HIIHFWRU FDQ EH FKRVHQ VR WKDW LW GLIIHUV IURP OLQN IUDPH )Q RQO\ E\ D URWDWLRQ RI DQJOH Q DERXW ]QBL ,Q RWKHU ZRUGV )Q FDQ EH VHOHFWHG VR WKDW GQ DQ DQ ZLWKRXW ORVV RI JHQHUDOLW\ :H SURYH WKLV SRLQW IRU Q EXW LW LV YDOLG IRU DQ\ UHOHYDQW YDOXH RI Q /HW XV DVVXPH WKDW (T f LV WR EH VROYHG ZLWK D '2) PDQLSXODWRU IRU ZKLFK G DJ RU DJ LV QRW HTXDO WR WKHQ WKH KRPRJHQHRXV PDWUL[ WUDQVIRUP $J GHFRPSRVHV LQWR

PAGE 27

DV JLYHQ LQ (T f (TXDWLRQ f LV WKHQ HTXLYDOHQW WR $ $ $ $ $ S % ZKHUH WKH ULJKW KDQG VLGH RI WKLV ODVW HTXDWLRQ LV VHHQ WR EH D FRQVWDQW SRVH PDWUL[ IRU D PDQLSXODWRU GHVFULEHG E\ WKH OHIW KDQG VLGH LH RQH IRU ZKLFK G D D VR WKDW $af§ Af :KHQ MRLQW LV QRW SULVPDWLF GA LV FRQVWDQW DQG WKH RULJLQ RI WKH EDVH IUDPH )4 FDQ EH SRVLWLRQHG VR WKDW GA LV HTXDO WR 7KH 5HGXFHG 6\VWHP RI (TXDWLRQV )RU D '2) DUP (T f EHFRPHV $ $ $ $ $ $ S f DQG LW \LHOGV WZHOYH QRQ WULYLDO VFDODU HTXDWLRQV LQ WKH VL[ XQNQRZQ YDULDEOHV ,W LV GHVLUDEOH WR UHGXFH WKLV V\VWHP WR D PLQLPDO QXPEHU RI HTXDWLRQV LQYROYLQJ DV IHZ RI WKH MRLQW YDULDEOHV DV SRVVLEOH )RU DOOUHYROXWH '2) PDQLSXODWRUV 7VDL DQG 0RUJDQ f KDYH HVWDEOLVKHFI WKDW ZLWK UHVSHFW WR IUDPH ) WKH ]FRPSRQHQW RI WKH SRVLWLRQ YHFWRU S DQG WKDW RI YHFWRU W DORQJ ZLWK WKH LQQHU SURGXFWV WSM DQG SSf SURYLGH HTXDWLRQV LQ RQO\ RI WKH XQNQRZQV WKHUHE\ UHGXFLQJ WKH FRPSOH[LW\ RI WKH SUREOHP 7KH SURFHVV RI REWDLQLQJ WKHVH IRXU HTXDWLRQV

PAGE 28

LQYROYHG PXOWLSO\LQJ WKH $PDWULFHV DQG VLPSOLI\LQJ WKH H[SUHVVLRQV REWDLQHG IRU WKH HOHPHQWV RI W DQG S %HVLGHV EHLQJ OHQJWK\ WKLV PHWKRG GRHV QRW DOORZ LQVLJKW LQWR WKH PHFKDQLVPV WKDW PDNH WKH VLPSOLILFDWLRQV SRVVLEOH 7KH DSSURDFK SUHVHQWHG KHUH SURYLGHV WKH VDPH UHVXOWV ZLWK PXFK OHVV HIIRUW DQG JUHDWHU LQVLJKW E\ WDNLQJ DGYDQWDJH RI WKH SURSHUWLHV RI URWDWLRQ WUDQVIRUPDWLRQV %\ ZULWLQJ WKH SURGXFW RI WZR $ PDWULFHVnLQ WKH IRUP 5M5M 5LOM OLf ZH GLYLGH (T f LQWR D SRVLWLRQ HTXDWLRQ S 55555ffff f DQG DQ RULHQWDWLRQ HTXDWLRQ 5 5MA 5 5 5 5J 5J f f :LWK WKH IUDPH DVVLJQPHQW FRQYHQWLRQV GLVFXVVHG ZKHQ MRLQW LV UHYROXWH (TXDWLRQ f WKHQ VLPSOLILHV WR S 5555fffL f 7KUHH LQGHSHQGHQW VFDODU HTXDWLRQV IRU S[ 3\ DQG S] FDQ EH REWDLQHG IURP (T f DQG PRUH HTXDWLRQV FDQ EH VHOHFWHG RXW RI WKH VFDODU HTXDWLRQV LPSOLHG E\ (T f

PAGE 29

6LQFH URWDWLRQV DUH RUWKRJRQDO WUDQVIRUPDWLRQV WKH\ OHDYH LQQHU SURGXFWV LQYDULDQW KHQFH 5 X 5 Y X Y f IRU DQ\ URWDWLRQ PDWUL[ 5 DQG DQ\ YHFWRUV X DQG Y $ VSHFLDO FDVH RI f WKDW LV YHU\ XVHIXO LV 5 X Y X 5BY f 7KHVH SURSHUWLHV DUH H[WUHPHO\ HIILFLHQW LQ HOLPLQDWLQJ DOJHEUDLF WHUPV DQG XQQHFHVVDU\ MRLQW YDULDEOHV ZKHQ DSSOLHG WR (TV f DQG f LI LW LV IXUWKHU UHFRJQL]HG WKDW rL AL > A LU @ DQG f 5L] >DLUL@ f ZKHUH ] > @7 DUH DOZD\V LQGHSHQGHQW GXH WR WKH IUDPH DVVLJQPHQWV GLVFXVVHG DERYH 5 ] 5B ] ] RI A $OVR LQ DOO FDVHV VLQFH IUDPH )J LV FKRVHQ WR IRUFH D rr %\ UHSHDWHG XVH RI (TV f DQG f UHGXFHG HTXDWLRQV IURP (TV f DQG f ZH REWDLQ IRXU W] HDXDWLRQ W W ] 5 ]f ] W 5L 5 5 5 5 ]f ] W] 5O U U U 5 ]f f ] W] ] 5 5B 5 5B 5M ]f f

PAGE 30

S] HTXDWLRQ S 5A 5 5 5A ZLWK T 5BO 5 5B 5OOOLfff VR WKDW S S ] T 5B 5 5 5Af ] f SW HTXDWLRQ S W 5T f ] f SS HTXDWLRQ SS S TT T f 6LQFH 5A A DQG 5 -] DUH LQGHSHQGHQW RI A (TV f DQG ff YHFWRU T DQG (TV ff DUH HDVLO\ VHHQ WR EH LQGHSHQGHQW RI WKH ILUVW DQG ODVW MRLQW YDULDEOHV DQG WKHUHIRUH IRUP D V\VWHP RI HTXDWLRQV LQ XQNQRZQV )LJXUH LOOXVWUDWHV WKLV GLVFXVVLRQ :LWK YHFWRU W ZKLFK FRLQFLGHV ZLWK ] DQG WKH SRVLWLRQ YHFWRU S RI WKH RULJLQ RI IUDPH ) DUH LQYDULDQW LQ WKH URWDWLRQ 5t URWDWLRQ DERXW ] ZKLFK FDQ RQO\ DIIHFW WKH HQGHIIHFWRU RULHQWDWLRQf 5RWDWLRQ DERXW ]4 KDV QR HIIHFW RQ WKH ] FRPSRQHQW RI DQ\ YHFWRU H[SUHVVHG LQ IUDPH )T +HQFH S] DQG W] DUH LQGHSHQGHQW RI DV ZHOO )LQDOO\ VLQFH URWDWLRQ DERXW =T PRYHV DOO WKH URERWLF VWUXFWXUH DV D EORF LW GRHV QRW DIIHFW WKH OHQJWK RI YHFWRU S RU WKH LQQHU SURGXFW RI W DQG S ‘

PAGE 31

n H )LJXUH 5RWDWLRQV DERXW ]4 RU ] GR QRW DIIHFW W] S] WS DQG SS

PAGE 32

7KH UHGXFHG V\VWHP RI HTXDWLRQV ff GHWHUPLQHV FDQGLGDWH VROXWLRQV IRU MRLQW YDULDEOHV DQG 2QFH WKLV V\VWHP RI HTXDWLRQV LV VROYHG WKH UHPDLQLQJ WZR YDULDEOHV FDQ EH IRXQG E\ XVLQJ PRUH HTXDWLRQV IURP f DQG WKHQ WHVWHG IRU FRQVLVWHQF\ 7KH SRZHU RI WKLV DSSURDFK ZLOO EHFRPH DSSDUHQW IRU VSHFLILF PDQLSXODWRUV DV IXUWKHU VLPSOLILFDWLRQ XVLQJ (TV ff EHFRPHV REYLRXV )XUWKHUPRUH VLPSOLILFDWLRQ E\ XVH RI URWDWLRQ LQQHUSURGXFW LQYDULDQFH LV FRPSXWDWLRQDOO\ HFRQRPLFDO DQG SURYLGHV JUHDWHU LQVLJKW LQWR WKH VWUXFWXUH DQG SURSHUWLHV RI WKH LQYHUVH NLQHPDWLF HTXDWLRQV $GGLWLRQDO ,QYHUVH .LQHPDWLFV (TXDWLRQV (TXDWLRQV ff DUH QHFHVVDU\ EXW QRW VXIILFLHQW $OWKRXJK WKH\ DUH VDWLVILHG E\ DOO VROXWLRQ VHWV RI (T f WKH\ DUH DOVR LQ JHQHUDO VDWLVILHG E\ H[WUDQHRXV VROXWLRQV 7KLV SUREOHP ZDV UHSRUWHG E\ 7VDL DQG 0RUJDQ f DV ZHOO $QRWKHU SUREOHP ZLWK FRQVLGHULQJ (TV ff DORQH LV WKH SUHVHQFH RI VLJQ DPELJXLWLHV ,Q PDQ\ SUDFWLFDO VLWXDWLRQV RQH RI WKH HTXDWLRQV ZLOO DOORZ D FORVHGIRUP VROXWLRQ IRU HLWKHU WKH VLQH RU WKH FRVLQH IXQFWLRQ RI D UHYROXWH YDULDEOH 7KH RWKHU IXQFWLRQ QHHGV WR EH FRPSXWHG XVLQJ WKH 3\WKDJRUHDQ LGHQWLW\ ZKLFK RIIHUV WZR YDOXHV RSSRVLWH LQ VLJQ $OWKRXJK ERWK VLJQV FDQ EH WULHG LQ WKH VHDUFK IRU D VROXWLRQ LQ VRPH FDVHV WKH QXPEHU

PAGE 33

RI VLJQ DPELJXLWLHV FDQ EH UHGXFHG E\ FRQVLGHULQJ PRUH FRQVWUDLQWV IURP (TV f DQG f $GGLWLRQDO HTXDWLRQV ZLOO DOVR KHOS ILOWHU RXW H[WUDQHRXV VROXWLRQV DQG LQ VRPH FDVHV ZLOO HDVH WKH VROXWLRQILQGLQJ SURFHVV UDWKHU WKDQ FRPSOLFDWH LW 7KH [ DQG \FRPSRQHQWV RI YHFWRUV W DQG S SURYLGH FRQYHQLHQW DGGLWLRQDO FRQVWUDLQWV DW WKH FRVW RI LQWURGXFLQJ WKH YDULDEOH (TXDWLRQV W[ 5M 5 5 5M ] [ f W\ 5 5 5 5 ] \ f 3[ A A A AArfA f f rA A Af r ;n f DQG 3\ A A A AAAf A f A A AA r < f ZKHUH ; DQG \ DUH WKH XVXDO FDQRQLFDO XQLW YHFWRUV DUH VWLOO LQGHSHQGHQW RI 6ROYLQJ ,QYHUVH .LQHPDWLF (TXDWLRQV 2QFH WKH UHGXFHG VHW RI HTXDWLRQV ff DQG WKH DGGLWLRQDO HTXDWLRQV ff KDYH EHHQ H[SDQGHG WKH SUREOHP EHFRPHV WKDW RI H[WUDFWLQJ WKH YDOXHV RI WKH MRLQW DQJOHV IURP WKH HTXDWLRQV ZKLFK DUH LQ WHUPV RI WKH VLQHV

PAGE 34

DQG FRVLQHV RI WKH DQJOHV ,Q WKLV VHFWLRQ ZH GHVFULEH VRPH RI WKH WHFKQLTXHV WKDW FDQ EH XVHG IRU WKLV WDVN &HUWDLQ VLPSOH DUP JHRPHWULHV DOORZ D FORVHG IRUP VROXWLRQ )RU VXFK DUPV RQH RI WKH HTXDWLRQV ZLOO KDYH WKH IRUP D 6 E & G ZKHUH 6 DQG & DUH WKH VLQH DQG FRVLQH UHVSHFWLYHO\ RI VRPH DQJOH ,I WKH FRQVWDQWV D E DQG G DUH NQRZQ WKHQ WKHUH DUH WZR SRVVLEOH VROXWLRQV ZKHQ D E G DWDQ>GsDEGf@ DWDQEDf ZKHUH DWDQYZf UHWXUQV WKH DQJOH DUFWDQYZf DGMXVWHG WR WKH SURSHU TXDGUDQW DFFRUGLQJ WR WKH VLJQ RI WKH UHDO QXPEHUV Y DQG Z $ VSHFLDO FDVH RFFXUV ZKHQ D RU E 7KH HTXDWLRQ FDQ WKHQ EH VROYHG IRU 6 RU & VHSDUDWHO\ 7KH RWKHU YDULDEOH FDQ EH REWDLQHG IURP WKH 3\WKDJRUHDQ LGHQWLW\ ‘‘ 6 F f ZLWK D VLJQ DPELJXLW\ $JDLQ WKLV OHDGV WR WZR SRVVLEOH YDOXHV IRU WKH DQJOH DWDQ6 s 6ff LI 6 LV FRPSXWHG RU DWDQs &f &f LI & LV WKH NQRZQ YDULDEOH $ YDOXH RI k FDQ EH GLUHFWO\ DQG XQLTXHO\ REWDLQHG ZKHQ WZR OLQHDU HTXDWLRQV LQ WKH VLQH DQG FRVLQHV RI RQH DQJOH

PAGE 35

DUH REWDLQHG ,Q WKLV FDVH WKH YDOXHV RI 6 DQG & DUH FRPSXWHG DQG WKH DQJOH LV WKHQ JLYHQ E\ DWDQ6 &f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f 7KH VDPH WHFKQLTXHV FDQ EH XVHG IRU D URERW ZKRVH ILUVW WKUHH D[HV LQWHUVHFW E\ UHYHUVLQJ WKH UROHV RI HQF/HIIHFWRU DQG EDVH IUDPH ,Q WKH QH[W FKDSWHUV ZH ZLOO XVH WKLV SUREOHP UHYHUVDO WHFKQLTXH WR DYRLG UHSHWLWLRXV GHYHORSPHQWV

PAGE 36

HQGHIIHFWRU )LJXUH ,QWHUFKDQJLQJ EDVH DQG HQGHIIHFWRU IUDPHV

PAGE 37

&+$37(5 62/9,1* '2) 0$1,38/$7256 5HGXFHG 6\VWHP RI (TXDWLRQV )RU '2) URERW DUPV WKH LQYHUVH NLQHPDWLF SUREOHP LV VROYHG ZKHQ MRLQW YDOXHV DUH IRXQG WKDW VDWLVI\ WKH HTXDWLRQ $O $ $ $ 3 f (TXDWLRQ f GHFRXSOHV LQWR D SRVLWLRQ HTXDWLRQ 5[ 5 5 f f S f DQG DQ RULHQWDWLRQ HTXDWLRQ JLYHQ E\ 5[ 5 5 5 5 f :KHQ WKH IRXUWK MRLQW LV UH\ROXWH LV REWDLQHG E\ SURSHU FKRLFH RI IUDPH ) DQG (T f VLPSOLILHV WR 5S 5 f A S f $ UHGXFHG V\VWHP RI IRXU HTXDWLRQV LQ WKH VLQHV DQG FRVLQHV RI MRLQW DQJOHV A DQG FDQ EH GHULYHG E\ FRQVLGHULQJ WKH TXDQWLWLHV W] S] DQG WKH LQQHU SURGXFWV WS DQG SS H[SUHVVHG LQIUDPH )A

PAGE 38

9HFWRU W LV JLYHQ E\ W 5 ] 5O 5 5 5 ] ZKHUH ] LV WKH WKLUG FDQRQLFDO XQLW YHFWRU ] > @7 6LQFH WZLVW DQJOH Dc LV HTXDO WR 5 ] >D 6 D & W@rA > O@A ] DQG WKH H[SUHVVLRQ IRU W VLPSOLILHV WR W 5[ 5 5 = f 0XOWLSO\LQJ E\ 5A \LHOGV 5A W 5 5 ] DQG WKH LQQHU SURGXFW RI HDFK VLGH RI WKLV HTXDOLW\ ZLWK YHFWRU ] SURYLGHV ] f 5@B A Wf ] 5 5 f (T f DSSOLHG WR ERWK VLGHV RI WKLV ODVW HTXDWLRQ JLYHV 5 ] W 5 ] 5 ] RU 5 ]fW 5B ]f5 ]f f 6LQFH 5B ] > FU W@7 GRHV QRW GHSHQG RQ k WKLV ODVW HTXDWLRQ LV LQGHSHQGHQW RI MRLQW YDULDEOHV DQG 6XEWUDFWLQJ YHFWRU IURP ERWK VLGHV RI (T f DQG PXOWLSO\LQJ E\ 5a \LHOGV

PAGE 39

5 ; 5O 3 Of f DQG WDNLQJ WKH LQQHUSURGXFW ZLWK YHFWRU ] SURYLGHV ] 5 f ] f ] 5O Sf ] 5 A $SSO\LQJ f WR WKH ILUVW WHUP RI ERWK VLGHV RI WKLV HTXDWLRQ JLYHV DIWHU UHDUUDQJLQJ WHUPVf 5[ = S 5A = = 5 O = f 7KH ULJKW KDQG VLGH RI (T f LV FRQVWDQW VLQFH 0XOWLSO\LQJ (T f E\ 5 JLYHV Af 5 A A 5 A 5 A 3 f AOf f DQG PXOWLSOLFDWLRQ RI (T f E\ 5 5Lf 5B 5L S OA@A 5M W@ 5HSHDWHG XVH RI SURSHUWLHV f DQG f DQG UHRUGHULQJ VLPSOLILHV WKLV ODVW HTXDWLRQ WR GL&7L GLUL DQG ]OA GM/ DUH LQGHSHQGHQW RI A

PAGE 40

AW a >5 f 5 =@ WS >5J A f =@ f (TXDWLRQ f LV DOVR LQGHSHQGHQW RI k DQG k 8VLQJ (T f WKH LQQHUSURGXFW SS VDWLVILHV SS f§ >5A 5 ,f A@ f >A A A AA A@ f ([SDQGLQJ WKH OHIW KDQG VLGH XVLQJ LQQHUSURGXFW LQYDULDQFH RI URWDWLRQV ZKHUH QHHGHG DQG UHDUUDQJLQJ WHUPV \LHOG ArA ; A 3rAO f§ W3r3 ArA ArA ArAAAr f (TXDWLRQV f f f DQG f IRUP D OLQHDU V\VWHP LQ WKH YDULDEOHV 6A &6 DQG & 7KH IRXU HTXDWLRQV REWDLQHG DUH DAW\ DAW[ tr A AAA A DOW[ 6 f DOIF\ & DD & &7O3[ 6 DO3\ & DD 6 DO3\ 6[ DO3[ &s DDG 6 DD & fDX OD U U f f f f ZLWK UO WS J GJ G[W] 77G 8 U7 f UOW= n UA UAGMA Wf G" UG Uf SS DGDGDGf GO3] UGG f f f f 7KH OLQHDU V\VWHP RI HTXDWLRQV IRUPHG E\ (TV ff ZLOO EH UHIHUUHG WR nDV WKH UHGXFHG V\VWHP IRU D IRXU'2)

PAGE 41

PDQLSXODWRU $OWKRXJK G FDQ EH DVVXPHG ]HUR ZLWKRXW ORVV RI JHQHUDOLW\ IRU D '2) PDQLSXODWRU WKLV V\VWHP RI HTXDWLRQV ZLOO EH XVHG IRU '2) VHFWLRQV RI ODUJHU PDQLSXODWRUV QH[W FKDSWHUVf IRU ZKLFK WKH SDUDPHWHU FRUUHVSRQGLQJ WR GA ZLOO LQ JHQHUDO QRW EH ]HUR +HQFH GA LV DVVXPHG QRW HTXDO WR ]HUR DW WKLV SRLQW $ XQLTXH VROXWLRQ WR WKH UHGXFHG V\VWHP LV JLYHQ E\ f ‘ 6 UL & f§ U + ; 6 U R ZKHUH + DOW\ DOW[ DO3[ DO3\ DO3\ DO3[ DD DDG FWFW DD DDG DD f f ZKHQ PDWUL[ + LV QRQVLQJXODU 8QLTXH YDOXHV RI A DQG k DUH WKXV REWDLQHG IURP WKH YDOXHV RI 6A &A 6 DQG & 7KH FDVH ZKHUH + LV QRW LQYHUWLEOH LV GLVFXVVHG LQ WKH QH[W VHFWLRQV EHFDXVH RI LWV LQWHUHVWLQJ LPSOLFDWLRQV :LWK AA DQG NQRZQ (T f SURYLGHV DnZD\ WR VROYH IRU kf ,QGHHG ZKHQ H[SDQGHG WKH ILUVW FRPSRQHQWV \LHOG AA 7DAA A DDA a AOA[ AO3\ DO f

PAGE 42

DQG DD&nA A f DA7DAA ra 7O6O3[ UO&O3\ DL3]fGLf f :KHQ WKH GHWHUPLQDQW RI WKLV OLQHDU V\VWHP RI HTXDWLRQV LQ 6 DQG & LV QRW D XQLTXH YDOXH RI FDQ EH FRPSXWHG 2WKHUZLVH ZH FDQ REWDLQ XQLTXHO\ IURP DQRWKHU OLQHDU V\VWHP RI HTXDWLRQV LQ 6 DQG & 7AHr7A A AA A AOA[ AOA\ f DQG FW6 6 ^Wr&-OUDUf & f§7WAABA\ DW] f GHULYHG IURP (T f 1RWH WKDW FDQ DOVR EH FRPSXWHG XVLQJ D V\VWHP RI WZR HTXDWLRQV IRUPHG E\ (T f RU (T f DQG RQH RI (TV f DQG f 7KH $SSHQGL[ VKRZV WKDW WKH GHWHUPLQDQWV RI WKH WZR V\VWHPV RI HTXDWLRQV DERYH DUH VLPXOWDQHRXVO\ ]HUR RQO\ ZKHQ MRLQW D[LV DOLJQV ZLWK DQRWKHU MRLQW D[LV ZKLFK SXWV WKH DUP LQ D GHJHQHUDWH FRQILJXUDWLRQ 7R FRPSOHWH WKH '2) VROXWLRQ VHW ZH XVH (T f ZKLFK FDQ EH UHZULWWHQ DV 5 5Mf 5B 5I 5 7KH ILUVW FROXPQ YHFWRU RI 5 REWDLQHG E\ PXOWLSO\LQJ ERWK VLGHV E\ WKH ILUVW FDQRQLFDO XQLW YHFWRU [ >@7

PAGE 43

f & Q[ 5 ; 6 5 UB 5A 5 ; 5 5 5[ Q\ Q] FDQ EH XVHG WR FRPSXWH WKH ODVW YDULDEOH k 7KLV VKRZV WKDW D '2) LQYHUVH NLQHPDWLF SUREOHP ZLOO LQ JHQHUDO JHQHUDO LQ WKH VHQVH WKDW PDWUL[ + LV QRQVLQJXODUf \LHOG D XQLTXH VROXWLRQ VHW +RZHYHU IRU VRPH PDQLSXODWRU JHRPHWULHV DQGRU VRPH SDUWLFXODU HQGHIIHFWRU SRVHV WKH SUREOHP PD\ KDYH PRUH WKDQ RQH VROXWLRQ 6SHFLDO '2) 0DQLSXODWRU *HRPHWULHV (TXDWLRQ f LV YDOLG RQO\ ZKHQ PDWUL[ + LV LQYHUWLEOH 7KH GHWHUPLQDQW RI PDWUL[ + FRPSXWHG IURP (T f LV JLYHQ E\ G+ DO D DOWD D : D Z! r r D G Z@ R D >D D D Gf R DU@ Z f ZKHUH WKH TXDQWLWLHV Z Z Z DQG Z DUH GHILQHG LQ WHUPV RI WKH FRPSRQHQWV RI SRVH YHFWRUV W DQG S DV : W[ Z S[ S< Z 3[ IF[ Z 3[ IF\ f 3\ IF[r ‘

PAGE 44

$QDO\]LQJ (TXDWLRQ f VKRZV WKDW WKH YDOXH RI G+ GHSHQGV RQ WKH VHYHQ URERW SDUDPHWHUV DD D DA D D DQG G DV ZHOO DV WKH SRVH TXDQWLWLHV Z Z Z DQG Z +RZHYHU IRU FHUWDLQ URERW VWUXFWXUHV G+ LV HTXDO WR ]HUR QR PDWWHU ZKDW WKH HQGHIIHFWRU SRVH LV 7KH H[SUHVVLRQ RI G+ DERYH SURYLGHV XV ZLWK D ZD\ WR ILQG DOO VXFK '2) URERW JHRPHWULHV 'XH WR RXU OLQN IUDPHV DVVLJQPHQW WKH RQO\ URERW SDUDPHWHU LQ WKH H[SUHVVLRQ RI GA WKDW FDQ EH QHJDWLYH LV G %\ H[SDQGLQJ (T f ZH JHW GD D D A r r D A D D DAA D G Z DA D D D Z D[ D D D G Z D D DA D Z f ZKHUH RQO\ WKH TXDQWLWLHV G Z DQG Z FDQ EH QHJDWLYH ,I DQ DUP VWUXFWXUH LV VXFK WKDW G+ LV ]HUR IRU HYHU\ SRVVLEOH HQGHIIHFWRU SRVH WKHQ G+ ZLOO EH ]HUR HYHQ IRU D SRVH ZLWK SRVLWLYH Z DQG Z ,I ZH DVVXPH Z DQG Z QRQ QHJDWLYH WKHQ ZLWK G QHJDWLYH G+ FDQ EH ]HUR LI WKH HTXDOLW\ f§ FUA A DO D G Z f§ r r DO D D ZA D D D DO D : D/ FW D D Z r r D D G Z D D D[ D Z KROGV +RZHYHU VXFK DQ HTXDOLW\ LV DFWXDOO\ D FRQGLWLRQ RQ SRVH TXDQWLWLHV Z Z Z DQG Z :H FRQFOXGH WKDW URERW

PAGE 45

VWUXFWXUHV IRU ZKLFK G+ LV DOZD\V ]HUR LQGHSHQGHQW RI WKH HQGHIIHFWRU SRVHf KDYH '+SDUDPHWHUV RA FU D D D D DQG G IRU ZKLFK HDFK RI WKH VL[ WHUPV LQ (T f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f '2) URERW JHRPHWULHV IRU ZKLFK PDWUL[ + ZLOO EH VLQJXODU ,Q WKH QH[W VHFWLRQ ZH VKRZ WKDW WKH LQYHUVH NLQHPDWLFV SUREOHP IRU VXFK URERWV FDQ VWLOO EH VROYHG E\ XVH RI WKH UHGXFHG V\VWHP RI HTXDWLRQV ff 6SHFLDO '2) $UP 6WUXFWXUHV $ WULYLDO FRQGLWLRQ RFFXUV ZKHQ WZR FRQVHFXWLYH MRLQW D[HV FRLQFLGH VRPHZKHUH DORQJ WKH DUP 6XFK D GHJHQHUDWH FRQGLWLRQ LV GHWHFWHG E\ DA IRU VRPH MRLQW L ,Q WKLV FDVH WKH PDQLSXODWRU ORVHV RQH GHJUHH RI IUHHGRP DQG EHFRPHV D UHGXQGDQW '2) DUP ,I D VROXWLRQ VHW H[LVWV IRU VXFK DQ DUP WKHUH ZLOO EH DQ LQILQLWH QXPEHU RI VROXWLRQ VHWV $ FDUHIXO DQDO\VLV RI 7DEOH VKRZV WKDW WKHUH DUH RQO\ WHQ PLQLPDO QRQWULYLDO FRQGLWLRQV RQ WKH DUP

PAGE 46

7DEOH 6SHFLDO H[SUHVVLRQV IRU G+ &RQGLWLRQ G+ R ,, U+ E U+ r r DO D Z 2 ,, &0 E &0 DO r D D Z D a r A D D AA R + ,, 2 DO r D D D Gf Z D D r D DO DO G Z DO &7 GfZA D 2f D FW DL D Z G 3 2 RL R D D D Z D ZA D D D Ds D Df Z JHRPHWU\ SRVHLQGHSHQGHQWf IRU G+ $OO WHQ FRQGLWLRQV DUH OLVWHG DQG GHVFULEHG LQ 7DEOH DQG LOOXVWUDWHG LQ )LJXUH 7KH ILUVW WKUHH FRQGLWLRQV LQ 7DEOH IROORZ IURP WKH ILUVW HQWU\ RI 7DEOH &RQGLWLRQV DQG DUH GHULYHG IURP WKH VHFRQG HQWU\ LQ 7DEOH DIWHU GURSSLQJ GXSOLFDWH FRQGLWLRQV DOUHDG\ HVWDEOLVKHG &RQWLQXLQJ LQ WKLV IDVKLRQ DOO RI 7DEOH FDQ EH FRPSOHWHG 2EVHUYH WKDW HQWU\ LQ 7DEOH GRHV QRW DGG DQ\ QHZ FRQGLWLRQV LQWR 7DEOH VLQFH DOO PLQLPDO VHWV RI ]HUR SDUDPHWHUV LPSOLHG E\ G+ LQ HQWU\ KDYH DOUHDG\ EHHQ DFFRXQWHG IRUn ,W PXVW EH QRWHG WKDW G+ FDQ VWLOO EH ]HUR IRU '2) DUP JHRPHWULHV QRW OLVWHG LQ WKH SUHFHGLQJ 7DEOH +RZHYHU IURP WKH GLVFXVVLRQ DERYH ZH VHH cWKDW VXFK D VLWXDWLRQ FDQ RQO\ KDSSHQ DW SDUWLFXODU HQGHIIHFWRU SRVHV ZKHUHDV G+ ZLOO

PAGE 47

7DEOH 6SHFLDO VWUXFWXUHV RI '2) PDQLSXODWRUV &RQGLWLRQ 'HVFULSWLRQ DO D )LUVW MRLQW D[HV f DUH SDUDOOHO )LUVW D[HV f DUH SDUDOOHO DQG /DVW D[HV f DUH SDUDOOHO D 2 )LUVW D[HV f DUH SDUDOOHO DQG ODVW D[HV f LQWHUVHFW R /DVW WKUHH D[HV f DUH SDUDOOHO D 2 0LGGOH D[HV f DUH SDUDOOHO DQG ODVW D[HV f LQWHUVHFW D 2 )LUVW D[HV f LQWHUVHFW t ODVW D[HV f DUH SDUDOOHO D r 0LGGOH D[HV f LQWHUVHFW t ODVW D[HV f DUH SDUDOOHO D D )LUVW D[HV f LQWHUVHFW t ODVW D[HV f LQWHUVHFW DO D G )LUVW D[HV f LQWHUVHFW D D 0LGGOH D[HV f LQWHUVHFW DQG ODVW D[HV f LQWHUVHFW EH ]HUR IRU WKH JHRPHWULHV GHVFULEHG LQ 7DEOH DW DQ\ SRVH :H QRZ H[DPLQH LQ GHWDLO WKH LQYHUVH NLQHPDWLFV RI HDFK RI WKH VSHFLDO '2) URERW DUFKLWHFWXUHV GHVFULEHG LQ 7DEOH DQG LOOXVWUDWHG LQ )LJXUH

PAGE 48

)LUVW WKUHH MRLQW D[HV DUH SDUDOOHO $[HV DQG DUH SDUDOOHO DQG DUH SDUDOOHO $[HV DQG DUH SDUDOOHO D[HV DQG LQWHUVHFW $[HV DQG DUH SDUDOOHO $[HV DQG DUH SDUDOOHO $[HV DQG LQWHUVHFW DQG LQWHUVHFW DQG DUH SDUDOOHO )LJXUH 6SHFLDO '2) VWUXFWXUHV

PAGE 49

$[HV DQG LQWHUVHFW DQG DUH SDUDOOHO $[HV DQG LQWHUVHFW DQG LQWHUVHFW $[HV DQG LQWHUVHFW $[HV DQG LQWHUVHFW DQG LQWHUVHFW )LJXUH f§&RQWLQXHG

PAGE 50

7KH UHGXFHG V\VWHP RI HTXDWLRQV ff FDQ VWLOO EH HIILFLHQWO\ XVHG WR ILQG DOO WKH VROXWLRQ VHWV RI (T f ZKHQ PDWUL[ + LV VLQJXODU &DVH D D )LUVW WKUHH MRLQW D[HV DUH SDUDOOHO (QWU\ LQ 7DEOH f7KH UHGXFHG V\VWHP RI HTXDWLRQV EHFRPHV DOW\ 6 DOIF[ FO D&7 6 DO3\ V DO3[ &[ DD & U U U U f f f f (TXDWLRQV f DQG f DUH FRQVWUDLQWV RQ SRVH SDUDPHWHUV W] DQG S] UHVSHFWLYHO\ 2QO\ HQGHIIHFWRU SRVHV WKDW VDWLVI\ 3] GL G G DQG W] U (TV f DQG ff DUH VROYDEOH ZLWK WKLV DUP JHRPHWU\ (TXDWLRQVn f DQG f VWLOO DOORZ D VROXWLRQ LQ WKH VW\OH RI 3LHSHU f E\ ILUVW HOLPLQDWLQJ 6 DQG & IURP WKH HTXDWLRQV 7KLV FDQ EH GRQH E\ VROYLQJ IRU 6 DQG & DQG VXEVWLWXWLQJ LQ WKH 3\WKDJRUHDQ LGHQWLW\ f WR JHW ^>U/ DMW\ 6 D[W[ &f@DD` ^>UO DO3\ 6O DO3[ &f@DD` f :LWK WKH WULJRQRPHWULF LGHQWLWLHV 6 W[O W[f DQG &s WfO W[f

PAGE 51

ZKHUH WA WDQ Af (T f \LHOGV D TXDUWLF SRO\QRPLDO HTXDWLRQ LQ WA :LWK A FRPSXWHG D YDOXH RI LV REWDLQHG DQG FDQ EH FRPSXWHG XQLTXHO\ IURP (TV f DQG f 7KH UHPDLQLQJ DQJOHV k DQG kf FDQ EH FRPSXWHG DV LQGLFDWHG HDUOLHU :H SURSRVH D PHWKRG WKDW DOORZV EHWWHU LQVLJKW ZLWKRXW WKH FRPSOH[LW\ RI D TXDUWLF SRO\QRPLDO HTXDWLRQ )RU VLPSOLFLW\ WKH VLQH DQG FRVLQH RI D VXP RI DQJOHV ZLOO EH UHSUHVHQWHG DFFRUGLQJ WR &AMMAFRV AMf DQG 6AM VLQAMf $V GHVFULEHG LQ FKDSWHU D VHW RI LQYHUVH NLQHPDWLF HTXDWLRQV FDQ EH REWDLQHG E\ H[SUHVVLQJ WKH FRPSRQHQWV RI YHFWRUV W DQG S DQG WKH LQQHU SURGXFWV WS DQG SS LQ WHUPV RI WKH MRLQW YDULDEOHV A REWDLQHG DUH L O 7KH HTXDWLRQV IF[ r 6 f IF\ ar & f S[ D & D & D/ &A f 3\ D 6 D 6 D[ 6s f WS Ds F DD 6 UGGf G f SS DD & DD & DsD &f FW f ZKHUH FW DA DA DA GA GA GA GG GAGM GGf (TXDWLRQV f DQG f \LHOG 6 DQG & GLUHFWO\! VR D XQLTXH YDOXH RI pO REWDLQHG

PAGE 52

:LWK k NQRZQ (TV f DQG f EHFRPH HOERZ HTXDWLRQVf S[ a D & D & DO & nf 3\ f D 6 D 6 DO 6 nf DQG FDQ EH VROYHG IRU & E\ & WAS; f D &A A3\ f D 6A D D[@ D[ Df ZKLFK LV REWDLQHG E\ DSSO\LQJ WKH FRVLQH ODZ WR WKH WULDQJOH KDYLQJ OLQNV DQG DV LWV VLGHV 7ZR YDOXHV RI k IROORZ IURP k DWDQs -^&f &f DQG D XQLTXH YDOXH RI kA FDQ WKHQ EH FRPSXWHG IURP (TV nf DQG nf ZKLFK \LHOG D OLQHDU V\VWHP LQ 6A DQG & ZKHQ 6 DQG & DUH H[SDQGHG XVLQJ VXP RI DQJOHV WULJRQRPHWULF LGHQWLWLHV -RLQW YDULDEOH k LV JLYHQ E\ H H f HO f H DQG WKH VROXWLRQ VHW LV FRPSOHWHG ZKHQ WKH ODVW DQJOH k LV FRPSXWHG DV VKRZQ HDUOLHU 7KLV GHYHORSPHQW SURYHV WKDW WKHUH FDQ EH DW PRVW VROXWLRQ VHWV IRU D '2) DUP ZLWK WKLV SDUWLFXODU JHRPHWU\ &DVH FU FU 7KH ILUVW WZR MRLQW D[HV DUH SDUDOOHO DQG WKH ODVW WZR MRLQW D[HV DUH SDUDOOHO 7KH UHGXFHG V\VWHP LV

PAGE 53

DnOW\ 6 DOW[ & UO f U f DD 6 U f DO3\ 6O DO3[ F[ &DG 6 DD A Ur f (TXDWLRQ f LPSRVHV D FRQVWUDLQW RQ SRVH SDUDPHWHU Wf W] W77 r WKLV FRQVWUDLQW LV VDWLVILHG (T f FDQ EH VROYHG DQG \LHOGV WZR GLVWLQFW YDOXHV IRU A 7KHQ (TV f DQG f IRUP D OLQHDU V\VWHP LQ 6 DQG & ZKLFK FDQ EH VROYHG XQLTXHO\ IRU H :LWK AA DQG H FRPSXWHG DQG k FDQ EH XQLTXHO\ REWDLQHG DV VKRZQ HDUOLHU +HUH DJDLQ ZH ILQG DW PRVW WZR VROXWLRQ VHWV &DVH FUM D )LUVW WZR MRLQW D[HV DUH SDUDOOHO DQG ODVW WZR MRLQW D[HV LQWHUVHFW 7KH UHGXFHG V\VWHP EHFRPHV DMW\ 6O W DAW[ DA 6 f§ AA A AL f DD A U f U f DO3\ 6 DA & U f )URP (T f WKH SRVH FRQVWUDLQW U WUDQVODWHV WR 3] GL G UG r )RU D SRVH PDWUL[ WKDW VDWLVILHV WKLV FRQVWUDLQW WZR SRVVLEOH YDOXHV RI FDQ EH REWDLQHG IURP (T f )RU HDFK RI WKRVH k YDOXHV D XQLTXH YDOXH RI k LV FRPSXWHG IURP WKH OLQHDU V\VWHP LQ 6 DQG &A IRUPHG E\ (TV f

PAGE 54

DQG f 7KH WZR VROXWLRQ VHWV DUH WKHQ FRPSOHWHG DV VKRZQ SUHYLRXVO\ &DVH &7 &7 r 7KH ODVW WKUHH MRLQW D[HV DUH SDUDOOHO 7KH UHGXFHG V\VWHP VLPSOLILHV WR DOW\ 6 DOW[ FL UL f &7OW[ 6 DOW\ FL U rf &7O3[ 6 DOS\ & U f DO3\ V[ DO3[ &D DD & U f 7ZR RXW RI WKH ILUVW WKUHH HTXDWLRQV (TV ff f FDQ EH XVHG WR VROYH XQLTXHO\ IRU A 7KH WKLUG XQXVHG HTXDWLRQf EHFRPHV D UHDOL]DELOLW\ FRQVWUDLQW RQ WKH SRVH :LWK A NQRZQ (T f \LHOGV D YDOXH IRU & ZKLFK LQ WXUQ JLYHV WZR SRVVLEOH YDOXHV IRU k 7ZR VROXWLRQ VHWV FDQ EH REWDLQHG DIWHU FRPSXWLQJ DQG k &DVH D D 7KH LQWHUPHGLDWH MRLQW D[HV DUH SDUDOOHO DQG WKH ODVW WZR D[HV LQWHUVHFW 7KH UHGXFHG V\WHP EHFRPHV DMAW\ 6O DLW[ DFU 6 f§ UA f DOW; 6 DOW\ & U f &7OS[ 6 &7OS\ & U f DOS\ 6 DOS[ & U f 7ZR RXW RI WKH ODVW WKUHH HTXDWLRQV f f FDQ EH VROYHG XQLTXHO\ IRU A 7KH WKLUG HTXDWLRQ EHFRPHV D UHDOL]DELOLW\ FRQVWUDLQW RQ SRVH HOHPHQW W] RU S] GHSHQGLQJ

PAGE 55

RQ WKH FKRVHQ HTXDWLRQ :LWK MA NQRZQ WZR YDOXHV RI FDQ EH FRPSXWHG IURP WKH YDOXH RI 6 GHULYHG IURP (T f &DVH D D 7KH ODVW WZR MRLQW D[HV DUH SDUDOOHO DQG WKH ILUVW WZR D[HV LQWHUVHFW 7KH UHGXFHG V\VWHP LV UO f DOW[ 6 a DOW\ & U f DO3[ 6 f DO3\ & f DD 6 U f &7DA A DD A U f (TXDWLRQ f LV D UHDOL]DELOLW\ FRQVWUDLQW RQ SDUDPHWHU W] (T ff )RU DQ HQGHIIHFWRU SRVH WKDW VDWLVILHV WKLV FRQVWUDLQW (T f \LHOGV WZR YDOXHV IRU A (TXDWLRQV f DQG f FDQ WKHQ EH VROYHG IRU k XQLTXHO\ &DVH &7 D r 7KH ODVW WZR MRLQW D[HV DUH SDUDOOHO DQG WKH LQWHUPHGLDWH WZR D[HV LQWHUVHFW 7KH UHGXFHG V\VWHP LV DOA\ s A UO f &7OW[ 6 f &7OW\ & U f DO3[ 6 f &7O3\ & DD 6 U rf DO3\ A DAS[ &[ 2DG 6 U f +HUH (TV f DQG f \LHOG D XQLTXH YDOX IRU WKHQ 2QH RI (TV f RU f FDQ EH XVHG WR VROYH IRU

PAGE 56

6 WKHUHE\ SURYLGLQJ WZR YDOXHV IRU k WKH UHPDLQLQJ HTXDWLRQ LV D SRVH FRQVWUDLQW &DVH D D G 7KH ILUVW WKUHH MRLQW D[HV LQWHUVHFW DQG WKH UHGXFHG V\VWHP EHFRPHV ,, r‘ + f DOW[ 6 f IILW\ &/ UD & a U f &7O3; 6 f &7O3\ & f DD 6 U f U f (TXDWLRQV f DQG f LPSRVH FRQVWUDLQWV RQ SDUDPHWHUV W] DQG S] +HUH DJDLQ D VROXWLRQ FDQ EH REWDLQHG LQ IRUP RI D TXDUWLF SRO\QRPLDO HTXDWLRQ LQ WAWDQAf E\ HOLPLQDWLQJ k IURP (TV f DQG f DV ZH GLG HDUOLHU LQ FDVH :LWK kA NQRZQ k FDQ EH XQLTXHO\ REWDLQHG IURP (TV f DQG f DQG WKH VROXWLRQ VHW FRPSOHWHG DV XVXDO 7KLV PHWKRG SXWV DQ XSSHU ERXQG RI RQ WKH QXPEHU RI VROXWLRQ VHWV VLQFH DW PRVW IRXU GLVWLQFW YDOXHV RI 4A FDQ EH REWDLQHG IURP WKH TXDUWLF SRO\QRPLDO HTXDWLRQ LQ WA $Q HDVLHU LQYHUVH NLQHPDWLF DQDO\VLV RI WKLV VWUXFWXUH FDQ EH REWDLQHG LI WKH UROHV RI HQGHIIHFWRU IUDPH DQG EDVH IUDPH DUH UHYHUVHG DQG WKH LQWHUPHGLDWH OLQNIUDPHV DUH UHDVVLJQHG DFFRUGLQJO\ 7KLV ZLOO SXW WKH WKUHH LQWHUVHFWLQJ D[HV DW WKH HQGHIIHFWRU SRVLWLRQ LQVWHDG RI DW WKH EDVH 7KH '2) VWUXFWXUH LV VHHQ WR EH HTXLYDOHQW WR RQH WKDW KDV D D ZKLFK LV GLVFXVVHG LQ FDVH

PAGE 57

:LWK WKLV DQDO\VLV ZH ILQG WKDW WKHUH FDQ EH DW PRVW WZR VROXWLRQ VHWV &DVH 7KH ILUVW WZR MRLQW D[HV LQWHUVHFW DQG WKH ODVW WZR MRLQW D[HV LQWHUVHFW 7KH UHGXFHG V\VWHP LV DD V f D&7G & UO f 7OW; 6 f DOW\ & DD & U f DO3[ 6 f DOS\ & U f U f +HUH U SRVHV D FRQVWUDLQW RQ SRVH SDUDPHWHU (TXDWLRQ f \LHOGV WZR GLVWLQFW YDOXHV RI A WKHQ (TV f DQG f ZLOO IRUP D OLQHDU V\VWHP LQ 6 DQG & ZKLFK FDQ EH VROYHG IRU D XQLTXH YDOXH RI k IRU HDFK YDOXH RI k} 7ZR VROXWLRQ VHWV DUH WKXV REWDLQHG &DVH D D 7KH LQWHUPHGLDWH WZR MRLQW D[HV LQWHUVHFW DQG WKH ODVW WZR MRLQW D[HV LQWHUVHFW 7KH UHGXFHG V\VWHP EHFRPHV DOW\ 6 DOIF[ & f &7DG & UO f FUOW[ 6 f DOW\ & DD & U f &7O3[ 6 DOS\ & U f DO3\ 6 DOS[ & Ur f (TXDWLRQV f DQG f \LHOG D XQLTXH VROXWLRQ IRU k 7KH YDOXH RI k REWDLQHGFDQ EH VXEVWLWXWHG LQ (T f RU f WR VROYH IRU & ZKLFK SURYLGHV WZR SRVVLEOH YDOXHV

PAGE 58

IRU k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

PAGE 59

&+$37(5 62/9,1* ),9('2) 0$1,38/$7256 2QH'LPHQVLRQDO ,WHUDWLYH 7HFKQLTXH :LWK ILYH GHJUHHV RI IUHHGRP (T f WDNHV WKH IRUP $I $ $A $A $A 3 f DQG DIWHU PXOWLSO\LQJ ERWK VLGHV RI WKLV HTXDWLRQ E\ $A ZH REWDLQ A A A A 4 f ZLWK 4 $ 3 f :KHQ k LV NQRZQ PDWUL[ 4 LV IXOO\ GHWHUPLQHG DQG FDQ EH YLHZHG DV D SRVH PDWUL[ IRU D '2) DUP ZKRVH VWUXFWXUH LV GHVFULEHG E\ WKH OHIW KDQG VLGH RI (T f ZKLFK PHUHO\ H[SUHVVHV D '2) SUREOHP ,Q &KDSWHU ZH KDYH VHHQ WKDW D '2) SUREOHP FDQ DOZD\V EH VROYHG LQ FORVHGIRUP KHQFH WKH UHPDLQLQJ MRLQW YDULDEOHV FDQ EH FRPSXWHG DV VKRZQ HDUOLHU 6LQFH ZH RQO\ QHHG WR NQRZ RQH RI WKH MRLQW YDULDEOHV WR VROYH IRU WKH ZKROH VROXWLRQ VHW WKH LQYHUVH NLQHPDWLFV SUREOHP RI ILYH'2) PDQLSXODWRUV UHGXFHV WR ILQGLQJ WKH

PAGE 60

YDOXH RI WKH ILUVW MRLQW YDULDEOH RQO\ DQG JHWWLQJ FORVHG IRUP YDOXHV IRU WKH UHPDLQLQJ YDULDEOHV /HW WKH FROXPQ YHFWRUV RI SRVH PDWUL[ 4 RI (T f EH 4 P F X DQG P[ F[ X[ A[ P\ F\ X\ T\ P] F] X] = L P F X T f WKHQ DQG X 5a 5 ] 5 W T 5O S 5A [ 5[ AS f f f )URP WKH OHIW KDQG VLGH RI (T f WZR YHFWRUV X/ DQG T/ FRUUHVSRQGLQJ WR YHFWRUV X DQG T DUH JLYHQ E\ 8/ 5 5 5 ] f DQG T/ B 5A5 A AA f $ QRQOLQHDU IXQFWLRQ RI A FDQ EH GHILQHG DV D GLIIHUHQFH EHWZHHQ FRUUHVSRQGLQJ TXDQWLWLHV IURP WKH OHIW DQG WKH ULJKW VLGH RI (T f )RU H[DPSOH WKH GLIIHUHQFH EHWZHHQ WKH LQQHUSURGXFWV X/T/f DQG XTf \LHOGV WKH IXQFWLRQ I kOf X/T/ XT f

PAGE 61

,I WKH YDOXH RI A XVHG WR FRPSXWH SRVH PDWUL[ 4 LQ (T f GRHV FRUUHVSRQG WR D VROXWLRQ VHW WKHQ (T f ZLOO KROG YHFWRUV X/ DQG T/ ZLOO EH H[DFWO\ HTXDO WR X DQG T UHVSHFWLYHO\ DQG IXQFWLRQ I ZLOO HTXDO ]HUR ,Q RWKHU ZRUGV VROXWLRQ VHWV RI (T f FRUUHVSRQG WR ]HURV RI IXQFWLRQ I GHILQHG LQ (T f +HQFH WKH LQYHUVH NLQHPDWLFV SUREOHP RI '2) URERW PDQLSXODWRUV UHGXFHV WR VROYLQJ WKH RQHGLPHQVLRQDO HTXDWLRQ IkLf 7KH ]HURV RI I FDQ EH IRXQG E\ XVH RI DQ\ VXLWDEOH RQHn GLPHQVLRQDO WHFKQLTXH VXFK DV 1HZWRQ5DSKVRQ RU WKH VHFDQW PHWKRG 2QFH A LV NQRZQ WKH VROXWLRQ VHW FDQ EH FRPSOHWHG E\ VROYLQJ (T f LQ FORVHG IRUP DV ZH VKRZHG LQ &KDSWHU 7KH VROXWLRQ VHW FDQ WKHQ EH FKHFNHG IRU FRQVLVWHQF\ ZLWK (T f WR GHWHUPLQH ZKHWKHU WKH RQH IRXQG LV H[WUDQHRXV RU QRW EHFDXVH WKH ]HURV RI I DUH QRW DOZD\V SDUW RI D VROXWLRQ VHW RI WKH PDQLSXODWRU &RPSXWLQJ ILkA 8VLQJ (TV f DQG f WKH LQQHUn SURGXFW X/T/ LV JLYHQ E\ 8O f O A A A r AA A AA AA n ,I ZH DSSO\ SURSHUWLHV f DQG f UHSHDWHGO\ WKLVODVW HTXDWLRQ EHFRPHV f 8MATA ] 5 fAf ]r A A AAf ]r A A A Af

PAGE 62

RU DIWHU FRPSXWLQJ WKH ]FRPSRQHQWV RI WKH WHUPV LQ SDUHQWKHVHV 8/rA/ 7G G DA A f§ -GF rf 7G W 6 A AG AA D F fDU 6 DG7 & UGDA W DD A f DGD A 7G7A r f 7KLV ODVW HTXDWLRQ VKRZV WKDW k DQG k PXVW EH NQRZQ EHIRUH ZH FDQ FRPSXWH X/T/ :LWK kA NQRZQ k DQG k FDQ EH REWDLQHG E\ VROYLQJ (T f DV GHVFULEHG LQ &KDSWHU 7KH FRRUGLQDWHV RI YHFWRUV X DQG T DQG WKH LQQHUSURGXFWV XT DQG TT DUH QHFHVVDU\ IRU WKH '2) LQYHUVH NLQHPDWLF PHWKRG RI &KDSWHU (TXDWLRQ f \LHOGV 8 5A W f DOW[ 6 DOW\ & 7OW] DQG IURP (T f S[ &[ 3\ 6[ D[ T UO3[ 6 UO3\ &s DA] rO3[ 6 DO3\ & 7O3] f ZKHUH ZH KDYH DVVXPHG 5A OMBf >DMB @7 VLQFH G E\ SURSHU SRVLWLRQLQJ RI IUDPH )

PAGE 63

7KH LQQHUSURGXFWV XT DQG TT FDQ WKHQ EH HDVLO\ FRPSXWHG E\ XrT X[T[ X\T\ X]T] DQG TT T[ T\ T ZKHQ WKH QXPHULF YDOXHV RI WKH FRPSRQHQWV RI X DQG T KDYH EHHQ REWDLQHG 7KHVH LQQHU SURGXFWV FDQ EH REWDLQHG IURP (TV f DQG f DV ZHOO XT 5O Wf 5S O[ff W S O[f XT W[ S[ DA/ &[f W\ S\ D/ 6[f f DQG TT 5ALS O[f 5AWH O[f 3 /f S O[f TT SS OLO/ SO[f TT SS DA S[D[ &s S\D[ 6sf f (TXDWLRQV ff FOHDUO\ VKRZ WKDW DOO FRPSRQHQWV RI X DQG T DQG WKH LQQHUSURGXFWV XT DQG TT DUH OLQHDU IXQFWLRQV RI DQG &A D UHVXOW WKDW ZLOO SURYH XVHIXO LQ WKH QH;W VHFWLRQ 7R VXPPDUL=H Ikf@Bf FDQ EH FRPSXWHG IRU D JLYHQ YDOXH RI A DFFRUGLQJ WR WKH IROORZLQJ VWHSV 6WHS )URP WKH FXUUHQW HVWLPDWH RI A &RPSXWH WKH FRPSRQHQWV RI X DQG T DQG WKH LQQHU SURGXFWV XT DQG TT DV VKRZQ LQ (TV ff

PAGE 64

6WHS &RPSXWH k DQG k IURP WKH UHGXFHG V\VWHP RI HTXDWLRQV DX\ 6 DX[ & DD 6 f &7DG & a UO f DX[ 6 DX\ & DD & U f FWF[ 6 DT\ & DD 6 U f DT\ 6 DT[ & DDG 6 DD & U f ZLWK UO TX UG GX] WWG f U WW f WX] f U UG A]f G 7G f U TTDGDGDGf £A] UGGn f GHULYHG IURP (TV ff E\ SURSHU LQGH[ VXEVWLWXWLRQ WKH LQGH[HV DUH LQFUHPHQWHG WR ILW WKH '2) SUREOHP RI (T ff 9HFWRUV X DQG T SOD\ WKH UROHV RI YHFWRUV W DQG S UHVSHFWLYHO\ 7KH ODVW V\VWHP RI HTXDWLRQV JLYHV WKH YDOXHV RI k DQG k (TXDWLRQV f DQG f ZLWK WKH SURSHU LQGH[ FKDQJHV AGf§ ADA f A DD&f & n &[ 6\ D DQG f DD&f 6 f§ &7GaWD6f & W6F[ 7&F\ n f

PAGE 65

FDQ EH VROYHG IRU k $QRWKHU ZD\ WR REWDLQ LV E\ XVLQJ WKH HTXDWLRQV 6 fL X 6 & &X[ W 6 8\ f DQG DA 6 7 FFU Uf A W6X; 7&8\ D8=n f GHULYHG IURP (TV f DQG f E\ LQFUHPHQWLQJ WKH LQGH[HV :LWK k k DQG NQRZQ X/T/ FDQ EH FRPSXWHG DV LQ (T f DQG ILkAf LV WKHQ JLYHQ E\ (T f 7KH DELOLW\ WR FRPSXWH Akf ZKHQ k LV JLYHQ LV VXIILFLHQW WR LPSOHPHQW D SUDFWLFDO 1HZWRQ5DSKVRQ DOJRULWKP IRU ILQGLQJ WKH ]HURV RI IXQFWLRQ I 7KH DOJRULWKP FDQ EH SURJUDPPHG DFFRUGLQJ WR WKH IROORZLQJ VWHSV 6WHS 2EWDLQ DQ LQLWLDO HVWLPDWH IRU k $V IRU DOO LWHUDWLYH PHWKRGV WKH FORVHU WKH LQLWLDO HVWLPDWH RI k LV WR D WUXH VROXWLRQ WKH IDVWHU WKH FRQYHUJHQFH ZLOO EH ,I WKH HQGHIIHFWRU RI WKH URERW LV WUDFNLQJ D WUDMHFWRU\ JLYHQ DV D ILQLWH VHW RI HQGHIIHFWRU SRVHV D JRRG HVWLPDWH IRU ILQGLQJ WKH VROXWLRQ VHW IRU D SRVH DORQJ WKH WUDMHFWRU\ LV WKH YDOXH RI k FRUUHVSRQGLQJ WR WKH SUHFHGLQJ SRVH RQ WKH WUDMHFWRU\ 6WHS &RPSXWH k DQG k DQG WKHQ Ikf DV GHVFULEHG HDUOLHU

PAGE 66

6WHS &RPSXWH WKH GHULYDWLYH GIGkA RI I ZLWK UHVSHFW WR A $ QXPHULF DSSUR[LPDWLRQ RI WKLV GHULYDWLYH LV JLYHQ E\ GIGHA >I fI 4f @64OI f ZKHUH MA LV D VPDOO LQFUHPHQW RI A 1RWH WKDW WKLV DSSUR[LPDWLRQ UHTXLUHV DQRWKHU IXQFWLRQ HYDOXDWLRQ DW AAf 6WHS 2EWDLQ D QHZ HVWLPDWH IRU A E\ WKH RQHn GLPHQVLRQDO 1HZWRQ5DSKVRQ PHWKRG LH *QHZf kL IHfGIG*f f 6WHSV WR PXVW EH UHSHDWHG XQWLO LV REWDLQHG WR WKH GHVLUHG DFFXUDF\ 7KH VROXWLRQ VHW FDQ WKHQ EH FRPSOHWHG E\ XVLQJ WKH YDOXHV RI DQG DV FRPSXWHG DW WKH ODVW LWHUDWLRQ DQG E\ FRPSXWLQJ XQLTXHO\ IURP 5B 5B 5 5 P DQG k DWDQ6&f 7KH RQHGLPHQVLRQDO PHWKRG MXVW GHVFULEHG LV IOH[LEOH LQ WHUPV RI WKH FKRLFH RI IXQFWLRQ I WR EH XVHG $ GLIIHUHQW IXQFWLRQ FDQ EH LPSOHPHQWHG 7KH RQO\ UHTXLUHPHQWV DUH WKDW IWkA EH FRPSXWDEOH IRU DQ\ YDOXH RI

PAGE 67

DQG VRPH NQRZQ UHODWLRQVKLS EHWZHHQ WKH ]HURV RI I DQG WKH VROXWLRQ VHWV RI (T f )RU H[DPSOH DQRWKHU FKRLFH RI I PD\ EH IkLf T/rT/ 6nrr f RU DQ\ GLIIHUHQFH EHWZHHQ FRUUHVSRQGLQJ TXDQWLWLHV IURP WKH OHIW DQG ULJKW VLGH RI (T f 7KH IXQFWLRQ FKRLFH LV LPSRUWDQW LQ WHUPV RI PLQLPDO FRPSXWDWLRQ FRPSOH[LW\ DQG ILOWHULQJ RI H[WUDQHRXV VROXWLRQV ZKLFK DUH GLVFXVVHG QH[W ,Q DOO SUDFWLFDO H[SHULPHQWV WKH IXQFWLRQ GHILQHG LQ (T f KDV JLYHQ JRRG UHVXOWV ([WUDQHRXV VROXWLRQV $Q H[WUDQHRXV VROXWLRQ VHW LV RQH WKDW WKH LWHUDWLYH PHWKRG FRQYHUJHV WR LH LW VDWLVILHV WKH UHGXFHG V\VWHP RI HTXDWLRQV ff DQG IH[f EXW \HW LW LV QRW D VROXWLRQ IRU (T f 7KH LWHUDWLYH PHWKRG MXVW GHVFULEHG PD\ FRQYHUJH WR VXFK Dn VHW 7KLV SUREOHP ZDV DOVR UHSRUWHG E\ 7VDL DQG 0RUJDQ f ZKR GHYHORSHG D GLIIHUHQW LQYHUVH NLQHPDWLF PHWKRG WKDW PDNHV XVH RI D VLPLODU UHGXFHG V\VWHP RI HTXDWLRQV ,Q GHULYLQJ WKH UHGXFHG V\VWHP RI HTXDWLRQV f f LQ FKDSWHU YHFWRUV X DQG TDQG WKH LQQHUSURGXFWV XT DQG TT DUH WKH RQO\ SRVH UHODWHG TXDQWLWLHV WKDW ZHUH LQYROYHG 7KLV PHDQV WKDW D VROXWLRQ VHW REWDLQHG E\ FRQYHUJHQFH RI WKH PHWKRG MXVW GHVFULEHG GRHV QRW QHFHVVDULO\ VDWLVI\ RWKHU SRVH UHTXLUHPHQWV IURP (T ([WUDQHRXV VROXWLRQV FDQ EH ILOWHUHG RXW E\ D FKRLFH RI

PAGE 68

IXQFWLRQ I WKDW FRQVWUDLQV PRUH RI WKH HQGHIIHFWRU SRVH HOHPHQWV DW WKH H[SHQVH RI FRPSXWDWLRQ WLPH RU E\ FKHFNLQJ DOO VROXWLRQV IRXQG IRU FRQVLVWHQF\ ZLWK RQH RU PRUH HQG HIIHFWRU SRVH HOHPHQWV ,WHUDWLQJ RQ k $Q HTXLYDOHQW RQHGLPHQVLRQDO LWHUDWLYH WHFKQLTXH FDQ EH LPSOHPHQWHG EDVHG RQ D IXQFWLRQ RI H LQVWHDG RI A 5HFDOO IURP &KDSWHU WKDW WKH KRPRJHQHRXV PDWUL[ $ GHFRPSRVHV LQWR A f§ A p ZKHUH % LV D KRPRJHQHRXV PDWUL[ IXOO\ GHWHUPLQHG E\ SDUDPHWHUV D G DQG D DQG LQGHSHQGHQW RI k 5LJKW PXOWLSOLFDWLRQ RI (T f E\ $%f \LHOGV A A A f§ a 4 f ZLWK 4 3 $% f :KHQ k LV JLYHQ PDWUL[ 4 EHFRPHV D NQRZQ SRVH PDWUL[ IRU WKH '2) SUREOHP H[SUHVVHG E\ HTXDWLRQ f 9HFWRUV Xn DQG T DUH JLYHQ E\ X 5 5JkA ] f DQG T 5 5B*Nf S nf ZKHUH LV WKH URWDWLRQ SDUW RI KRPRJHQHRXV PDWUL[%

PAGE 69

% 2 2 U 2 D f U U 2 2 2 G A f§ 2 G LV WKH SRVLWLRQ YHFWRU RI % DQG 5 DQG S DUH WKH XVXDO URWDWLRQ PDWUL[ DQG SRVLWLRQ YHFWRU RI HQGHIIHFWRU SRVH 3 ([SOLFLWO\ ZH JHW DQG T Q[D 6 E[D LQ 2 E;7 X Q\D 6 E\D & UW ‘V r Q]D 6 ED & IW 1 Q[&7G E;D! 6 Q[D E[DGf & W[UG S[ Q\DG E\Df 6 f Q\D E\DGf & W\UG S\ Q]DG E]Df 6 f§ Q]D E]DGf f§ W]7G S] f f ([SUHVVLRQV RI LQQHUSURGXFWV XT DQG TT LQ WHUPV RI 6 DQG & FDQ EH REWDLQHG IURP (TV f DQG f nXT n>5n5* ]@ >5 5*Nf S@ DQG

PAGE 70

TT >5 5*aNf S@ >5 5B*BNf S@ :LWK WKH XVH RI SURSHUWLHV f DQG f DV QHFHVVDU\ DQG UHDUUDQJLQJ WHUPV WKH HTXDWLRQV \LHOG XT D >QSf 6 ESf F@ UWSf G f DQG TT >DG QSf D ESf@ 6 >D QSf DG ESf@ F UG WSf D G SS f ZKHUH ZH XVHG WKH IDFW WKDW QS Q[3[ Q\3\ Q]3] ES E[3[ E\3\ E]3] WS IF[3[ W\3\ E]3] +HUH DJDLQ ZH QRWH WKDW X] S] XT DQG TT DUH OLQHDU IXQFWLRQV RI 6 DQG & :LWK WKH FRPSRQHQWV RI X DQG T DQG WKH LQQHU SURGXFWV XT DQG TT FRPSXWHG D RQHGLPHQVLRQDO LWHUDWLYH PHWKRG FDQ EH LPSOHPHQWHG DV GHVFULEHG HDUOLHU ZLWK D IXQFWLRQ Ikf JLYHQ E\ Ikf X/T/ a XT f ZKLFK ZLOO FRQYHUJH WR D YDOXH RI k

PAGE 71

'2) 5RERWV ZLWK &ORVHG)RUP 6ROXWLRQ &HUWDLQ )LYHGHJUHHRIIUHHGRP URERWV ZLWK VLPSOH JHRPHWULHV PD\ \LHOG LQYHUVH NLQHPDWLF HTXDWLRQV WKDW FDQ EH VROYHG GLUHFWO\ DQG ZLWKRXW QHHG IRU D QXPHULF WHFKQLTXH VXFK DV 1HZWRQ5DSKVRQ ,Q &KDSWHU VRPH SDUWLFXODU '2) URERW VWUXFWXUHV ZHUH IRXQG IRU ZKLFK WKH UHGXFHG V\VWHP RI HTXDWLRQV ff ZDV RYHUVSHFLILHG LH WKH PDWUL[ + RI WKH OLQHDU V\VWHP ZDV VLQJXODU 7KH DQDO\VLV RI WKHVH VSHFLDO JHRPHWULHV VKRZHG WKDW RQH RU WZR RI WKH IRXU HTXDWLRQV RI WKH UHGXFHG V\VWHP EHFDPH FRQVWUDLQW HTXDWLRQV RQ SRVH HOHPHQWV SDUWLFXODUO\ HOHPHQWV WB Sf WS DQG SS ,Q WKH FDVH RI '2) URERWV WKH TXDQWLWLHV X] T XT DQG TT X SOD\LQJ WKH UROH RI W DQG T WKDW RI Sf DUH HLWKHU OLQHDU IXQFWLRQV RI DQG &A DV VKRZQ LQ (TV f f RU OLQHDU IXQFWLRQV RI DQG &A DV VKRZQ LQ (TV ff (LWKHU ZD\ WKH FRQVWUDLQW HTXDWLRQV GHVFULEHG LQ WKH WHQ FDVHV RI FKDSWHU FDQ EH XVHG WR VROYH IRU WKH FRUUHFW YDOXH RI A RU k GLUHFWO\ ZLWKRXW QHHG IRU DQ LWHUDWLYH WHFKQLTXH 7KLV UHVXOW PHDQV WKDW LI D '2) URERW PDQLSXODWRU KDV D '2) VHFWLRQ ZLWK RQH RI WKH VSHFLDO JHRPHWULHV GLVFXVVHG LQ &KDSWHU WKHQ WKH DUP FDQ EH VROYHG LQ FORVHG IRUP :H QRZ SURFHHG WR SURYH WKLV SRLQW 7KH '2) LQYHUVH NLQHPDWLFV SUREOHP RI (T f FDQ EH UHGXFHG WR WKH '2) RQH RI (T f ,Q WKLV FDVH WKH

PAGE 72

UHGXFHG V\VWHP RI HTXDWLRQV ff PXVW EH VROYHG %\ VXEVWLWXWLQJ WKH H[SUHVVLRQV RI X] T] XT DQG TT IURP (TV ff LQWR (TV ff DQG UHDUUDQJLQJ ZH JHW UO AOW\ DOGr[f 6 GDOW\f & IF[3[ W\3\ f UOGW= UGU f UGn fr U &7O7W; 6 DOUW\ & 7O7W] 7Un f U DUS[ RA3\ WW3] UG G 7Gn rf DQG U fDL3\ DLG3[! 6 fDO3[ G&7O3\f & 7LG3 7GG SS DDGfDGDGf f 7KHVH ODVW IRXU HTXDWLRQV SURYH WKDW WKH WHUPV UA U U DQG U DUH DOO RI WKH IRUP UL ULO 6 UL & ULn L n f f r n n ZKHUH WKH FRQVWDQWV UAM DUH IXOO\ GHWHUPLQHG E\ WKH DUP SDUDPHWHUV DQG WKH HQGHIIHFWRU SRVH HOHPHQWV $QRWKHU FKRLFH LV WR XVH (T f 7KH UHGXFHG V\VWHP RI HTXDWLRQV LV JLYHQ E\ (TV ff ZLWK DOO HOHPHQWV RI SRVH PDWUL[ 3 UHSODFHG E\ FRUUHVSRQGLQJ HOHPHQWV RI PDWUL[ 4 RI (T f 7KH UA TXDQWLWLHV EHFRPH OLQHDU H[SUHVVLRQV LQ 6 DQG & DQG KDYH WKH IRUP

PAGE 73

UL ULO V UL & ULn Ln f f f r ,QGHHG LI ZH UHSODFH X] T] XT DQG TT E\ WKHLU H[SUHVVLRQV LQ WHUPV RI 6 DQG & DV JLYHQ E\ (TV f f DQG VXEVWLWXWH LQ (TV ff VXEVWLWXWH IRU W] S] WS DQG SS DQG OHW G[ f ZH REWDLQ UO DQSf V 3f 7WSf UG f UG W f f U a7 ,HUQ] 6 U ODE] & 7 UE] WW n f U 7O&7G Q] UO DE]f 6 7 ODQ] 7 ]f & UO7 GIF] UO3=f G 7G n f DQG U >DG QSf D E A 3f 6 >D Q f3f DG E 3f F f UG WSf UG G SS D DO D G D G D G f f f ,Q WKH DQDO\VLV RI VSHFLDO IRXU '2) JHRPHWULHV &KDSWHU ZH IRXQG FDVHV ZKHUH WKH UHGXFHG V\VWHP RI HTXDWLRQV LQFOXGHG D FRQVWUDLQW RI WKH IRUP UA 6XFK D FRQVWUDLQW DSSOLHG WR RQH RI (TV ff ZLOO XVXDOO\ \LHOG D YDOXH RI A RU k ZKLFK LQ WXUQ PDNHV WKH '2) LQYHUVH NLQHPDWLFV SUREOHP VROYDEOH LQ FORVHG IRUP &DVH O 7KUHH MRLQW D[HV DUH SDUDOOHO :KHQ WKH SDUDOOHO D[HV DUH WKH ILUVW WKUHH LH D[HV DQG f (T f FDQ EH XVHG &DVH RI &KDSWHU VKRZV WKDW

PAGE 74

U DQG U 7KHVH WZR FRQVWUDLQWV DQG (TV f DQG f \LHOG D V\VWHP RI HTXDWLRQV LQ 6 DQG & U 6 U & fU U 6 U & BUn ZKLFK FDQ EH VROYHG IRU k ZKHQ WKH GHWHUPLQDQW JLYHQ E\ ULU ULU A LV QRW ]HUR RWKHUZLVH WKHUH LV QR VROXWLRQ :LWK k NQRZQ WKH UHPDLQLQJ DQJOHV FDQ EH REWDLQHG LQ FORVHGIRUP ,I WKH ODVW WKUHH D[HV DUH SDUDOOHO D VLPLODU UHVXOW LV REWDLQHG E\ H[FKDQJLQJ WKH UROHV RI EDVH DQG HQGHIIHFWRU IUDPHV :KHQ WKH LQWHUPHGLDWH D[HV DUH SDUDOOHO (T f VKRXOG EH XVHG 7KH FRQVWUDLQWV U U WKHQ \LHOG D YDOXH RI A DQG WKH LQYHUVH NLQHPDWLF SUREOHP FDQ EH VROYHG LQ FORVHG IRUP DV ZHOO &DVH WZR FRQVHFXWLYH VHWV RI WZR SDUDOOHO D[HV ,I D[HV DQG DUH SDUDOOHO DQG D[HV DQG DUH SDUDOOHO (T f DQG &KDSWHU FDVH \LHOG U ZKLFK FDQ EH XVHG WR VROYH IRU k IURP (T f ,I D[HV DQG DUH SDUDOOHO DQG D[HV DQG DUH SDUDOOHO WKHQ XVLQJ (T f DQG (T f ZLOO \LHOG D YDOXH RI A &DVH 7ZR SDUDOOHO D[HV IROORZHG E\ WZR LQWHUVHFWLQJ D[HV :KHQ WKLV VSHFLDO JHRPHWU\ FRQFHUQV WKH ILUVW IRXU MRLQW D[HV RI WKH '2) DUP XVLQJ (T f DQG &KDSWHU FDVH \LHOGV U 7KLV FRQVWUDLQW DSSOLHG WR (T f JLYHV D YDOXH RI k ,I WKH XSSHU SDUW RI WKH '2) URERW

PAGE 75

KDV WKH VSHFLDO VWUXFWXUH (T f FDQ EH XVHG DQG WKH FRQVWUDLQW U DSSOLHV WR (T f $QJOH A FDQ EH GLUHFWO\ FRPSXWHG &DVH 7ZR LQWHUVHFWLQJ D[HV IROORZHG E\ WZR SDUDOOHO D[HV 7KLV VWUXFWXUH FRUUHVSRQGV WR &KDSWHU FDVH +HUH WKH FRQVWUDLQW LV DQG DV LQ WKH SUHFHGLQJ FDVHV A RU k FDQ EH GLUHFWO\ FRPSXWHG IURP (T f RU f UHVSHFWLYHO\ &DVH 7KUHH LQWHUVHFWLQJ D[HV 3LHSHU f KDV VKRZQ WKDW D VL['2) PDQLSXODWRU ZLWK WKUHH LQWHUVHFWLQJ D[HV FDQ DOZD\V EH VROYHG LQ FORVHG IRUP 7KLV UHVXOW DSSOLHV WR WKH VLPSOHU FDVH RI ILYH'2) URERWV 7KLV VWUXFWXUH FRUUHVSRQGV WR FDVH RI &KDSWHU ZKHUH WKH FRQVWUDLQWV DUH U DQG U ,I WKH WKUHH LQWHUVHFWLQJ D[HV DUH WKH ILUVW WKUHH (TXDWLRQ f VKRXOG EH XVHG :LWK (TV f DQG f D YDOXH RI k FDQ EH REWDLQHG GLUHFWO\ 7KLV VDPH PHWKRG FDQ EH XVHG ZKHQ WKH ODVW WKUHH D[HV LQWHUVHFW E\ ILUVW H[FKDQJLQJ WKH UROHV RI HQGHIIHFWRU DQG EDVH IUDPHV :KHQ WKH LQWHUPHGLDWH WKUHH D[HV DUH LQWHUVHFWLQJ XVH RI (TV f f DQG f ZLOO \LHOG D YDOXH RI A &DVH 7ZR FRQVHFXWLYH VHWV RI WZR LQWHUVHFWLQJ f! D[HV 7KLV VWUXFWXUH LV DQDO\]HG LQ FDVH RI &KDSWHU 7KH FRQVWUDLQW HTXDWLRQ LV U 'HSHQGLQJ RQ ZKHUH WKLV VSHFLDO VWUXFWXUH LV ORFDWHG DORQJ WKH ILYH'2) DUP AA RU

PAGE 76

k FDQ EH GLUHFWO\ FRPSXWHG E\ XVH RI (TV f RU f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f VKRXOG EH XVHG )URP &KDSWHU FDVH DQG FDVH ZH VHH WKDW WKH ODVW WZR HTXDWLRQV RI WKH UHGXFHG V\VWHP (TV f DQG f KDYH WKH IRUP DA[ 6 \ &f U DA\ 6 T[ &A U ZKHUH T[ TA U DQG U DUH DOO OLQHDU H[SUHVVLRQV LQ DQG &A $ TXDUWLF SRO\QRPLDO HTXDWLRQ LQ WMB WDQkf LV UHDGLO\ REWDLQHG E\ VTXDULQJ DQG DGGLQJ WKH ODVW WZR HTXDWLRQV T[ T\ UD A UDfn DQG VXEVWLWXWLQJ WOWf DQG &A WAfOWMf 7KLV SRO\QRPLDO FDQ EH VROYHG IRU A DQG WKH VROXWLRQ VHW

PAGE 77

FRPSOHWHG DV GHVFULEHG HDUOLHU 6LPLODUO\ IURP FDVH RI &KDSWHU ZH JHW WKH HTXDWLRQV DX\ 6 X[ &f U[ DX[ 6 X\ F! U FRUUHVSRQGLQJ WR (TVf DQG f RI WKH UHGXFHG V\VWHP RI HTXDWLRQV +HUH DJDLQ D TXDUWLF SRO\QRPLDO HTXDWLRQ LQ WA LV REWDLQHG E\ HOLPLQDWLQJ 6 DQG & :KHQ WKH WKUHH D[HV ZLWK WKH VSHFLDO JHRPHWU\ DUH ORFDWHG HOVHZKHUH DORQJ WKH ILYH'2) VWUXFWXUH D VLPLODU UHVXOW FDQ EH REWDLQHG E\ XVLQJ HTXDWLRQ f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‘

PAGE 78

&+$37(5 62/9,1* 6,;'2) 0$1,38/$7256 5HGXFWLRQ WRD '2) 3UREOHP $W OHDVW VL[ GHJUHHV RI IUHHGRP DUH UHTXLUHG IRU D URERW PDQLSXODWRU WR EH DEOH WR DUELWUDULO\ SRVLWLRQ DQG RULHQW LWV HQGHIIHFWRU ZLWKLQ LWV ZRUNVSDFH (TXDWLRQ f ZLWK Q HTXDO WR VL[ \LHOGV $L $ $ $ $J $J 3 f ,I ERWK VLGHV RI WKLV HTXDOLW\ DUH PXOWLSOLHG E\ $MA $f WKH HTXDWLRQ EHFRPHV $ $ $J $J f§ 4 f ZLWK 4 $ $ 3 f :KHQ k DQG k DUH NQRZQ PDWUL[ 4 LV IXOO\ GHWHUPLQHG DQG FDQ EH YLHZHG DV D SRVH PDWUL[ IRU D '2) DUP ZKRVH VWUXFWXUH LV GHVFULEHG E\ WKH OHIW KDQG VLGH RI (T f ZKLFK PHUHO\ H[SUHVVHV D '2) SUREOHP ,Q &KDSWHU ZH KDYH VHHQ WKDW D '2) SUREOHP FDQ DOZD\V EH VROYHG LQ FORVHGIRUP KHQFH WKH UHPDLQLQJ MRLQW YDULDEOHV FDQ EH FRPSXWHG IURP (T f

PAGE 79

)LUVW ZH VKRZ WKDW D VLPLODU UHVXOW FDQ EH REWDLQHG LI k DQG k DUH NQRZQ RU LI kA DQG k DUH NQRZQ LQVWHDG RI WKH ILUVW WZR MRLQW YDULDEOHV ,Q WKH GHYHORSPHQW RI WKH '2) LQYHUVH NLQHPDWLFV VROXWLRQ ZH KDYH XVHG WKH VLPSOLI\LQJ DVVXPSWLRQ WKDW WKH ODVW IUDPH KDG '+SDUDPHWHUV G D DQG D DOO HTXDO WR ]HUR $OWKRXJK WKLV DVVXPSWLRQ LV REYLRXVO\ FRUUHFW LQ WKH FDVH RI (T f ZH PXVW VKRZ WKDW LW FDQ EH REWDLQHG LQ RWKHU FDVHV $V VKRZQ LQ (T f D KRPRJHQHRXV PDWUL[ $A GHFRPSRVHV LQWR $A $A ZKHUH $A DQG DUH JLYHQ E\ (TV f DQG f DQG $A LV D KRPRJHQHRXV PDWUL[ IRU ZKLFK '+ SDUDPHWHUV D G DQG D DUH ]HUR ,I WKH YDOXHV RI kA DQG k DUH NQRZQ (TXDWLRQ f QRZ UHGXFHV WR WKH '2) SUREOHP $ $ $ f§ 4 f ZKHUH 4 $O 3 $ %B f 6LPLODUO\ ,I k DQG k DUH NQRZQ WKH LQYHUVH NLQHPDWLF WDVN EHFRPHV WKDW RI VROYLQJ WKH '2) FDVH $ $ $ f§ a f ZLWK 4 3 $J $ f ,Q WKH IROORZLQJ VHFWLRQ ZH ZLOO VKRZ KRZ D WZR GLPHQVLRQDO LWHUDWLYH WHFKQLTXH FDQ EH LPSOHPHQWHG WR VROYH WKH LQYHUVH NLQHPDWLFV SUREOHP RI VL['2) URERW

PAGE 80

PDQLSXODWRUV $OWKRXJK WKLV WHFKQLTXH FDQ HTXDOO\ EH GHYHORSHG XVLQJ (TV f RU f LW ZLOO EH EDVHG RQ (T f IRU FRQYHQLHQFH 7ZR'LPHQVLRQDO ,WHUDWLYH 7HFKQLTXH 6LQFH ZH RQO\ QHHG WR NQRZ RI WKH MRLQW YDULDEOHV WR VROYH IRU WKH ZKROH VROXWLRQ VHW WKH LQYHUVH NLQHPDWLFV SUREOHP RI VL['2) PDQLSXODWRUV FDQ EH UHGXFHG WR ILQGLQJ WKH YDOXHV RI WKH ILUVW WZR MRLQW YDULDEOHV RQO\ DQG JHWWLQJ FORVHGIRUP YDOXHV IRU WKH UHPDLQLQJ YDULDEOHV $ QXPHULFDO WHFKQLTXH DLPHG DW ILQGLQJ WKH YDOXHV RI 4MA DQG FDQ EH LPSOHPHQWHG E\ GHILQLQJ D V\VWHP RI WZR QRQOLQHDU HTXDWLRQV LQ A DQG If f JkOf f WKDW FDQ EH VROYHG XVLQJ DQ LWHUDWLYH PHWKRG VXFK DV D WZR GLPHQVLRQDO 1HZWRQ5DSKVRQ )URP WKH OHIW KDQG VLGH RI (T f WZR YHFWRUV X/ DQG T/ FRUUHVSRQGLQJ WR YHFWRUV X DQG T YHFWRUV X DQG T UHODWH WR SRVH 4 DV VKRZQ LQ (T ff DUH JLYHQ E\ 8/ 5 U U ] f AA A L f  f f DQG

PAGE 81

:H GHILQH WZR QRQOLQHDU IXQFWLRQV RI A DQG k DV GLIIHUHQFHV EHWZHHQ WKH LQQHUSURGXFWV XATA TATA DQG WKH LQQHUSURGXFWV XT DQG TT UHVSHFWLYHO\ IAkf XOrA/ f§ ff Jkf T/T/ TT f ,I WKH YDOXHV RI MB DQG k XVHG WR FRPSXWH SRVH PDWUL[ 4 LQ (T f GR FRUUHVSRQG WR D VROXWLRQ VHW WKHQ (T f ZLOO KROG DQG YHFWRUV X/ DQG T/ ZLOO EH H[DFWO\ HTXDO WR X DQG T IRUFLQJ ERWK IXQFWLRQV I DQG J WR EH HTXDO WR ]HUR ,Q RWKHU ZRUGV VROXWLRQ VHWV RI (T f FRUUHVSRQG WR ]HURV RI WKH IXQFWLRQV I DQG J GHILQHG LQ (TV f DQG f &RPSXWLQJ II4A42 DQG TI4A4Rf ,Q RUGHU WR FRPSXWH WKH YDOXHV RI I DQG J IRU D JLYHQ SDLU Akf WKH FRPSRQHQWV RI YHFWRUV X T DQG WKH LQQHU SURGXFWV XT DQG TT DUH QHHGHG WR VROYH WKH '2) HTXDWLRQ f ZKLFK LQ WXUQ DOORZV FRPSXWDWLRQ RI LQQHU SURGXFWV 8/r DQG T/T/ DQG ILQDOO\ WKH YDOXHV RI I DQG J 9HFWRUV X DQG T FRPSXWHG IURP (T f DUH X 5B UA 5 ] UB 5B W f DQG T 5B >5I S O[f @ f ,I ZH FRQVLGHU WKH FRPSRQHQWV RI YHFWRU W DV H[SUHVVHG ZLWK UHVSHFW WR IUDPH )A

PAGE 82

;W[ & IF[ 6 IF\ OIF< W 7OVO IF[ 7OFO IF\ rO IF] OW] DO6O IF[ a DOFO 3\ 7 W] WKHQ YHFWRU X LV JLYHQ E\ X 9 [W & ;W[ 6 W\ W6 ;W[ W& A\ D D6 OIF[ D& OIF\ 7 f 7R REWDLQ WKH FRPSRQHQWV RI YHFWRU T ILUVW ZH UHZULWH (T f DV T A A5O 3 f A ALf a A A A DQG ZH GHILQH K ;3[ OS\ 5rf S & 3[ 6 3\ 7OVO 3[ 7OFO 3\ DO 3] B 9 DOVO 3[ DOFO 3\ 7 3] YHFWRU T LV WKHQ JLYHQ E\ T & ;3[ DOf 6 ;3\ D 76 ;3[ 7& ;3\ II A] DG D6 ;3[ D& 3\ U A] a 7G f 7KH LQQHU SURGXFW XT FDQ EH GHULYHG IURP (TV f DQG f XT 5A 5I Wf paUa W3 a ;Lf a 5nL @f

PAGE 83

8VLQJ f DQG f DV QHHGHG DQG H[SDQGLQJ \LHOGV XT W S f§ Af f§ AW RU XT WS WO WO ZKLFK JLYHV XT WS & 6 D W[ & D W\ 6 G W] f 6LPLODUO\ WKH VTXDUH RI WKH OHQJWK RI YHFWRU T TT LV JLYHQ E\ TT 5f5L >S LOfr A r UaU W3 A f§ f§ ZKHUH ZH IDFWRUHG RXW 5 5A P WKH H[SUHVVLRQ RI T IURP (T f 8VLQJ f DQG f DQG H[SDQGLQJ DJDLQ OHDGV WR TT SS AA m f§ S 5A S 5A A Amf RU TT D >D/ S[f & S\ 6@ G S D SS Ds D G f (TXDWLRQ f JLYHV ULVH WR D UHGXFHG V\VWHP VLPLODU WR WKDW RI (TV ff ZLWK WKH UHTXLUHG VKLIW LQ LQGH[HV DX\ 6 DX[ & DD 6 FUDG & UA f &78; A f DX\ r &A U f

PAGE 84

&7T[ 6 f DT\ & f DD 6 U DT\ 6 DT[ & FUDG 6 DD & U ZLWK U/ TX UG GX] UUG U WW UX] U UG T]f G UG W TT D G D G D G n 7GG f 6ROYLQJ WKLV V\VWHP RI HTXDWLRQV ZLOO \LHOG WKH YDOXHV RI DQG k 7KH YDOXH RI FDQ WKHQ EH FRPSXWHG IURP WKH WZR HTXDWLRQV AGa7DAA A DDAnf FT[ 6T\ D f DQG DDJ&Jf 6 f§ FUFAfD6f & UVT[ n&T\ DT]aGf f f GHULYHG IURP (TV f DQG f RU IURP WKH HTXDWLRQV U r Ar 7 f A W AA A a A8; 6X\ f DQG &7A A f§ 7Dar7 f A W6X[ U&X\ FUX] FRUUHVSRQGLQJ WR (TV f DQG f f f f f f f f f

PAGE 85

:H FDQ QRZ FRPSXWH WKH LQQHU SURGXFWV X/T/ DQG T/T/ %\ LQFUHPHQWLQJ WKH LQGH[HV LQ (T f ZH GHULYH 8/r 7G D&7 6 &7GD & 7G FU 6 D & DG 6f a D A DU A DGU A 7GDf A D&7 A f FGF r 7G7fr f 9HFWRU T/ REWDLQHG IURP WKH OHIW KDQG VLGH RI (T f LV AA AnAfO f DQG WKH VTXDUH RI LWV OHQJWK LV JLYHQ E\ 6O6O AA A Af Af f AA ,f ,f RU a A AA A AAf f AA AAA A Af! DIWHU IDFWRULQJ RXW 5 5f DQG XVLQJ LQQHU SURGXFW LQYDULDQFH RI URWDWLRQV 0XOWLSO\LQJ RXW WKH WHUPV LQ SDUHQWKHVHV DQG XVLQJ f DQG f ZKHUH QHFHVVDU\ WKH ODVW HTXDWLRQ \LHOGV WD &AD D & DG 6A D 6DU6 DGU & UGD DGf GJ WG DD 6 f§ FUGD & G7Uf DD A eAD A UAG@ D GA D G DA GV f

PAGE 86

*LYHQ D SDLU k kf WKH FRUUHVSRQGLQJ YDOXHV RI Ikkf DQG Jkkf DUH REWDLQHG E\ WKH IROORZLQJ VWHSV 6WHS )RU LQLWLDO YDOXHV RI k DQG k FRPSXWH WKH FRRUGLQDWHV RI YHFWRUV X DQG T DV JLYHQ E\ (TV f DQG f 7KH LQQHU SURGXFWV XT DQG TT FDQ EH FRPSXWHG XVLQJ WKH UHJXODU LQQHU SURGXFW IRUPXOD XT X[T[ X\T\ X]T] DQG TT T[ T\ T] 6WHS 6ROYH WKH UHGXFHG V\VWHP RI (TV ff IRU k DQG k 6WHS &RPSXWH WKH YDOXH RI k IURP (TV f DQG f RU (TV f DQG f 6WHS &RPSXWH WKH LQQHU SURGXFWV XATA DQG T/T/ JLYHQ E\ (TV f DQG f UHVSHFWLYHO\ DQG FRPSXWH WKH YDOXHV RI I DQG J DV JLYHQ E\ (TV f DQG f 7ZR'LPHQVLRQDO 1HZWRQ5DSKVRQ 7KH ]HURV RI I DQG J FDQ EH LWHUDWLYHO\ FRPSXWHG DQG FKHFNHG IRU FRQVLVWHQF\ ZLWK (T f ,I D FRPSXWHU SURJUDP IRU HYDOXDWLQJ WKH WZR IXQFWLRQV LV DYDLODEOH WKH SDUWLDO GHULYDWLYHV RI I DQG J ZLWK UHVSHFW WR k DQG k GHQRWHG IOI I DQG JOI J UHVSHFWLYHO\ FDQ EH QXPHULFDOO\ DSSUR[LPDWHG E\ IOkOf GIDH/ >I k kfI kf @ f

PAGE 87

IpLLpfa %IDk >IkkfIkf@k f DQG LpLpf %Jk >JfJf@k f Jkf DJDk &JkfJkHf@mk f ZKHUH DQG k DUH VPDOO LQFUHPHQWV RI k DQG UHVSHFWLYHO\ 7KH WZRGLPHQVLRQDO 1HZWRQ5DSKVRQ WHFKQLTXH IRU VROYLQJ WKH LQYHUVH NLQHPDWLFV SUREOHP IRU D VL[UHYROXWH '2) URERW DUP RI DUELWUDU\ DUFKLWHFWXUH SURFHHGV DFFRUGLQJ WR WKH IROORZLQJ VWHSV 6WHS $VVXPH DQ LQLWLDO HVWLPDWH RI k DQG k DQG FRPSXWH k k DQG k 6WHS )URP WKH YDOXHV RI k k k k DQG k FRPSXWH Ikkf DQG Jkf DV LQ (TV f DQG f 6WHS &RPSXWH WKH SDUWLDO GHULYDWLYHV RI I DQG J ZLWK UHVSHFW WR k DQG k E\ QXPHULF DSSUR[LPDWLRQV DV VKRZQ HDUOLHU 6WHS 2EWDLQ D QHZ HVWLPDWH IRU k DQG k E\ WKH WZRGLPHQVLRQDO 1HZWRQ5DSKVRQ PHWKRG LH HO p I I IkHf B B J[ J B JIkf QHZ 6WHS 5HSHDW VWHSV WR XQWLO GHVLUHG DFFXUDF\ LV DWWDLQHG

PAGE 88

6WHS &RPSOHWH WKH VROXWLRQ VHW E\ XQLTXHO\ FRPSXWLQJ k IURP r f & Q; 5 [ 6 f§ 7f S f S % a S f N N N N N Q\ B Q] B 6WHS &KHFN WKH VROXWLRQ VHW IRU FRQVLVWHQF\ ZLWK (T f &KRLFH RI IXQFWLRQV I DQG J 7KH IXQFWLRQV I DQG J GHILQHG E\ (TV f DQG f DUH FRPSXWDWLRQDOO\ HFRQRPLFDO VLQFH WKH\ GR QRW UHTXLUH FRPSXWDWLRQ RI WKH IRUZDUG NLQHPDWLFV RU WKH LQYHUVH MDFRELDQ RI WKH PDQLSXODWRU ,Q IDFW HYHQ WKH YDOXH RI H LV QRW UHTXLUHG WR FRPSXWH I DQG J VLQFH (T f LV FRQVLGHUHG RQO\ DIWHU FRQYHUJHQFH 8QIRUWXQDWHO\ D SDLU A f IRU ZKLFK ERWK I DQG J DUH ]HUR LV QRW JXDUDQWHHG WR FRUUHVSRQG WR D VROXWLRQ VHW RI (T f ([WUDQHRXV VROXWLRQ VHWV FDQ EH FRQYHUJHG WR DV ZHOO 7KHVH DUH MRLQW YDOXHV WKDW ZLOO \LHOG DQ HQGHIIHFWRU SRVH WKDW LV QRW H[DFWO\ WKH GHVLUHG RQH 2WKHU IXQFWLRQV WKDW IXOO\ FRQVWUDLQW WKH HQGHIIHFWRU SRVH FDQ EH GHILQHG DW WKH FRVW RI JUHDWHU FRPSXWDWLRQDO FRPSOH[LW\ :X DQG 3DXO f KDYH VKRZQ WKDW WKH GLIIHUHQFH EHWZHHQ DFWXDO DQG GHVLUHG HQGHIIHFWRU SRVHV FDQ EH H[SUHVVHG DV D [ YHFWRU [H JLYHQ E\

PAGE 89

; Q/ r 3 f 3OA [ E/ r 3 3/f [ E/ f 3 3O! [ W/E WE/f [ Q/W QW/f B [ E/Q EQ/f f ZKHUH QI E W DQG S DUH WKH YHFWRUV WKDW GHVFULEH WKH GHVLUHG HQGHIIHFWRU SRVH 3 DV GHILQHG LQ f DQG YHFWRUV Q/n E/n IF/n DQFA 3/ DUH WAHLU FRUUHVSRQGLQJ YHFWRUV IURP WKH OHIW KDQG VLGH RI HTXDWLRQ f 7ZR IXQFWLRQV FDQ EH GHILQHG DV IHkf [A [ [ f Jf [ [ [ f $ SDLU f WKDW LV D ]HUR RI ERWK I DQG J LV JXDUDQWHHG WR FRUUHVSRQG WR D VROXWLRQ VHW RI (T f VR WKDW WKH LWHUDWLYH PHWKRG GHVFULEHG DERYH ZLOO RQO\ FRQYHUJH WR D WUXH VROXWLRQ +RZHYHU QRZ WKH IRUZDUG NLQHPDWLFV PXVW EH FRPSXWHG DW HDFK LWHUDWLRQ VLQFH WKH FRPSRQHQWV RI YHFWRUV Q/n E/n A/n DQFA "O DUH DOO QHHGHG WR HYDOXDWH IXQFWLRQV I DQG J DV GHILQHG E\ (TV f DQG f

PAGE 90

2QH'LPHQVLRQDO 0HWKRG 7KH LQYHUVH NLQHPDWLF SUREOHP IRU VL['2) PDQLSXODWRUV UHGXFHV WR D ILYH'2) RQH ZKHQ WKH ILUVW RU WKH ODVW MRLQW YDULDEOH LV NQRZQ (TXDWLRQ f EHFRPHV $ $ $ $ $ n f ZLWK 4 $O 3 f ZKHQ kL LV NQRZQ DQG $L A A A 4 f ZLWK 4 3 $JAJ f LI kJ LV NQRZQ ,Q ERWK FDVHV D ILYH'2) SUREOHP LV REWDLQHG :KHQ WKH UHVXOWLQJ ILYH'2) SUREOHP LV VROYDEOH LQ FORVHG IRUP NQRZOHGJH RI kA RU k LV WKHQ VXIILFLHQW WR \LHOG D FRPSOHWH VROXWLRQ VHW 7KH LQYHUVH NLQHPDWLF SUREOHP WKHQ UHGXFHV WR ILQGLQJ RQH MRLQW DQJOH ZKLFK FDQ EH DFFRPSOLVKHG E\ D RQHGLPHQVLRQDO LWHUDWLYH WHFKQLTXH ,Q FKDSWHU ZH IRXQG WKDW D VXIILFLHQW FRQGLWLRQ IRU FORVHG IRUP VROXWLRQV RI '2) PDQLSXODWRUV LV WKDW WKH\ KDYH RQH RI WKH VSHFLDO VWUXFWXUHV OLVWHG DW WKH HQG RI &KDSWHU 6L['2) DUPV WKDW LQFOXGH D ILYH'2) VHJPHQW ZLWK WKLV W\SH RI JHRPHWU\ FDQ WKHQ EH VROYHG XVLQJ D RQHn GLPHQVLRQDO LWHUDWLYH PHWKRG 7KLV LWHUDWLYH WHFKQLTXH FDQ EH LPSOHPHQWHG LQ PXFK WKH VDPH ZD\ DV GHVFULEHG LQ &KDSWHU

PAGE 91

IRU ILYHGHJUHHRIIUHHGRP DUPV $VVXPLQJ (T f LV WR EH VROYHG ZH GHILQH D IXQFWLRQ I RI MA E\ IOf 8OA/ n X n n f ZKHUH YHFWRUV X/ T/ X DQG T DUH GHILQHG DV HDUOLHU *LYHQ D YDOXH RI A YHFWRUV X DQG T DUH FRPSXWHG IURP (T f WKH YDOXHV RI WKH UHPDLQLQJ MRLQW YDULDEOHV DUH FRPSXWHG LQ FORVHG IRUP IURP (T f DV LQGLFDWHG LQ &KDSWHU DQG $SSHQGL[ % WKH LQQHU SURGXFW X/T/ FDQ WKHQ EH REWDLQHG DV LQ (T f ZLWK WKH SURSHU LQGH[ DGMXVWPHQWV DQG WKH YDOXH RI I LV WKHQ JLYHQ E\ (T f $V ZH KDYH VHHQ EHIRUH WKH DELOLW\ WR FRPSXWH WKH IXQFWLRQ I IRU D JLYHQ YDOXH RI A DOORZV WKH LPSOHPHQWDWLRQ RI D SUDFWLFDO RQHGLPHQVLRQDO 1HZWRQ5DSKVRQ DOJRULWKP 7KHUHIRUH ZH FDQ FRQFOXGH WKDW D VL[GHJUHHRIIUHHGRP PDQLSXODWRU ZLWK WZR FRQVHFXWLYH SDLUV RI LQWHUVHFWLQJ RU SDUDOOHO MRLQW D[HV RU WKUHH FRQVHFXWLYH MRLQW D[HV WKDW DUH SDUDOOHO DQGRU LQWHUVHFWLQJ WZR DW D WLPH FDQ EH VROYHG E\ XVH RI D RQHGLPHQVLRQDO LWHUDWLYH WHFKQLTXH LQVWHDG RI WKHn WZRGLPHQVLRQDO PHWKRG UHTXLUHG IRU DQ DUP RI DUELWUDU\ DUFKLWHFWXUH &ORVHG)RUP 6ROXWLRQ 6RPH VL[GHJUHHRIIUHHGRP PDQLSXODWRUV ZLWK VLPSOH JHRPHWULHV GR QRW UHTXLUH DQ\ LWHUDWLYH PHWKRG VLQFH WKH\ FDQ EH VROYHG LQ FORVHGIRUP 3LHSHU f KDV VKRZQ WKDW

PAGE 92

D VXIILFLHQW FRQGLWLRQ IRU FORVHGIRUP VROXWLRQV LV WKDW WKUHH FRQVHFXWLYH D[HV EH LQWHUVHFWLQJ 7KH LQYHUVH NLQHPDWLFV SUREOHP WKHQ UHGXFHV WR ILQGLQJ WKH ]HURV RI D TXDUWLF SRO\QRPLDO ,Q WKH OLWHUDWXUH ,W VHHPV WR EH FRPPRQ NQRZOHGJH WKDW WKUHH FRQVHFXWLYH MRLQW D[HV WKDW DUH SDUDOOHO LV DQRWKHU VXIILFLHQW JHRPHWULF FRQGLWLRQ IRU FORVHG IRUP VROXWLRQV 7KH DQDO\VLV RI &KDSWHU DQG $SSHQGL[ $ VKRZHG WKDW XQGHU FHUWDLQ FRQGLWLRQV WKH UHGXFHG V\VWHP RI HTXDWLRQV ff LQFOXGHG FRQVWUDLQW HTXDWLRQV RI WKH IRUP U f 7KH TXDQWLWLHV UA L O DUH IXQFWLRQV RI kAA DQG k DV ZH KDYH VHHQ HDUOLHU %\ ORRNLQJ IRU FRQGLWLRQV XQGHU ZKLFK D MRLQW YDULDEOH YDOXH FDQ EH GLUHFWO\ REWDLQHG IURP DQ HTXDWLRQ KDYLQJ WKH IRUP RI (T f ZH ILQG WZR PRUH VXIILFLHQW VL['2) URERW VWUXFWXUH FRQGLWLRQV IRU FORVHGIRUP VROXWLRQV H[FOXGLQJ WKH DOUHDG\ NQRZQ FRQGLWLRQV RI WKUHH SDUDOOHO RU WKUHH LQWHUVHFWLQJ D[HVf :KHQ WKH ILUVW WZR MRLQW D[HV RI D PDQLSXODWRU DUH SDUDOOHO VR WKDW A WKHQ FU A DQG WKH ]FRPSRQHQWV RI YHFWRUV X DQG T JLYHQ E\ (TV f DQG f EHFRPH X] D W\ & IF[ 6 f 7LW] f T] D 3[ 6 3\ &f U 3]fGf f DQG

PAGE 93

7KLV VKRZV WKDW U DQG U DV JLYHQ LQ f DQG f EHFRPH OLQHDU IXQFWLRQV RI 6 DQG FLf :KHQ MRLQW D[HV DQG DUH SDUDOOHO DQG MRLQW D[HV DQG DUH SDUDOOHO WKH UHGXFHG V\VWHP RH HTXDWLRQV f f EHFRPHV VLPLODU WR WKDW RI FDVH RI &KDSWHU DX\ 6 DX[ & f U f DD 6 U f DA\ 6 D[ & &7DG 6 DD & Un f (TXDWLRQ f \LHOGV WZR SRVVLEOH YDOXHV IRU kn HDFK RI ZKLFK ZLOO SURYLGH WZR SRVVLEOH YDOXHV RI k IURP (T f ZKHQ VXEVWLWXWHG LQ WKH H[SUHVVLRQ RI U 7KH UHPDLQLQJ MRLQW YDOXHV FDQ WKHQ EH FRPSXWHG LQ FORVHG IRUP $ VLPLODU GHYHORSPHQW RFFXUV ZKHQ D[HV DQG DUH SDUDOOHO DQG D[HV DQG LQWHUVHFW 7R VXPPDUL]H D VL['2) PDQLSXODWRU KDV D FORVHGIRUP VROXWLRQ LI RQH RI WKH IROORZLQJ FRQGLWLRQV LV VDWLVILHG 7KUHH FRQVHFXWLYH MRLQW D[HV DUH SDUDOOHO 7KUHH FRQVHFXWLYH MRLQW D[HV LQWHUVHFW DW RQH SRLQW 7KH DUP LV IRUPHG RI WKUHH VHWV RI WZR SDUDOOHO D[HV 7KLV VWUXFWXUH LV LOOXVWUDWHG LQ )LJXUH Df 7KH DUP KDV WZR VHWV RI WZR SDUDOOHO MRLQW D[HV IROORZHG RU SUHFHGHG E\ WZR LQWHUVHFWLQJ D[HV 7KHVH VWUXFWXUHV DUH LOOXVWUDWHG LQ )LJXUH Ef DQG Ff

PAGE 94

E SDLUV RI SDUDOOHO MRLQW D[HV IROORZHG E\ LQWHUVHFWLQJ MRLQW D[HV F SDLUV RI SDUDOOHO MRLQW D[HV SUHFHGHG E\ LQWHUVHFWLQJ MRLQW £[HV )LJXUH '2) PDQLSXODWRUV ZLWK FORVHGIRUP LQYHUVH NLQHPDWLFV

PAGE 95

&+$37(5 257+2*21$/ 0$1,38/$7256 'HILQLWLRQ $Q QD[HV VHULDO NLQHPDWLF FKDLQ RI UHYROXWH RU SULVPDWLF MRLQWV LV RUWKRJRQDO LI DOO WZLVW DQJOHV D L O Q DORQJ WKH FKDLQ DUH RU LU $Q RSHQ RUWKRJRQDO NLQHPDWLF FKDLQ ZLOO EH FDOOHG DQ RUWKRJRQDO PDQLSXODWRU 'RW\ f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re DQG ELW L LV LI DA Q )RU H[DPSOH D PDQLSXODWRU ZLWK WZLVW DQJOHV

PAGE 96

D U D WW D DQG R/ U EHORQJV WR WKH FODVV RI RUWKRJRQDO PDQLSXODWRUV 6LQFH PRVW LQGXVWULDO URERW DUPV DUH RUWKRJRQDO LW LV ZRUWKZKLOH WR FRQVLGHU WKH LQYHUVH NLQHPDWLFV SUREOHP ZLWK UHVSHFW WR WKHVH PDQLSXODWRUV 7KH $PDWULFHV DVVRFLDWHG $LD f RU $LD Uf KDYH RQH RI WKH FL 6L DL&L VL &L DL6L GL &L VL DLFL 6L FL DLVL f f )XUWKHU FRPSXWDWLRQDO VLPSOLILFDWLRQ LV REWDLQHG LQ WKH LQYHUVH NLQHPDWLF HTXDWLRQV ZLWK RUWKRJRQDO PDQLSXODWRUV VLQFH 5A] 5MB] ] LI DA DQG f f§n ‘ n n f B 5A] \ LI DA Z 'RW\ f KDV VKRZQ WKDW RI WKH FODVVHV RI QRQWULYLDO RUWKRJRQDO PDQLSXODWRUV WKRVH ZLWK QRQ]HUR

PAGE 97

WZLVW DQJOHV FODVVHV DQG f KDYH FORVHGIRUP VROXWLRQV 7KH LQYHUVH NLQHPDWLF DQDO\VLV RI &KDSWHU VKRZV WKDW WKH PRVW FRPSOH[ VL['2) URERW VWUXFWXUH FDQ EH VROYHG E\ XVH RI D WZRGLPHQVLRQDO LWHUDWLYH WHFKQLTXH 6LPSOHU VWUXFWXUHV RQO\ UHTXLUH D RQHn GLPHQVLRQDO QXPHULFDO WHFKQLTXH DQG VRPH HYHQ VLPSOHU VWUXFWXUHV FDQ EH VROYHG LQ FORVHGIRUP ,Q 7DEOH ZH SURYLGH D OLVW RI DOO WKLUW\WZR RUWKRJRQDO PDQLSXODWRU FODVVHV LQ ZKLFK ZH LQGLFDWH WKH GHJHQHUDWH JHRPHWULHV DQG IRU WKH WZHQW\ IRXU QRQn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

PAGE 98

7DEOH ,QYHUVH NLQHPDWLFV RI RUWKRJRQDO PDQLSXODWRUV &ODVV 0HWKRG -XVWLILFDWLRQ $OO VL[ D[HV DUH SDUDOOHO )LYH FRQVHFXWLYH SDUDOOHO D[HV )RXU FRQVHFXWLYH SDUDOOHO D[HV )RXU FRQVHFXWLYH SDUDOOHO D[HV &) 7KUHH FRQVHFXWLYH SDUDOOHO D[HV &) 7KUHH FRQVHFXWLYH SDUDOOHO D[HV &) 7KUHH FRQVHFXWLYH SDUDOOHO D[HV &) 7KUHH FRQVHFXWLYH SDUDOOHO D[HV )RXU FRQVHFXWLYH SDUDOOHO D[HV &) 7KUHH FRQVHFXWLYH SDUDOOHO D[HV &) f7KUHH SDLUV RI SDUDOOHO D[HV 7ZR SDLUV RI SDUDOOHO D[HV &) 7KUHH FRQVHFXWLYH SDUDOOHO D[HV '

PAGE 99

7DEOH f§&RQWLQXHG &ODVV 0HWKRG -XVWLILFDWLRQ )LYH FRQVHFXWLYH SDUDOOHO c D[HV )RXU FRQVHFXWLYH SDUDOOHO c D[HV &) 7KUHH FRQVHFXWLYH SDUDOOHO D[HV &) 7KUHH FRQVHFXWLYH SDUDOOHO D[HV &) 7KUHH FRQVHFXWLYH SDUDOOHO D[HV 7ZR SDLUV RI SDUDOOHO D[HV )RXU FRQVHFXWLYH SDUDOOHO D[HV &) 7KUHH FRQVHFXWLYH SDUDOOHO D[HV 7ZR SDLUV RI SDUDOOHO D[HV &) 7KUHH FRQVHFXWLYH SDUDOOHO D[HV 1RWDWLRQ 'HJHQHUDWH JHRPHWU\ &) &ORVHG)RUP 2QH 'LPHQVLRQDO LWHUDWLYH PHWKRG 7ZR'LPHQVLRQDO LWHUDWLYH PHWKRG

PAGE 100

&+$37(5 $33/,&$7,21 (;$03/(6 ([DPSOH 7KH 380$ $ SRSXODU RUWKRJRQDO PDQLSXODWRU JHRPHWU\ WKH 380$ LV GHVFULEHG E\ WKH NLQHPDWLF SDUDPHWHUV JLYHQ LQ 7DEOH DQG LOOXVWUDWHG LQ )LJXUH 7KLV PDQLSXODWRU KDV D VSKHULFDO ZULVW DQG WKHUHIRUH DOORZV FORVHGIRUP VROXWLRQV 3LHSHU f ,QYHUVH NLQHPDWLF VROXWLRQV KDYH EHHQ SURSRVHG E\ QXPHURXV DXWKRUV IRU WKLV W\SH RI DUP /HH DQG =LHJOHU &UDLJ 3DXO DQG =KDQJ f 7DEOH 380$ NLQHPDWLF SDUDPHWHUV -RLQW G D D UGf ; 7 D G k D Q G k LW WW p 7KLV H[DPSOH LV LQFOXGHG KHUH WR GHPRQVWUDWH WKH XWLOLW\ RI WKH DSSURDFK DOUHDG\ RXWOLQHG DQG WR FRQWUDVW LW

PAGE 101

, )LJXUH 7KH 380$ NLQHPDWLF VWUXFWXUH

PAGE 102

ZLWK WKH JHRPHWULF DQG DOJHEUDLF DSSURDFKHV WDNHQ E\ WKH SUHYLRXV DXWKRUV :LWKRXW FRPSXWLQJ WKH IRUZDUG NLQHPDWLFV ZH ZLOO LOOXVWUDWH KRZ (TV ff PD\ EH HDVLO\ REWDLQHG )RU TXLFN UHIHUHQFH LQ WKH IROORZLQJ GLVFXVVLRQ ZH ZULWH WKH HTXDWLRQV LPPHGLDWHO\ W= a 6 & 6 & & f 3= D 6 D 6 G & f SW DD&f AA A AA f A A DAA f SaDAfDAfGAf§GAfDf G 6 D & f 7R LOOXVWUDWH WKH VLPSOLILFDWLRQ REWDLQHG E\ WKH IUDPH DVVLJQPHQW GHVFULEHG HDUOLHU DQG LQQHUSURGXFW LQYDULDQFH XQGHU URWDWLRQV ZH JLYH LQ GHWDLO WKH GHYHORSPHQW RI HTXDWLRQ f :LWK / O OJ DQG O G] (T f \LHOGV 3 A 5 A5A Af AA RU S 5A55 > 5 A 5 AA AA n %\ RUWKRJRQDOLW\ WKH LQQHUSURGXFW SS KDV WKH VDPH YDOXH DV WKH LQQHUSURGXFW RI WKH WHUP LQ EUDFNHWV KHQFH SS > 5B 55B @ r > 5OO 5Ua/@

PAGE 103

7KH LQQHUSURGXFW RI HDFK WHUP LQ WKH VTXDUH EUDFNHWV ZLWK LWVHOI LV WKH VTXDUH RI WKH OHQJWK RI WKDW YHFWRU )RU H[DPSOH 5 5 A r A AA A ArA DA AA AAr 7KHVH LQQHUSURGXFW PDQLSXODWLRQV UHSUHVHQW D FRQVLGHUDEOH DOJHEUDLF VLPSOLILFDWLRQ WKDW UHTXLUHV OLWWOH RU QR PHQWDO HIIRUW )XUWKHU WKH\ SURYLGH D PHWKRGRORJ\ DQG FRQVLGHUDEOH LQVLJKW LQWR KRZ WR ILQG RWKHU DOJHEUDLF UHGXFWLRQV 6RPH RI WKH FURVV WHUPV DOVR UHGXFH IRU LQVWDQFH 5 A f 5 A AA A r A AAn &RPSOHWH H[SDQVLRQ RI (T f DQG DSSOLFDWLRQ RI WKH UHGXFWLRQ WHFKQLTXHV MXVW GLVFXVVHG OHDG WR SA AAAf r>A AA 5 AA AA ArA AAr )RU WKLV PDQLSXODWRU YHFWRUV >G@7 G] f§ r7 5 >DG@ DQG 5/ >D! rn r@7 D [ DOORZ XV WR VLPSOLI\ WKH ODVW HTXDWLRQ WR SA f§A f§A f§Af G= >5 AA D 5 DA f [ HJ O5BO LV REYLRXVO\ ZKLFK HOLPLQDWHV k IURP WKLV HTXDWLRQf

PAGE 104

:LWKRXW DQ\ PDWUL[ PXOWLSOLFDWLRQ UHTXLUHG ZH REWDLQ WKH IXOO\ VLPSOLILHG UHODWLRQ LQYROYLQJ k RQO\ 3DDGGf D G 6 D &f 7KH ODVW HTXDWLRQ \LHOGV DW PRVW WZR VROXWLRQV IRU k $IWHU DSSO\LQJ WULJRQRPHWULF LGHQWLWLHV IRU DQJOH VXPV WR f ZH JHW 3] D A D AaAa A a A AnAnf§ 6Af n DQG JURXSLQJ WHUPV ZH REWDLQ DD&G6f 6 D6G&f & S] :LWK HDFK YDOXH RI k YDOXHV IRU k FDQ EH REWDLQHG IURP WKLV ODVW HTXDWLRQ :LWK k DQG k NQRZQ f DQG f EHFRPH IXQFWLRQV RI k DQG k RQO\ $OWKRXJK WKLV V\VWHP RI HTXDWLRQV LQ WZR XQNQRZQV FDQ WKHRUHWLFDOO\ EH VROYHG LWV VROXWLRQ LV QRW REYLRXV $ VLPSOHU VROXWLRQ H[LVWV LI (TV f WR f DUH FRQVLGHUHG S [ 5A55 f f [ S[ f S \ 555 f f f \ 3\ f RU G 6 WfD & WfD & f A A A 3[ G &A G6D&D&f 6A S

PAGE 105

7KH ODVW WZR (TV IRUP D OLQHDU V\VWHP LQ 6A DQG DQG SURYLGH D XQLTXH YDOXH IRU k (TV f DQG f DORQJ ZLWK f SURYLGH D ZD\ WR VROYH IRU DQG %\ VROYLQJ IRU & LQ f DQG VXEVWLWXWLQJ LQ W[ & & &6 6 A 66 & 6 & f W\ s & &6 & 66 6 6 &n f REWDLQ & &6 6O F 66 & W] & 6 f 6 &6 f & & 66 W\ & W] 6 6 r f 7KLV OLQHDU V\VWHP FDQ EH VROYHG IRU WKH SURGXFWV &6 DQG 66 XQLTXHO\ :KHQ 6 LV QRW WZR VROXWLRQV IRU k DUH WKHQ REWDLQHG E\ k $WDQ66A&6Af RU k $WDQa66A&6f :KHQ 6 MRLQW D[HV ] DQG ]A DUH DOLJQHG DQG WKH PDQLSXODWRU ORVHV RQH '2) 2QO\ WKH VXP kk FDQ EH IRXQG E\ XVH RI (TV f DQG f :LWK k NQRZQ WKH W[ DQG W\ HTXDWLRQV DERYH FRQVWLWXWH D OLQHDU V\VWHP RI HTXDWLRQV ZKLFK \LHOGV D XQLTXH VROXWLRQ IRU 7KH ODVW MRLQW YDULDEOH k FDQ WKHQ EH REWDLQHG IURP PRUH HTXDWLRQV IURP f VXFK DV WKH Q] DQG E] HTXDWLRQV

PAGE 106

7KLV SURFHGXUH ZLOO \LHOG VROXWLRQV ZKLFK PXVW WKHQ EH FKHFNHG IRU MRLQW YDULDEOH UDQJH OLPLWDWLRQV :H HQG WKH GLVFXVVLRQ RI WKH 380$ H[DPSOH ZLWK WKH REVHUYDWLRQ WKDW WKH IRUZDUG NLQHPDWLFV ZHUH QHYHU GHWHUPLQHG WR REWDLQ WKH LQYHUVH NLQHPDWLF VROXWLRQ ([DPSOH 7KH *3 &RQVLGHU WKH PDQLSXODWRU JHRPHWU\ ZLWK NLQHPDWLF SDUDPHWHUV JLYHQ LQ 7DEOH 7KLV URERW DUP LV DQ H[LVWLQJ LQGXVWULDO PDQLSXODWRU WKDW EHORQJV WR WKH OL FODVV RI RUWKRJRQDO DUPV DQG GRHV QRW DOORZ FORVHGIRUP VROXWLRQV $OWKRXJK 7DEOH VSHFLILHV WKDW WKH PRVW FRPSOH[ DUP VWUXFWXUH ZLWKLQ WKH RUWKRJRQDO PDQLSXODWRUV FODVV FDQ EH VROYHG ZLWK D WZRGLPHQVLRQDO LWHUDWLYH WHFKQLTXH EXW WKH *3 )LJXUH f KDV WZR FRQVHFXWLYH SDLUV RI LQWHUVHFWLQJ D[HV DQG D SULVPDWLF MRLQW DQG LW FDQ EH VROYHG ZLWK D RQHGLPHQVLRQDO LWHUDWLYH WHFKQLTXH $QRWKHU UHDVRQ IRU GLVFXVVLQJ WKLV DUP KHUH LV WR VKRZ WKDW WKH WHFKQLTXHV GHYHORSHG LQ WKLV WH[W DSSO\ WR PDQLSXODWRUV ZLWK SULVPDWLF MRLQWV DV ZHOO $Q LWHUDWLYH PHWKRG WKDW H[DFWO\ FRPSXWHV WKH SRVLWLRQ EXW DSSUR[LPDWHV WKH RULHQWDWLRQ ZDV SURSRVHG IRU WKLV W\SH RI JHRPHWU\ E\ /XPHOVN\ f 7KH WHFKQLTXH SUHVHQWHG KHUH GLIIHUV LQ WKDW LW VROYHV IRU ERWK WKH RULHQWDWLRQ DQG WKH SRVLWLRQ ZLWK WKH VDPHn SUHFLVLRQ DQG LWLV DSSOLFDEOH WR D ODUJHU YDULHW\ RI PDQLSXODWRUV

PAGE 107

)LJXUH 7KH *3 NLQHPDWLF VWUXFWXUH

PAGE 108

7DEOH *3 PDQLSXODWRU NLQHPDWLF SDUDPHWHUV MRLQW G k D D 7 D 7 G 7 G 7 p -RLQW LV SULVPDWLF :H ZLOO LPSOHPHQW DQ LWHUDWLYH WHFKQLTXH WR ILQG WKH ]HURV RI D UHDO YDOXHG IXQFWLRQ RI A )LUVW ZH VKRZ WKDW NQRZOHGJH RI A LV VXIILFLHQW WR GHWHUPLQH D VROXWLRQ VHW $VVXPLQJ D JXHVV RI A WKH FRUUHVSRQGLQJ [ DQG ] FRPSRQHQWV RI S 5fS DQG FDQ EH FRPSXWHG 3; 3[ &L3\ VL f S] S[ 6M3\ &O f +[ W[ &MAW\ 6[ f OIF] 6OW\ &f f f f§ r )RU WKLV URERW A 5 D[ ‘rf A= G] DQG 5 :LWK WKHVH YDOXHV f \LHOGV S 5A5 A AA A A D A RU DIWHU PXOWLSOLFDWLRQ E\ 5A

PAGE 109

S 5G 5] G = D [f f 9HFWRU W LV JLYHQ E\ W 5A W 555 ] f $ UHGXFHG V\VWHP RI HTXDWLRQV FDQ EH REWDLQHG IURP WKH ODVW r WZR HTXDWLRQV E\ FRQVLGHULQJ WKH H[SUHVVLRQV RI AS W W ] W S DQG -SS &RPSXWLQJ f§W] )URP (T f DQG XVLQJ (TV f DQG f DV QHFHVVDU\ \LHOGV W] W ] 5] 5B\ 6LQFH 5] >6 & @7 DQG 5B\ >6 &@7 WKH SUHFHGLQJ HTXDWLRQ EHFRPHV W] 6 6 f &RPSXWLQJ f§S] 6LQFH S] S ] IURP (T f SURSHUWLHV f DQG f DQG 5fA] \ ZH REWDLQ A‘3] G U] G ] D [f \ ZKLFK LV HDVLO\ VHHQ WR SURGXFH S] G & f &RPSXWLQJ f§Wf§S (TV f fDQG XVH RI (TV f DQG f \LHOG WS WS 5] G ] G 5B] D 5[f

PAGE 110

:LWK 5] \ DQG 5]G] WKLV HTXDWLRQV UHGXFHV WR WS G &J D & 6A f &RPSXWLQJ SS 7KH LQQHU SURGXFW GLUHFWO\ SURGXFHV SS SS G G D D G 6 f ZLWKRXW DQ\ PDWUL[ RSHUDWLRQV (TXDWLRQ f FDQ EH XVHG WR GHILQH D UHDO IXQFWLRQ RI HOI LH I@f G & D & 6 WS f 9DOXHV RI k WKDW \LHOG D VROXWLRQ WR WKH ,QYHUVH NLQHPDWLFV RI WKLV PDQLSXODWRU PXVW EH ]HURV RI IXQFWLRQ I (TV f f DQG f SURYLGH D ZD\ WR FRPSXWH I JLYHQ k‘@B :LWK NQRZQ ;3[ AW[ DQG W] DUH IXOO\ GHWHUPLQHG (TXDWLRQ f WKHQ \LHOGV & f DQG 6 X 7ULJ&f f ZKHUH X RU H[SUHVVHV D VLJQ DPELJXLW\ DQG WKH IXQFWLRQ 7ULJ LV GHILQHG E\ 7ULJ[f O[f 7KH SULVPDWLF YDULDEOH G FDQ WKHQ EH IRXQG IURP (T f G SS D G D G 6f f

PAGE 111

DQG IURP f WKH YDOXH RI FDQ EH FRPSXWHG LI V LV QRW ]HUR 6 W6nn f DQG & X 7ULJ6f f ZKHUH X RU LV DQRWKHU VLJQ DPELJXLW\ 7KLV DGGLWLRQDO VLJQ DPELJXLW\ FDQ EH DYRLGHG LI PRUH HTXDWLRQV LQYROYLQJ k DUH FRQVLGHUHG ,QGHHG (T f DOORZV XV WR GHULYH H[SUHVVLRQV IRU S[ DQG 3\ DV OS[ 5AG 5] G ] D [f [ DQG 3\ 5G U] G ] D ;f \ 7KHVH ODVW HTXDOLWLHV \LHOG D V\VWHP RI WZR HTXDWLRQV WKDW FDQ UHDGLO\ EH VROYHG IRU 6 DQG & 6 G 3; N 3]fGNRfn f & NR 3[ fG 3fGNfn f ZKHUH N4 D G 6f 7KH YDOXH RI & FDQ WKHQ EH REWDLQHG IURP HLWKHU WKH H[SUHVVLRQ IRU W\ AW\ 5 5 5 ] \ ZKLFK DIWHU XVLQJ SURSHUWLHV f DQG f JLYHV

PAGE 112

& W]6 & 6f& f RU IURP f W-L 5 5 5 5 ] [ ZKLFK \LHOGV & & & 6 f f :LWK WKH FRPSXWHG YDOXHV RI G & & DQG 6 WKH YDOXH RI IWkf LV IXOO\ GHWHUPLQHG 7KH GHULYDWLYH RI I FDQ DOVR EH HYDOXDWHG %\ GLIIHUHQWLDWLQJ f ZLWK UHVSHFW WR A G&Gk FDQ EH REWDLQHG G&Gk 3[G f ZKHUH ZH VXEVWLWXWHG GS]fGk S[r 7KH YDOXH RI G6Gk FDQ EH REWDLQHG E\ GLIIHUHQWLDWLQJ WKH 3\WKDJRUHDQ LGHQWLW\ G>6 &@Gk GOfGk DQG VROYLQJ IRU G6Gk G6Gk[ & G&Gkf6 & 3;6 Gf f 'LIIHUHQWLDWLQJ f \LHOGV GGfGk D G G6GkfG 6XEVWLWXWLQJ IURP (T f JLYHV GGGk aD A A A3[p GA Gf f

PAGE 113

DQG IURP f G6JGkA FDQ EH REWDLQHG G6Gk 6 G6Gkf@6 f 2QFH DJDLQ IURP GLIIHUHQWLDWLRQ RI WKH 3\WKDJRUHDQ LGHQWLW\ G&Gk 6 G6Gkf& f DQG GIGkA LV ILQDOO\ REWDLQHG E\ GLIIHUHQWLDWLQJ HTXDWLRQ f GIGH/ D >& G6Gkf 6 G&AGkA @ >G G&Gkf & GGGkfM f %\ XVH RI WKH RQHGLPHQVLRQDO 1HZWRQ5DSKVRQ LWHUDWLYH PHWKRG D QHZ HVWLPDWH IRU kAA LV JLYHQ E\ nQHZ HL IkLf GIGkf 2QFH A LV REWDLQHG WR WKH GHVLUHG DFFXUDF\ WKH UHPDLQLQJ MRLQW YDULDEOHV k k DQG k DUH WKHQ FRPSXWHG IURP WKH YDOXHV RI WKHLU VLQHV DQG FRVLQHV DV REWDLQHG DORQJ ZLWK G IURP WKH ODVW LWHUDWLRQ $ YHFWRU HTXDWLRQ LQ k FDQ EH REWDLQHG IURP f ZKLFK JLYHV 5 5B 5 5O 5 [ f 7KLV HTXDWLRQ FDQ EH VROYHG XQLTXHO\ IRU k

PAGE 114

,Q SUDFWLFDO VLWXDWLRQV WKH GHULYDWLYH RI I FDQ EH HVWLPDWHG E\ GIGH/ > I MA 6kBf Ikf @ kA ZKHUH kA LV D VPDOO LQFUHPHQW RI kA :KHQ WKH YDOXH RI kA LV FORVH HQRXJK WR D VROXWLRQ WKH FRPSOH[LW\ FDQ EH UHGXFHG E\ FRPSXWLQJ GIGkA QXPHULFDOO\ DW DQ\ LWHUDWLRQ XVLQJ WKH YDOXHV RI kA DQG Ikf LQ WKH SUHFHGLQJ LWHUDWLRQ GIGkA ILB ILf4La kLf ZKHUH WKH VXSHUVFULSW UHSUHVHQWV WKH LWHUDWLRQ QXPEHU DW ZKLFK WKH YDULDEOH LV FRPSXWHG 7KLV VDYHV WKH FRPSXWDWLRQDO FRVW RI (TV f WKURXJK f DQG DYRLGV WKH SUREOHP RI VSHFLDO FDVHV WKDW RFFXU ZKHQ GLYLVLRQ E\ D QXPEHU FORVH WR ]HUR LV QHHGHG LQ DQ\ RI WKRVH HTXDWLRQV 7KH SURFHGXUH MXVW GHVFULEHG ZDV SURJUDPPHG WR FRPSXWH WKH MRLQW YDULDEOHV IRU HTXLGLVWDQW SRLQWV RQ D OLQHDU WUDMHFWRU\ ZLWK FRQVWDQW RULHQWDWLRQ WKDW ZLOO PRYH WKH HQG HIIHFWRU IURP WKH LQLWLDO SRVH 3

PAGE 115

WR WKH SRVLWLRQ > @7 7DEOH VKRZV WKH RXWSXW RI WKH SURJUDP 7DEOH *3 WUDMHFWRU\ SRLQWV LQ MRLQW FRRUGLQDWHVf HO p G k LQ k f $OO DQJOHV DUH LQ GHJUHHV k Ff§L n R ,, LQ 77KH PD[LPXP QXPEHU RI LWHUDWLRQV QHHGHG SHU SRLQW ZDV VL[ 7KH JXHVV IRU HDFK SRLQW ZDV WKH YDOXH RI k DW WKH SUHFHGLQJ SRLQW 7KH H[SHULPHQW ZDV VWDUWHG ZLWK D JXHVV RI r &RQYHUJHQFH DW HYHU\ SRLQW ZDV REWDLQHG ZKHQ RU PRUH SRLQWV ZHUH WDNHQ DORQJ WKH WUDMHFWRU\ $OWKRXJK 7DEOH JLYHV WKH MRLQW YDULDEOHV WR RQO\ GHFLPDO SODFHV WKH\ ZHUH FRPSXWHG ZLWK D SUHFLVLRQ RI 7KH SURJUDP ZDV ZULWWHQ LQ & DQG UXQ RQ DQ $7t7 % GHVNWRS FRPSXWHU

PAGE 116

7KH NLQHPDWLF LQYHUVLRQ IRU WKH SRLQWV WRRN VHFRQGV ZKLFK JLYHVDQDYHUDJH WLPH SHU NLQHPDWLF LQYHUVLRQ RI PLOOLVHFRQGV FOHDUO\ VXLWDEOH IRU UHDOWLPH KLJK SUHFLVLRQ LQYHUVH NLQHPDWLFV ([DPSOH 7KH 20 0DQLSXODWRU ,Q WKLV H[DPSOH ZH SUHVHQW DQ RUWKRJRQDO PDQLSXODWRU WKDW FDQ DFKLHYH D JLYHQ HQGHIIHFWRU SRVH LQ VL[WHHQ GLVWLQFW FRQILJXUDWLRQV %HVLGHV WKLV LQWHUHVWLQJ SRLQW ZH GLVFXVV WKLV PDQLSXODWRU WR VKRZ WKH SRZHU RI WKH VLPSOH LWHUDWLYH WHFKQLTXHV GHYHORSHG LQ WKLV GLVVHUWDWLRQ QRW RQO\ DV IDVW LQYHUVH NLQHPDWLF PHWKRGV EXW DOVR DV LQWHUDFWLYH LQYHUVH NLQHPDWLF DQDO\VLV WRROV FDSDEOH RI ILQGLQJ DOO WKH VROXWLRQV IRU FHUWDLQ PDQLSXODWRUV 7KH 20 PDQLSXODWRU LV LOOXVWUDWHG LQ )LJXUH DQG LWV '+SDUDPHWHUV DUH OLVWHG LQ 7DEOH 7DEOH '+SDUDPHWHUV RI 20 PDQLSXODWRU -RLQW G D D R H k DA/ 7 D 7 D 7

PAGE 117

ORJ

PAGE 118

7KLV URERW FRPELQHV WKH IROORZLQJ VSHFLDO VWUXFWXUHV LGHQWLILHG LQ &KDSWHU &DVH LV VDWLVILHG E\ WKH IRXU'2) VHJPHQW ]WR ] &DVH LV VDWLVILHG E\ WKH VHJPHQW ]4 WR ] DQG E\ WKH VHJPHQW ] WR ] &DVH LV VDWLVILHG E\ WKH VHJPHQW WR ] &DVH LV VDWLVILHG E\ WKH VHJPHQW ] WR ] 7KH VWUXFWXUH RI WKLV DUP DOWKRXJK VLPSOH HQRXJK WR DOORZ WKH XVH RI D RQHGLPHQVLRQDO LWHUDWLYH WHFKQLTXH DV VHHQ LQ &KDSWHU f VWLOO GRHV QRW DOORZ FORVHG IRUP VROXWLRQV 2QFH DJDLQ ZH ZLOO LPSOHPHQW DQ LWHUDWLYH WHFKQLTXH EDVHG RQ ILQGLQJ WKH ]HURV RI D UHDO YDOXHG IXQFWLRQ RI A :LWK O NQRZQ LQLWLDO HVWLPDWHf WKH YHFWRUV W ] f DQG &O3; 6O3\ ;S 5S S] VL3[ FO3\ f DUH IXOO\ GHWHUPLQHG ([SUHVVLRQV IRU W] 3= WS DQG fASAS SS DUH JLYHQ LQ WHUPV RI WKH UHPDLQLQJ MRLQW DQJOHV E\ f

PAGE 119

,OO D 6 G f D 6 6 D & DO &f f W S f D & D A DL Af G D 6 DO D & >SS D[ D D G@ f ([SUHVVLRQV IRU W[ AW\ ASA DQG AS\ SURYLGH WKH DGGLWLRQDO HTXDWLRQV f f f f ,I WKH IXQFWLRQ I LV GHILQHG IURP HTXDWLRQ f WKHQ IRU D JLYHQ YDOXH RI kf@B WKH FRUUHVSRQGLQJ YDOXHV RI 6 6 & DQG & PXVW EH FRPSXWHG EHIRUH IkOf D 6 6 D & D &f WS f FDQ EH HYDOXDWHG &RPSXWLQJI *LYHQ MB WKH FRPSRQHQWV RI W DQG S DUH FRPSXWHG IURP (TV f DQG f 7KH YDOXH RI & DQG 6 DUH WKHQ JLYHQ E\ (T f F DQG 6 a X 7ULJ&f

PAGE 120

ZKHUH X LV D VLJQ DPELJXLW\ DQG 7ULJ[f [f (TXDWLRQ f DQG f \LHOG & V W[6 f 6 W] 6 f UHVSHFWLYHO\ ZKHQ 6 LV QRW ]HUR ,Q WKH VSHFLDO FDVH ZKHUH 6 KDV DQ H[WUHPHO\ VPDOO DEVROXWH YDOXH k RU U DQG WKH FXUUHQW YDOXH RI k ZLOO DOORZ FORVHG IRUP VROYLQJ RI WKH UHPDLQLQJ MRLQW YDULDEOHV DQG ILkAf FDQ VWLOO EH FRPSXWHG 1H[W 6 DQG & DUH FRPSXWHG IURP HTXDWLRQ f &2 ,, ;3= a GfD f ,, X X 7ULJ6f f ZKHUH X LV DQRWKHU VLJQ DPELJXLW\ DQG & DQG 6 DUH REWDLQHG IURP f DQG f UHVSHFWLYHO\ F ;S[ D & & DLfD f 6 3] f D F VADr f )LQDOO\ WKH YDOXHV RI k DQG k DUH FRPSXWHG DV k DWDQ6&f DWDQ6&f DQG LL LQ &' DADQ 6&f DWDQ 6&f

PAGE 121

DQG IWkA FDQ EH FRPSXWHG $ IHZ SRLQWV DUH QRWHZRUWK\ LQ WKLV GHULYDWLRQ )LUVW VLQFH WKH DERYH SURFHGXUH FRPSXWHV I IURP D JXHVV kA DQG QRW D YDOXH RI A WKDW FRUUHVSRQGV WR D WUXH VROXWLRQ VHW WKH YDOXHV RI VLQH DQG FRVLQH RI DQ\ DQJOH FRPSXWHG IURP GLIIHUHQW HTXDWLRQV VXFK DV f DQG f IRU k RU f DQG f IRU kf ZLOO LQ JHQHUDO QRW VDWLVI\ WKH 3\WKDJRUHDQ LGHQWLW\ f $Q DOWHUQDWH SRVVLELOLW\ LV WR XVH RQH RI WKH WZR HTXDWLRQV WR VROYH IRU RQH WULJRQRPHWULF IXQFWLRQ HLWKHU VLQH RU FRVLQH FRPSXWH WKH RWKHU IURP HTXDWLRQ f DQG XVH WKH VHFRQG HTXDWLRQ WR DYRLG D VLJQ DPELJXLW\ RQO\ )RU H[DPSOH LQVWHDG RI XVLQJ (T f ZH FDQ FRPSXWH 6 IURP 6 VLQW]Vf 7UL"Ff f WR LQVXUH FRPSDWLELOLW\ ZLWK LGHQWLW\ f $V GLVFXVVHG LQ &KDSWHUV DQG WKH DELOLW\ WR FRPSXWH WKH IXQFWLRQ I4Af SURYHG VXIILFLHQW WR SURYLGH D SUDFWLFDO RQHGLPHQVLRQDO LQYHUVH NLQHPDWLF DOJRULWKP IRU WKH PDQLSXODWRU RI )LJXUH 7KH VLJQ DPELJXLWLHV X DQG X DERYH JLYH IRXU SRVVLEOH FRPELQDWLRQV WKDW PXVW DOO EH WULHG LQ WKH VHDUFK IRU D URRW RI WKH IXQFWLRQ I 2QFH D URRW RI I LV IRXQG WKH YDOXHV RI kA k DQG k DUH NQRZQ DORQJ ZLWK D YDOXH IRU & IURP WKH ODVW LWHUDWLRQ

PAGE 122

$ RQHGLPHQVLRQDO 1HZWRQ5DSKVRQ DOJRULWKP ZDV SURJUDPPHG WR ILQGn WKH ]HURV RI WKH IXQFWLRQ I GHILQHG LQ (T f ZLWK G D D D 2QFH D YDOXH RI kA IRU ZKLFK Ikf ZDV IRXQG WKH FRUUHVSRQGLQJ VROXWLRQ VHW ZDV FRPSOHWHG DQG FKHFNHG IRU FRQVLVWHQF\ E\ YHULI\LQJ WKDW WKH VROXWLRQ VHW VDWLVILHV WKH H[SUHVVLRQ IRU W\ DV REWDLQHG IURP (T f 7KLV VLPSOH WHVW SURYHG HIIHFWLYH LQ ILOWHULQJ RXW H[WUDQHRXV VROXWLRQV IRU WKLV SDUWLFXODU H[SHULPHQW $ PRUH LQYROYHG FRQVLVWHQF\ YHULILFDWLRQ SURFHGXUH PD\ EH UHTXLUHG IRU GLIIHUHQW PDQLSXODWRUV +RZHYHU &RPSXWLQJ WKH FRPSOHWH IRUZDUG NLQHPDWLFV IURP (T f DQG YHULI\LQJ DOO SRVH HOHPHQWV FRQVWLWXWHV D ZRUVW FDVH FRQGLWLRQ ,QYHUVH .LQHPDWLF 6ROXWLRQ 6HDUFK $OJRULWKP 7KH RQHGLPHQVLRQDO LQYHUVH NLQHPDWLF PHWKRG MXVW GHVFULEHG ZDV SURJUDPPHG LQ SDVFDO RQ D SHUVRQDO PLFURFRPSXWHU $ VLPSOH VHDUFK DOJRULWKP ZDV WKHQ LPSOHPHQWHG E\ VHOHFWLQJ UHJXODUO\ VSDFHG YDOXHV RI WKH LQLWLDO HVWLPDWH RI A ZLWKLQ WKH LQWHUYDO > Uf ZLWK WKH SUREOHP SRVH

PAGE 123

7KLV VHDUFK SURJUDP ZDV UXQ IRU HDFK RI WKH IRXU SRVVLEOH FRPELQDWLRQV RI YDOXHV IRU WKH VLJQ DPELJXLWLHV X DQG X f f f DQG ff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f 5HQDXG Df )RU WKH URERW GHVFULEHG KHUH WKH PLGIUDPH -DFRELDQ FRPSXWHG XVLQJ WDEOHV JLYHQ LQ 'RW\ f LV & 6 6 af& & 6 & & 6 6 LQ D &

PAGE 124

7DEOH 0 PDQLSXODWRU FRQILJXUDWLRQV IRU SRVH 3 RI HTXDWLRQ f k HO k k k G

PAGE 125

7KH GHWHUPLQDQW RI WKH PDQLSXODWRU -DFRELDQ GM LV LQGHSHQGHQW RI WKH IUDPH RI H[SUHVVLRQ DQG FDQ EH HDVLO\ REWDLQHG IURP PDWUL[ G& 6 W6 r &A f rn & &6 6 & & 6f` f 7KH YDOXHV RI GM OLVWHG LQ 7DEOH SURYH WKDW DOO VL[WHHQ VROXWLRQV IRXQG FRUUHVSRQG WR QRQGHJHQHUDWH FRQILJXUDWLRQV RI WKH 0 URERW DUP )LJXUH VKRZV SKRWRJUDSKV RI D FRPSXWHU JUDSKLFV VLPXODWLRQ RI WKH 20 PDQLSXODWRU LQ WKH VL[WHHQ FRQILJXUDWLRQV OLVWHG LQ 7DEOH )LJXUH LV D KDQG GUDZLQJ RI WKLV PDQLSXODWRU LQ FRQILJXUDWLRQ RI 7DEOH LH FRUUHVSRQGLQJ WR WKH ILUVW VROXWLRQ VHWf ZLWK DOO OLQN IUDPHV FOHDUO\ LQGLFDWHG DQG WR KHOS GLIIHUHQWLDWH EHWZHHQ VROXWLRQV ZLWK FRPPRQ YDOXHV RI O DQG k ZH KDYH DWWHPSWHG WR LQGLFDWH WKH GLUHFWLRQ RI D[LV YHFWRUV ] ] ] DQG ] RQ WKH SKRWRJUDSKV 7KH SRVLWLRQ DQG RULHQWDWLRQ RI WKH HQGHIIHFWRU DQG WKH EDVH IUDPH DV VKRZQ RQ )LJ f DUH WKH VDPH IRU DOO VL[WHHQ UHSUHVHQWDWLRQV RI )LJXUH ,W LV DOVR LQWHUHVWLQJ WR QRWH WKDW WKLV ODUJH QXPEHU RI VROXWLRQV FDQ EH UHDOL]HG E\ DQ RUWKRJRQDO PDQLSXODWRU ZLWK D IDLUO\ VLPSOH JHRPHWU\ )LQDOO\ WKLV H[DPSOH VKRZV WKDW WKH WHFKQLTXHV GHYHORSHG LQ WKLV GLVVHUWDWLRQ FDQ EH XVHG WR LPSOHPHQW VLPSOH DQG HIILFLHQW LQYHUVH NLQHPDWLF DQDO\VLV WRROV VXFK DV WKH VHDUFK DOJRULWKP MXVW GLVFXVVHG

PAGE 126

f f )LJXUH &RPSXWHU VLPXODWLRQ RI WKH VL[WHHQ FRQILJXUDWLRQV RI 7DEOH

PAGE 127

f f f f )LJXUH f§&RQWLQXHG

PAGE 128

f f f )LJXUH f§&RQWLQXHG

PAGE 129

f f )LJXUH f§&RQWLQXHG

PAGE 130

ZKLFK KDV IRXQG DOO WKH VROXWLRQV WR WKH LQYHUVH NLQHPDWLFV SUREOHP RI WKH 20 PDQLSXODWRU ([DPSOH 20 0DQLSXODWRU 7KH PRVW NLQHPDWLFDOO\ FRPSOH[ RUWKRJRQDO PDQLSXODWRU FODVV LV WKH RFWDO f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f

PAGE 131

7DEOH '+SDUDPHWHUV RI 20 0DQLSXODWRU -RLQW GL kL DL DLUGf k DO 7 G k D 77 G k D ,7 G k D ,7 G k D WW k 7KH VROXWLRQ VHW k k k k k kf r r r r r rf ZDV IRXQG LQ LWHUDWLRQV ZLWK DQ LQLWLDO HVWLPDWH RI f IRU A kf DQG DQ DFFXUDF\ RI 7KH VDPH VROXWLRQ VHW ZDV REWDLQHG LQ LWHUDWLRQV IURP DQ LQLWLDO HVWLPDWH RI r rf ZLWK WKH VDPH SUHFLVLRQ $ VHFRQG VROXWLRQ VHW JLYHQ E\ r r r r r rf ZDV IRXQG ZLWK WKH LQLWLDO HVWLPDWH r rf LQ LWHUDWLRQV ZLWK WKH VDPH DFFXUDF\

PAGE 132

([DPSOH $ *HQHUDO *HRPHWU\ '2) 0DQLSXODWRU 7KH SUHYLRXV H[DPSOHV ZHUH DOO DERXW RUWKRJRQDO PDQLSXODWRUV +HUH ZH H[DPLQH D QRQRUWKRJRQDO PDQLSXODWRU ZLWK QR VSHFLDO VWUXFWXUH DV GHVFULEHG E\ WKH NLQHPDWLF SDUDPHWHUV RI 7DEOH DQG WKH &DUWHVLDQ HQGHIIHFWRU SRVH 3 f 7DEOH '+SDUDPHWHUV RI D QRQRUWKRJRQDO PDQLSXODWRU -RLQW G D D p r &0 f 2 R LQ &' R R r r p r 7KH WZRGLPHQVLRQDO 1HZWRQ5DSKVRQ PHWKRG RI &KDSWHU ZDV SURJUDPPHG WR ILQG D VROXWLRQ VHW :LWK WKH LQLWLDO HVWLPDWH f rf WKH VROXWLRQ VHW r r r r r rf

PAGE 133

ZDV IRXQG LQ LWHUDWLRQV ZLWK DQ DFFXUDF\ RI fA 7KLV UHVXOW LV W\SLFDO RI WKH HIILFLHQF\ RI WKH WZRGLPHQVLRQDO 1HZWRQ5DSKVRQ LQYHUVH NLQHPDWLFV WHFKQLTXH

PAGE 134

&+$37(5 &21&/86,21 $1' )8785( :25. 7KLV UHVHDUFK KDV DGGUHVVHG WKH LQYHUVH NLQHPDWLFV SUREOHP RI QRQUHGXQGDQW DOOUHYROXWH URERW PDQLSXODWRUV :H ILUVW VKRZHG KRZ D FRQYHQLHQW FKRLFH RI PDQLSXODWRU IUDPHV DQG SURSHU XVH RI LQQHUSURGXFW LQYDULDQFH RI URWDWLRQ WUDQVIRUPDWLRQV FDQ EH XVHG WR HDVLO\ UHGXFH WKH LQYHUVH NLQHPDWLF SUREOHP WR IRXU VLPSOH HTXDWLRQV LQGHSHQGHQW RI WZR RI WKH MRLQW YDULDEOHV 'RW\ f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

PAGE 135

,Q WKH DQDO\VLV RI VL[GHJUHHRIIUHHGRP PDQLSXODWRUV ZH LGHQWLILHG WKUHH PDMRU FODVVHV RI '2) DUPV DFFRUGLQJ WR WKHLU NLQHPDWLF FRPSOH[LW\ )LUVW WKH FODVV RI FORVHGIRUP DUPV LQ ZKLFK ZH ILQG DOO '2) RSHQ NLQHPDWLF FKDLQV ZLWK WKUHH DGMDFHQW MRLQW D[HV LQWHUVHFWLQJ DW D FRPPRQ SRLQW 3LHSHU f RU ZLWK WKUHH SDUDOOHO DGMDFHQW MRLQW D[HV :H GLVFRYHUHG WKDW WKH '2) DUPV ZLWK WKUHH SDLUV RI WZR SDUDOOHO D[HV DQG WKRVH ZLWK WZR LQWHUVHFWLQJ D[HV IROORZLQJ RU SUHFHGLQJ WZR SDLUV RI SDUDOOHO MRLQW D[HV KDG FORVHG IRUP VROXWLRQV DV ZHOO 7KH QH[W FODVV FRQWDLQV DOO '2) PDQLSXODWRUV WKDW GR QRW DOORZ FORVHGIRUP VROXWLRQV EXW DUH VXFK WKDW NQRZOHGJH RI RQH MRLQW YDULDEOH LV VXIILFLHQW WR REWDLQ D FRPSOHWH VROXWLRQ VHW LQ FORVHGIRUP :H GHWHUPLQHG WKDW DOO '2) DUPV WKDW LQFOXGH RQH RI WKH WHQ VSHFLDO '2) VWUXFWXUHV GLVFXVVHG HDUOLHU ZHUH LQ WKLV VHFRQG FODVV 7KH LQYHUVH NLQHPDWLFV SUREOHP IRU WKHVH PDQLSXODWRUV UHGXFHV WR ILQGLQJ WKH ]HURV RI D UHDO YDOXHGIXQFWLRQ $ RQHGLPHQVLRQDO 1HZWRQ5DSKVRQ RU RWKHU LWHUDWLYH PHWKRG FDQ EH XVHG WR VROYH WKH LQYHUVH NLQHPDWLFV SUREOHP IRU WKHVH URERWV 7KH WKLUG FODVV RI VL[GHJUHHRIIUHHGRP URERWV FRQWDLQV DOO WKH PDQLSXODWRUV WKDW GR QRW IDOO LQWR WKH WZR SUHFHGLQJ FODVVHV )RU WKH PRVW NLQHPDWLFDOO\ FRPSOH[ VL[ '2) URERW PDQLSXODWRUV WKH LQYHUVH NLQHPDWLFV SUREOHP UHGXFHV WR VROYLQJ D QRQOLQHDU V\VWHP RI RQO\ WZR HTXDWLRQV LQ RQO\ WZR RI WKH MRLQW YDULDEOHV 7KHUHIRUH WKH URERWV

PAGE 136

LQ WKLV WKLUG FODVV FDQ VWLOO EH VROYHG XVLQJ D IDVW WZR GLPHQVLRQDO LWHUDWLYH WHFKQLTXH 7KLV WZRGLPHQVLRQDO PHWKRG VWLOO FRQVWLWXWHV D ODUJH UHGXFWLRQ LQ FRPSXWDWLRQDO FRPSOH[LW\ ZLWK UHVSHFW WR WKH XVXDO VL[GLPHQVLRQDO LWHUDWLYH WHFKQLTXHV IRXQG LQ WKH OLWHUDWXUH 7KH GLYLVLRQ RI WKH VHW RI DOO '2) PDQLSXODWRUV LQWR WKH WKUHH FODVVHV MXVW GLVFXVVHG LV LOOXVWUDWHG LQ )LJXUH ,Q &KDSWHU ZH SURYLGH D IRUPDO GHILQLWLRQ RI RUWKRJRQDO PDQLSXODWRUV 'RW\ f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

PAGE 137

'2) PDQLSXODWRUV VROYDEOH LQ FORVHGIRUP '2) PDQLSXODWRUV IRU ZKLFK WKH RQHGLPHQVLRQDO PHWKRG LV DSSOLFDEOH 6HW RI DOO '2) 0DQLSXODWRUV 6ROYDEOH E\ XVH RI D WZRGLPHQVLRQDO LWHUDWLYH PHWKRG )LJXUH O 6XEGLYLVLRQ RI '2) PDQLSXODWRUV LQ WHUPV RI LQYHUVH NLQHPDWLF WHFKQLTXHV

PAGE 138

UHDVRQV )LUVW WR VKRZ WKDW WKH RQHGLPHQVLRQDO 1HZWRQ 5DSKVRQ DOJRULWKP FDQ SHUIRUP NLQHPDWLF LQYHUVLRQV LQ UHDOn WLPH PLOOLVHFRQGV SHU NLQHPDWLF LQYHUVLRQ RQ DQ $7t7 % GHVNWRS FRPSXWHU WKLV ILJXUH GURSV WR PLOOLVHFRQGV RQ DQ $7t7 % FRPSXWHUf 7KH VHFRQG UHDVRQ LV WKH SUHVHQFH RI D SULVPDWLF MRLQW $OWKRXJK WKLV GLVVHUWDWLRQ ZDV RQO\ FRQFHUQHG ZLWK DOOUHYROXWH PDQLSXODWRUV WKH *3 URERW H[DPSOH VKRZV WKDW WKH WHFKQLTXHV GHYHORSHG KHUHLQ DUH DSSOLFDEOH WR PDQLSXODWRUV ZLWK SULVPDWLF MRLQWV DV ZHOO ,Q H[DPSOH ZH GLVFXVVHG DQ RUWKRJRQDO PDQLSXODWRU RI VLPSOH JHRPHWU\ \HW QRW VLPSOH HQRXJK WR DOORZ FORVHGIRUP VROXWLRQV 7KH 20 URERW LOOXVWUDWHV WKH XVH RI WKH RQHn GLPHQVLRQDO WHFKQLTXH DV DQ RIIOLQH DQDO\VLV WRRO %\ LQWHUDFWLYHO\ YDU\LQJ WKH URERW SDUDPHWHUV DQG WKH HQG HIIHFWRU SRVH SDUDPHWHUV LQ VHDUFK RI D PD[LPXP QXPEHU RI LQYHUVH NLQHPDWLF VROXWLRQV WKH PDQLSXODWRU DQG WKH HQG HIIHFWRU SRVH RI &KDSWHU H[DPSOH ZHUH GLVFRYHUHG ,I WKH ZRUN RI /HH DQG /LDQJ LQ SUHVVf SXWV DQ XSSHU ERXQG RI VL[WHHQ RQ WKH QXPEHU RI SRVVLEOH LQYHUVH NLQHPDWLF VROXWLRQV RI VL['2) URERWV WKH 20 PDQLSXODWRU ZLWK WKH VL[WHHQ VROXWLRQV IRXQG HVWDEOLVKHV DV WKH OHDVW XSSHU ERXQG RQ WKH QXPEHU RI VROXWLRQV WR WKH LQYHUVH NLQHPDWLFV SUREOHP 7KH LWHUDWLYH WHFKQLTXHV GHVFULEHG LQ WKLV GLVVHUWDWLRQ KDYH VHYHUDO DGYDQWDJHV RYHU RWKHU H[LVWLQJ QXPHULFDO

PAGE 139

WHFKQLTXHV 7KH\ GR QRW UHTXLUH WKH FRPSXWDWLRQ RI WKH PDQLSXODWRU -DFRELDQ RU LWV LQYHUVH QRU GR WKH\ UHTXLUH FRPSXWDWLRQ RI WKH IRUZDUG NLQHPDWLFV DW DQ\ WLPH EHIRUH FRQYHUJHQFH LV DFKLHYHG :KHQ WKH LQLWLDO HVWLPDWH RI WKH YDULDEOH LV UHDVRQDEO\ FORVH WR D VROXWLRQ W\SLFDOO\ ZLWKLQ GHJUHHVf FRQYHUJHQFH LVDFKLHYHG ZLWKLQ VL[ ‘ LWHUDWLRQV HYHQ ZLWK DFFXUDFLHV EHWWHU WKDQ ZKLFK PDNHV UHDOWLPH KLJK SUHFLVLRQ LQYHUVH NLQHPDWLFV D UHDOLW\ )LQDOO\ WKH DOJRULWKPV VLPSOLFLW\ LV VXFK WKDW WKH\ FDQ EH SURJUDPPHG DQG H[HFXWHG YHU\ UDSLGO\ HYHQ RQ D SHUVRQDO PLFURFRPSXWHU IRU DQ\ PDQLSXODWRU ZLWK DW PRVW VL[ GHJUHHV RI IUHHGRP 7KLV SURJUDPPLQJ VLPSOLFLW\ PDNHV WKH RQH DQG WZR GLPHQVLRQDO LWHUDWLYH WHFKQLTXHV VXLWDEOH IRU XVH DV RIIn OLQH LQWHUDFWLYH LQYHUVH NLQHPDWLF WRROV DV ZHOO 7KH GLVDGYDQWDJHV RI WKH LQYHUVH NLQHPDWLFV PHWKRGV GHYHORSHG KHUH DUH WKRVH LQKHUHQW WR LWHUDWLYH DOJRULWKPV 1R WKHRUHWLFDO JXDUDQWHH RI FRQYHUJHQFH FDQ EH JLYHQ DQG WKH QXPEHU RI LWHUDWLRQV UHTXLUHG IRU FRQYHUJHQFH GHSHQGV KLJKO\ RQ WKH LQLWLDO HVWLPDWH RI WKH YDULDEOHV $ GLVDGYDQWDJH RYHU WKH KRPRWRS\ PDS PHWKRG 7VDL DQG 0RUJDQ f LV WKDW FRQYHUJHQFH WR RQO\ RQH VROXWLRQ LV SRVVLEOH DJDLQ D FRPPRQ GLVDGYDQWDJH RI LWHUDWLYH WHFKQLTXHV 6HDUFKLQJ IRU PRUH WKDQ RQH VROXWLRQ UHTXLUHV WU\LQJ YDULRXV HVWLPDWHV RI WKH YDULDEOHV 2QH SUREOHP WKDW WKH PHWKRGV GHYHORSHG LQ WKLV WH[W VKDUH ZLWK 7VDL DQG 0RUJDQnV KRPRWRS\ PDS PHWKRG f LV

PAGE 140

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

PAGE 141

$33(1',; 62/9,1* )25 ,Q WKH LQYHUVH NLQHPDWLF VROXWLRQ IRU IRXUGHJUHHRI IUHHGRP URERW PDQLSXODWRUV GHVFULEHG LQ &KDSWHU DIWHU VROYLQJ IRU A DQG k ZH VDLG WKDW WKH DQJOH k FDQ EH REWDLQHG E\ VROYLQJ RQH WKH WZR OLQHDU V\VWHPV RI HTXDWLRQV LQ 6 DQG & JLYHQ E\ (TV f DQG f RU (TV f DQG f ,Q WKLV $SSHQGL[ ZH VKRZ WKDW k FDQ DOZD\V EH FRPSXWHG IURP RQH RI WKRVH WZR V\VWHPV ZKHQ WKH '2) DUP LV QRW LQ D GHJHQHUDWH FRQILJXUDWLRQ 7KH DEVROXWH YDOXH RI WKH GHWHUPLQDQW RI WKH V\VWHP RI (TV f DQG f 'A LV JLYHQ E\ A rUFA 7DAA D D&nA $Of DQG LQ DEVROXWH YDOXH WKH GHWHUPLQDQW RI WKH V\VWHP RI (TV f DQG f \LHOGV f§ AU WFA AAfA $f $ YDOXH RI k LV REWDLQHG DV ORQJ DV 'A DQG DUH QRW VLPXOWDQHRXVO\ HTXDO WR ]HUR +RZHYHU ERWK GHWHUPLQDQWV HTXDO ]HUR UHTXLUHV WKDW WKH IRXU FRQGLWLRQV DA f 7DA a $f D D& $f

PAGE 142

R $f $f D7 7D& D6 2 EH VLPXOWDQHRXVO\ VDWLVILHG :H QRZ H[DPLQH WKH FRQVHTXHQFHV RI KDYLQJ DOO IRXU FRQGLWLRQV VDWLVILHG &RQGLWLRQ $f UHTXLUHV WKDW D RU 6 &DVH X DQG 6 WKHQ U sO DQG IURP FRQGLWLRQ $f ZH VHH WKDW R PXVW EH HTXDO WR DQG U sOf &RQGLWLRQ $f WKHQ UHTXLUHV WKDW D EXW D D PHDQV WKDW MRLQW D[HV DQG FRLQFLGH DQG WKH DUP LV GHJHQHUDWH &DVH D 6 WKHQ & sO DQG DV LQ WKH SUHYLRXV FDVH X &RQGLWLRQ $f WKHQ EHFRPHV DD LI & O RU DfD LI & O :KHQ & O k f WKHQ DD PHDQV WKDW D D VLQFH ERWK OLQN OHQJWKV D DQG D DUH QRQQHJDWLYH QXPEHUV 7KHUHIRUH WKLV FDVH PHDQV WKDW D D DQG D D VR WKDW MRLQW D[HV DQG FRLQFLGH DQG WKH DUP LV GHJHQHUDWH :KHQ & O k Uf WKHQ DaD DQG D D 6LQFH D D D[HV DQG DUH SDUDOOHO 7KH YDOXHV D D DQFA k U IRUFH MRLQW D[HV DQG WR EH DOLJQHG WKHUHE\ IRUFLQJ WKH DUP LQ D GHJHQHUDWH FRQILJXUDWLRQ &DVH 6 DQG FU IR 7KLV FDVH RFFXUV ZKHQ k RU ZKHQ k 7 :KHQ k WKHQ & O DQG FRQGLWLRQ $f \LHOGV D D DV LQ WKH SUHYLRXV FDVH &RQGLWLRQ $f \LHOGV D RU G 6LQFH D WKH FDVH FU PHDQV WKDW MRLQW D[HV DQG DUH DOLJQHG DQG WKH DUP LV GHJHQHUDWH $VVXPLQJ G DQG

PAGE 143

D &RQGLWLRQ $f JLYHV FWW WFW RU VLQDDf 7KH YDOXHV DcD RU LU DQG G D D PHDQ WKDW MRLQW D[HV DQG DUH DOLJQHG VR WKH DUP LV GHJHQHUDWH :KHQ k U WKHQ & O DQG &RQGLWLRQ $f IRUFHV D D ZKLOH &RQGLWLRQ $f QRZ EHFRPHV D77&7 RU VLQDDf 7KH FRPELQDWLRQ RI YDOXHV DDf D D DQG k 7 DJDLQ IRUFH D[LV WR DOLJQ ZLWK D[LV ZKLFK SXWV WKH DUP LQ D GHJHQHUDWH FRQILJXUDWLRQ 6LQFH WKH GHWHUPLQDQWV DQG DUH VLPXOWDQHRXVO\ ]HUR RQO\ ZKHQ WKH PDQLSXODWRU LV GHJHQHUDWH :H FDQ FRQFOXGH WKDW D XQLTXH YDOXH RI k FDQ DOZD\V EH FRPSXWHG DV LQGLFDWHG LQ &KDSWHU ZKHQ WKH DUP LV QRQGHJHQHUDWH

PAGE 144

5()(5(1&(6 $QJHOHV 2Q WKH QXPHULFDO VROXWLRQ WR WKH LQYHUVH NLQHPDWLFV SUREOHPn ,QW 5RERWLFV 5HV a f a $QJHOHV ,WHUDWLYH NLQHPDWLF LQYHUVLRQ RI JHQHUDO ILYHD[LV URERW PDQLSXODWRUV ,QW 5RERWLFV 5HV f &UDLJ -,QWURGXFWLRQ WR URERWLFV PHFKDQLFV DQG FRQWURO $GGLVRQ:HVOH\ 5HDGLQJ 0DVVDFKXVHWWV 'HQDYLW DQG +DUWHQEHUJ 56 $ NLQHPDWLF QRWDWLRQ IRU ORZHUSDLU PHFKDQLVPV EDVHG XSRQ PDWULFHV $60( $SSO 0HFK 'RW\ / 0DFKLQH ,QWHOOLJHQFH /DE UHSRUW 0,/ f (OHFWULFDO (QJLQHHULQJ 'HSW 8QLYHUVLW\ RI )ORULGD *DLQHVYLOOH )ORULGD 'RW\ / 7DEXODWLRQ RI WKH V\PEROLF PLGIUDPH -DFRELDQ RI D URERW PDQLSXODWRU ,QW 5RERWLFV 5HV f 'XII\ $QDO\VLV RI PHFKDQLVPV DQG URERW PDQLSXODWRUV -RKQ :LOH\ 1HZ
PAGE 145

*ROGHQEHUJ $$ %HQKDELE % DQG )HQWRQ 5* $ FRPSOHWH JHQHUDOL]HG VROXWLRQ WR WKH LQYHUVH NLQHPDWLFV RI URERWV ,((( RI 5RERWLFV DQG $XWR 5$ *ROGHQEHUJ $ $ DQG /DZUHQFH '/ $ JHQHUDOL]HG VROXWLRQ WR WKH LQYHUVH NLQHPDWLFV RI URERWLF PDQLSXODWRUV ,((( RI '\QDPLF 6\VWHPV 0HDVXUHPHQW DQG &RQWURO +ROOHUEDFK -0 DQG 6DKDU :ULVWSDUWLWLRQHG LQYHUVH NLQHPDWLF DFFHOHUDWLRQV a DQG PDQLSXODWRU G\QDPLFV ,QW 5RERWLFV 5HV f .D]HURXQLDQ 2Q WKH QXPHULFDO LQYHUVH NLQHPDWLFV RI URERWLF PDQLSXODWRUV 7UDQV $60( 0HFK 7UDQV DQG $XWR LQ 'HV /HH &6* DQG =LHJOHU 0 $ JHRPHWULF DSSURDFK LQ VROYLQJ WKH LQYHUVH NLQHPDWLFV RI 380$ URERWV ,((( 7UDQV $HURVSDFH DQG (OHFWURQLF 6\VWHPV $(6f /HH + < DQG /LDQJ & LQ SUHVVf 'LVSODFHPHQW DQDO\VLV RI WKH JHQHUDO VSDWLDO /LQN 5 PHFKDQLVP 0HFK 0DFK 7KHRU\ /LQDUHV DQG 3DJH $ 3RVLWLRQ DQDO\VLV RI VSHFLDO PHFKDQLVPV 7UDQV $60( 0HFK 7UDQV DQG $XWR LQ 'HV /RZ .+ DQG 'XEH\ 51 $ FRPSDUDWLYH VWXG\ RI JHQHUDOL]HG FRRUGLQDWHV IRU VROYLQJ WKH LQYHUVH NLQHPDWLFV SUREOHP RI D 5 URERW PDQLSXODWRU ,QW 5RERWLFV 5HV f /XPHOVN\ 9,WHUDWLYH FRRUGLQDWH WUDQVIRUPDWLRQ SURFHGXUH IRU RQH FODVV RI URERWV ,((( 7UDQV 6\V 0DQ DQG &\EHU 60&f 0DQVHXU 5 DQG 'RW\ ./ $ IDVW DOJRULWKP IRU LQYHUVH NLQHPDWLF DQDO\VLV RI URERW PDQLSXODWRUV ,QW 5RERWLFV 5HV f 0DQVHXU 5 DQG 'RW\ ./ LQ SUHVVf $ URERW PDQLSXODWRU ZLWK UHDO LQYHUVH NLQHPDWLF VROXWLRQ VHWV ,QW 5RERWLFV 5HV 3DXO 53 5RERW PDQLSXODWRUV PDWKHPDWLFV SURJUDPPLQJ DQG FRQWURO 0,7 3UHVV &DPEULGJH 0DVVDFKXVHWWV

PAGE 146

3DXO 53 DQG =KDQJ + &RPSXWDWLRQDOO\ HIILFLHQW NLQHPDWLFVIRU PDQLSXODWRUVBZLWK VSKHULFDO ZULVWV EDVHG RQ WKHKRPRJHQHRXV WUDQVIRUPDWLRQ UHSUHVHQWDWLRQ ,QW 5RERWLFV 5HV f 3LHSHU '/ 7KH NLQHPDWLFV RI PDQLSXODWRUV XQGHU FRPSXWHU FRQWURO 3K' 'LVVHUWDWLRQ 6WDQIRUG 8QLYHUVLW\ 6WDQIRUG &DOLIRUQLD 5HQDXG 0 D &DOFXO GH OD PDWULFH MDFRELHQQH QHFHVVDLUH DUOD FRPPDQGH FRRUGRQQHH Gn XQ PDQLSXODWHXU 0HFK 0DFK 7KHRU\ f 5HQDXG 0 E &RQWULEXWLRQ D OD PRGHOLVDWLRQ HW D OD FRPPDQGH G\QDPLTXH GHV URERWV PDQLSXODWHXUV 7KHVH GH 'RFWHXU Gn(WDW 8QLYHUVLWH 3DXO 6DEDWLHU GH 7RXORXVH 6FLHQFHVf 7RXORXVH )UDQFH 5RWK % 5DVWHJDU DQG 6KHLQPDQ 9 2Q WKH GHVLJQ RI FRPSXWHU FRQWUROOHG PDQLSXODWRUV 2Q WKH WKHRU\ DQG SUDFWLFH RI URERWV DQG PDQLSXODWRUV 6SULQJHU9HUODJ 1HZ
PAGE 147

%,2*5$3+,&$/ 6.(7&+ a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

PAGE 148

, FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI?eKLORVRSK\ n‹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 FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ f§F\A\IF]eA FI a *HUIIDUG ; 5LWWHU ;3URIHVVRU RI &RPSXWHU DQG A ,QIRUPDWLRQ 6FLHQFHV FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ SL 6HOMIULF 5DOSK S 6HOMIULG5MH 3URIHVVRU RI &RPSXWHU DQG ,QIRUPDWLRQ 6FLHQFHV

PAGE 149

7KLV GLVVHUWDWLRQ ZDV VXEPLWWHG WR WKH *UDGXDWH )DFXOW\ RI WKH &ROOHJH RI (QJLQHHULQJDQGWR WKH*UDGXDWH 6FKRRO DQG :DV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ $XJXVW 'HDQ &ROOHJH RI (QJLQHHULQJ 'HDQ *UDGXDWH 6FKRRO

PAGE 150

,QWHUQHW 'LVWULEXWLRQ &RQVHQW $JUHHPHQW r n ,Q UHIHUHQFH WR WKH IROORZLQJ GLVVHUWDWLRQ $87+25 0DQVHXU 5DFKLG 7,7/( ,QYHUVH .LQHPDWLF $QDO\VLV RI 5RERW 0DQLSXODWRUV 38%/,&$7,21 '$7( -F c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f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