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UMFPACK Version 4.1 Quick Start Guide

Timothy A. Davis
Dept. of Computer and Information Science and Engineering
Univ. of Florida, Gainesville, FL

April 30, 2003


Abstract
UMFPACK is a set of routines for solving unsymmetric sparse linear systems, Ax = b,
using the Unsymmetric-pattern MultiFrontal method and direct sparse LU factorization. It is
written in ANSI/ISO C, with a MATLAB interface. UMFPACK relies on the Level-3 Basic
Linear Algebra Subprograms (dense matrix multiply) for its performance. This code works on
Windows and many versions of Unix (Sun Solaris, Red Hat Linux, IBM AIX, SGI IRIX, and
Compaq Alpha). This is a "quick -i n i guide for Unix users of the C interface.

Technical Report TR-03-009.
UMFPACK Version 4.1 (Apr. 30, 2003), Copyright@2003 by Timothy A. Davis. All Rights
Reserved. Refer to the UMFPACK V4.1 User Guide for the License. See
http://www.cise.ufl.edu/research/sparse/umfpack for the code and full documentation.


1 Overview

UMFPACK Version 4.1 is a set of routines for solving systems of linear equations, Ax = b, when
A is sparse and unsymmetric. The sparse matrix A can be square or rectangular, singular or
non-singular, and real or complex (or any combination). Only square matrices A can be used to
solve Ax = b or related systems. Rectangular matrices can only be factorized.
UMFPACK 4.0 is a built-in routine in MATLAB 6.5, used by the forward and backlash oper-
ator, and the lu routine. The following is a short introduction to Unix users of the C interface of
UMFPACK Version 4.1.
The C-callable UMFPACK library consists of 31 user-callable routines and one include file.
Twenty-seven of the routines come in four versions, with different sizes of integers and for real
or complex floating-point numbers. This Quick Start Guide assumes you are working with real
matrices (not complex) and with into's as integers (not long's). Refer to the User Guide for
information about the complex and long integer versions. The include file umfpack.h must be
included in any C program that uses UMFPACK.


2 Primary routines, and a simple example

Five primary UMFPACK routines are required to factorize A or solve Ax = b. An overview
of the primary features of the routines is given in Section 5. Additional routines are available
for passing a different column ordering to UMFPACK, changing default parameters, manipulating
sparse matrices, getting the LU factors, save and loading the LU factors from a file, and reporting
results. See the User Guide for more information.














umfpack_di_symbolic:
Pre-orders the columns of A to reduce fill-in and performs a l-, il,.lii. ;.I I, i-- Returns an
opaque Symbolic object as a void pointer. The object contains the E-, i,_li ;i .1,-i, and
is needed for the numerical factorization.

umfpack_di_numeric:
Numerically scales and then factorizes a sparse matrix PAQ, PRAQ, or PR-1AQ into the
product LU, where P and Q are permutation matrices, R is a diagonal matrix of scale factors,
L is lower triangular with unit diagonal, and U is upper triangular. Requires the symbolic
ordering and .111 -i, computed by umfpack_di_symbolic. Returns an opaque Numeric ob-
ject as a void pointer. The object contains the numerical factorization and is used by
umfpack_di_solve.

umfpack_di_solve:
Solves a sparse linear -, -I. 1 (Ax = b, ATx = b, or systems involving just L or U), using
the numeric factorization computed by umfpack_dinumeric.

umfpack_di_free_symbolic:
Frees the Symbolic object created by umfpack_di_symbolic.

umfpack_di_free_numeric:
Frees the Numeric object created by umfpack_di_numeric.

The matrix A is represented in compressed column form, which is identical to the sparse matrix
representation used by MATLAB. It consists of three arrays, where the matrix is m-by-n, with nz
entries:

int Ap [n+1] ;
int Ai [nz] ;
double Ax [nz] ;

All nonzeros are entries, but an entry may be numerically zero. The row indices of entries in
column j are stored in Ai [Ap[j] ... Ap[j+1]-1]. The corresponding numerical values are stored
in AxEApEj] ... Ap[j+1]-1].
No duplicate row indices may be present, and the row indices in any given column must be
sorted in ascending order. The first entry Ap [0] must be zero. The total number of entries in the
matrix is thus nz = Ap En]. Except for the fact that extra zero entries can be included, there is
thus a unique compressed column representation of any given matrix A.
Here is a simple main program, umfpack_simple.c, that illustrates the basic usage of UMF-
PACK.

#include
#include "umfpack.h"

int n = 5 ;
int Ap [ ] = {0, 2, 5, 9, 10, 12} ;
int Ai [ ] = { 0, 1, 0, 2, 4, 1, 2, 3, 4, 2, 1, 4} ;
double Ax [ ] = {2., 3., 3., -1., 4., 4., -3., 1., 2., 2., 6., 1.} ;
double b [ ] = {8., 45., -3., 3., 19.} ;
double x [5] ;















int main (void)
{
double *null = (double *) NULL ;
int i ;
void *Symbolic, *Numeric ;
(void) umfpack_di_symbolic (n, n, Ap, Ai, Ax, &Symbolic, null, null)
(void) umfpack_di_numeric (Ap, Ai, Ax, Symbolic, &Numeric, null, null)
umfpackdifree_symbolic (&Symbolic) ;
(void) umfpack_di_solve (UMFPACK_A, Ap, Ai, Ax, x, b, Numeric, null, null)
umfpackdifree_numeric (&Numeric) ;
for (i = 0 ; i < n ; i++) printf ("x [%d] = %g\n", i, x [i]) ;
return (0) ;
I

The Ap, Ai, and Ax arrays represent the matrix


2
3
A= 0
0
0


3 0
0 4


0 0
0 6
2 0
0 0
0 1


and the solution is x = [12 3 4 5]T. The program uses default control settings and does not return
any statistics about the ordering, factorization, or solution (Control and Info are both (double
*) NULL).
For routines to manipulate a simpler "triplet-form" data structure for your sparse matrix A,
refer to the UMFPACK V4.1 User Guide.


3 Synopsis of primary C-callable routines

The matrix A is m-by-n with nz entries. The optional umfpack_di_defaults routine loads the
default control parameters into the Control array. The settings can then be modified before
passing the array to the other routines. Refer to Section 5 for more details.

#include "umfpack.h"
int status, sys, n, m, nz, Ap [n+1], Ai [nz] ;
double Control [UMFPACK_CONTROL], Info [UMFPACK_INFO], Ax [nz], X [n], B [n]
void *Symbolic, *Numeric ;


status
status =
status
umfpack_
umfpack_
umfpack_


umfpack_di_symbolic (m, n, Ap, Ai, Ax, &Symbolic, Control, Info) ;
umfpack_di_numeric (Ap, Ai, Ax, Symbolic, &Numeric, Control, Info)
umfpack_di_solve (sys, Ap, Ai, Ax, X, B, Numeric, Control, Info)
di_free_symbolic (&Symbolic)
di_free_numeric (&Numeric)
di_defaults (Control) ;


4 Installation

You will need to install both UMFPACK v4.1 and AMD vl.0 to use UMFPACK. The UMFPACK
and AMD subdirectories must be placed side-by-side within the same parent directory. AMD is
a stand-alone package that is required by UMFPACK. UMFPACK can be compiled without the
BLAS but your performance will be much less than what it should be.













System-dependent configurations are in the AMD/Make and UMFPACK/Make directories. You can
edit the Make. include files in those directories to customize the compilation. The default settings
will work on most systems, except that UMFPACK will be compiled so that it does not use the
BLAS. Sample configuration files are provided for Linux, Sun Solaris, SGI IRIX, IBM AIX, and
the DEC/Compaq Alpha.
To compile and install both packages, go to the UMFPACK directory and type make. This will
compile the libraries (AMD/Lib/libamd.a and UMFPACK/Lib/libumfpack.a). A demo of the AMD
ordering routine will be compiled and tested in the AMD/Demo directory, and five demo programs will
then be compiled and tested in the UMFPACK/Demo directory. The outputs of these demo programs
will then be compared with output files in the distribution. Expect to see a few differences, such
as residual norms, compile-time control settings, and perhaps memory usage differences. The
AMD and MATLAB mexFunctions for use in MATLAB will also be compiled. If you do not have
MATLAB, type make lib instead.
If you compile UMFPACK and AMD and then later change the Make. include file or your
system-specific configuration file such as Make. linux, then you should type make purge and then
make to recompile.
Here are the various parameters that you can control in your Make. include file:

CC = your C compiler, such as cc.

RANLIB = your --,-I i 's ranlib program, if needed.

CFLAGS = optimization flags, such as -0.

CONFIG = configuration settings, for the BLAS, memory allocation routines, and timing
routines.

LIB = your libraries, such as -lm or -lblas.

RM = the command to delete a file.

MV = the command to rename a file.

MEX = the command to compile a MATLAB mexFunction.

F77 = the command to compile a Fortran program (optional).

F77FLAGS = the Fortran compiler flags (optional).

F77LIB = the Fortran libraries (optional).

The CONFIG string can include combinations of the f .11,. h.in. most deal with how the BLAS
are called:

-DNBLAS if you do not have any BLAS at all.

-DCBLAS if you have the C-BLAS.

-DNSUNPERF if you are on Solaris but do not have the Sun Performance Library.

-DNSCSL if you on SGI IRIX but do not have the SCSL BLAS library.

-DLONGBLAS if your BLAS can take long integer input arguments.













Options for controlling how C calls the Fortran BLAS: -DBLAS_BY_VALUE, -DBLASNO_UNDERSCORE,
and -DBLAS_CHAR_ARG. These are set automatically for Windows, Sun Solaris, SGI Irix, Red
Hat Linux, Compaq Alpha, and AIX (the IBM RS 6000).

-DGETRUSAGE if you have the getrusage function.

-DLP64 if you are compiling in the LP64 model (32 bit into's, 64 bit long's, and 64 bit
pointers).

-DNUTIL if you wish to compile the MATLAB-callable UMFPACK mexFunction with the
mxMalloc, mxRealloc and mxFree routines, instead of the undocumented (but superior)
utMalloc, utRealloc, and utFree routines.

-DNPOSIX if you do not have the POSIX-compliant sysconf and times routines.

-DNRECIPROCAL controls a trade-off between speed and accuracy. This is normally off by
default, except when the gcc compiler is used.

When you compile your program that uses the C-callable UMFPACK library, you need to
add the both UMFPACK/Lib/libumfpack. a and AMD/Lib/libamd.a libraries, and you need to tell
your compiler to look in the directories UMFPACK/Include and AMD/Include for include files. See
UMFPACK/Demo/Makefile for an example. You do not need to directly include any AMD include
files in your program, unless you directly call AMD routines. You only need the

#include "umfpack.h"

statement, as described in Section 3.
















5 The primary UMFPACK routines


5.1 umfpack_di_symbolic

int umfpack_di_symbolic


int n_row,
int n_col,
const int Ap [ ],
const int Ai [ ],
const double Ax [ ],
void **Symbolic,
const double Control
double Info [UMFPACK_


[UMFPACK_CONTROL],
INFO]


Purpose:

Given nonzero pattern of a sparse matrix A in column-oriented form,
umfpack_di_symbolic performs a column pre-ordering to reduce fill-in
(using COLAMD or AMD) and a symbolic factorization. This is required
before the matrix can be numerically factorized with umfpack_di_numeric.

For the following discussion, let S be the submatrix of A obtained after
eliminating all pivots of zero Markowitz cost. S has dimension
(n_row-nl-nempty_row) -by- (n_col-nl-nempty_col), where
nl = Info [UMFPACK_COL_SINGLETONS] + Info [UMFPACK_ROW_SINGLETONS],
nempty_row = Info [UMFPACK_NEMPTY_ROW] and
nempty_col = Info [UMFPACK_NEMPTY_COL].

Returns:

The status code is returned. See Info [UMFPACK_STATUS], below.

Arguments:


Int nrow
Int ncol


Input argument, not modified.
Input argument, not modified.


A is an n_row-by-n_col matrix. Restriction: n_row > 0 and n_col > 0.

Int Ap [n_col+l] ; Input argument, not modified.

Ap is an integer array of size n_col+1. On input, it holds the
"pointers" for the column form of the sparse matrix A. Column j of
the matrix A is held in Ai [(Ap [j]) ... (Ap [j+1]-1)]. The first
entry, Ap [0], must be zero, and Ap [j] <= Ap [j+1] must hold for all
j in the range 0 to n_col-l. The value nz = Ap [n_col] is thus the
total number of entries in the pattern of the matrix A. nz must be
greater than or equal to zero.


Int Ai [nz] ;


Input argument, not modified, of size nz = Ap [n_col].


The nonzero pattern (row indices) for column j is stored in
Ai [(Ap [j]) ... (Ap [j+1]-1)]. The row indices in a given column j
must be in ascending order, and no duplicate row indices may be present.
Row indices must be in the range 0 to n_row-1 (the matrix is O-based).


Optional input argument, not modified.


double Ax [nz] ;


















The numerical values of the sparse matrix A. The nonzero pattern (row
indices) for column j is stored in Ai [(Ap [j]) ... (Ap [j+1]-1)], and
the corresponding numerical values are stored in
Ax [(Ap [j]) ... (Ap [j+1]-1)]. Used only by the 2-by-2 strategy to
determine whether entries are "large" or "small". You do not have to
pass the same numerical values to umfpackdinumeric. If Ax is not
present (a (double *) NULL pointer), then any entry in A is assumed to
be "large".

void **Symbolic ; Output argument.

**Symbolic is the address of a (void *) pointer variable in the user's
calling routine (see Syntax, above). On input, the contents of this
variable are not defined. On output, this variable holds a (void *)
pointer to the Symbolic object (if successful), or (void *) NULL if
a failure occurred.

double Control [UMFPACK_CONTROL] ; Input argument, not modified.

If a (double *) NULL pointer is passed, then the default control
settings are used. Only the primary parameters are listed below:

Control [UMFPACK_STRATEGY]: This is the most important control
parameter. It determines what kind of ordering and pivoting
strategy that UMFPACK should use. It is new to Version 4.1
There are 4 options:

UMFPACK_STRATEGY_AUTO: This is the default. The input matrix is
analyzed to determine how symmetric the nonzero pattern is, and
how many entries there are on the diagonal. It then selects one
of the following strategies. Refer to the User Guide for a
description of how the strategy is automatically selected.

UMFPACK_STRATEGY_UNSYMMETRIC: Use the unsymmetric strategy. COLAMD
is used to order the columns of A, followed by a postorder of
the column elimination tree. No attempt is made to perform
diagonal pivoting. The column ordering is refined during
factorization. This strategy was the only one provided with
UMFPACK V4.0.

In the numerical factorization, the
Control [UMFPACK_SYM_PIVOT_TOLERANCE] parameter is ignored. A
pivot is selected if its magnitude is >=
Control [UMFPACK_PIVOT_TOLERANCE] (default 0.1) times the
largest entry in its column.

UMFPACK_STRATEGY_SYMMETRIC: Use the symmetric strategy (new to
Version 4.1). In this method, the approximate minimum degree
ordering (AMD) is applied to A+A', followed by a postorder of
the elimination tree of A+A'. UMFPACK attempts to perform
diagonal pivoting during numerical factorization. No refinement
of the column preordering is performed during factorization.

In the numerical factorization, a nonzero entry on the diagonal
is selected as the pivot if its magnitude is >= Control
[UMFPACK_SYM_PIVOT_TOLERANCE] (default 0.001) times the largest
entry in its column. If this is not acceptable, then an
off-diagonal pivot is selected with magnitude >= Control

















[UMFPACK_PIVOT_TOLERANCE] (default 0.1) times the largest entry
in its column.

UMFPACK_STRATEGY_2BY2: a row permutation P2 is found that places
large entries on the diagonal. The matrix P2*A is then
factorized using the symmetric strategy, described above.
Refer to the User Guide for more information.

Control [UMFPACK_2BY2_TOLERANCE]: a diagonal entry S (k,k) is
considered "small" if it is < tol max (abs (S (:,k))), where S a
submatrix of the scaled input matrix, with pivots of zero Markowitz
cost removed.

Control [UMFPACK_SCALE]: This parameter is new to V4.1. See
umfpack_numeric.h for a description. Only affects the 2-by-2
strategy. Default: UMFPACK_SCALE_SUM.

double Info [UMFPACK_INFO] ; Output argument, not defined on input.

Contains statistics about the symbolic analysis. If a (double *) NULL
pointer is passed, then no statistics are returned in Info (this is not
an error condition). The entire Info array is cleared (all entries set
to -1) and then the following statistics are computed (only the
primary statistics are listed):

Info [UMFPACK_STATUS]: status code. This is also the return value,
whether or not Info is present.

UMFPACK OK

Each column of the input matrix contained row indices
in increasing order, with no duplicates. Only in this case
does umfpack_di_symbolic compute a valid symbolic factorization.
For the other cases below, no Symbolic object is created
(*Symbolic is (void *) NULL).

UMFPACK_ERROR_n_nonpositive

n is less than or equal to zero.

UMFPACK ERROR invalid matrix

Number of entries in the matrix is negative, Ap [0] is nonzero,
a column has a negative number of entries, a row index is out of
bounds, or the columns of input matrix were jumbled (unsorted
columns or duplicate entries).

UMFPACK_ERRORout of memory

Insufficient memory to perform the symbolic analysis. If the
analysis requires more than 2GB of memory and you are using
the 32-bit ("int") version of UMFPACK, then you are guaranteed
to run out of memory. Try using the 64-bit version of UMFPACK.

UMFPACK_ERROR_argument_missing

One or more required arguments is missing.

UMFPACK ERROR internal error


















Something very serious went wrong. This is a bug.
Please contact the author (davis@cise.ufl.edu).

Info [UMFPACK_SIZEOFUNIT]: the number of bytes in a Unit,
for memory usage statistics below.

Info [UMFPACK_SYMBOLIC_PEAK_MEMORY]: the amount of memory (in Units)
required for umfpack_di_symbolic to complete. This count includes
the size of the Symbolic object itself, which is also reported in
Info [UMFPACK_SYMBOLIC_SIZE].

Info [UMFPACK_NUMERIC_SIZE_ESTIMATE]: an estimate of the final size (in
Units) of the entire Numeric object (both fixed-size and variable-
sized parts), which holds the LU factorization (including the L, U,
P and Q matrices).

Info [UMFPACK_PEAK_MEMORY_ESTIMATE]: an estimate of the total amount of
memory (in Units) required by umfpack_di_symbolic and
umfpackdinumeric to perform both the symbolic and numeric
factorization. This is the larger of the amount of memory needed
in umfpack_di_numeric itself, and the amount of memory needed in
umfpack_di_symbolic (Info [UMFPACK_SYMBOLIC_PEAK_MEMORY]). The
count includes the size of both the Symbolic and Numeric objects
themselves. It can be a very loose upper bound, particularly when
the symmetric or 2-by-2 strategies are used.

Info [UMFPACK_FLOPS_ESTIMATE]: an estimate of the total floating-point
operations required to factorize the matrix. This is a "true"
theoretical estimate of the number of flops that would be performed
by a flop-parsimonious sparse LU algorithm. It assumes that no
extra flops are performed except for what is strictly required to
compute the LU factorization. It ignores, for example, the flops
performed by umfpackdinumeric to add contribution blocks of
frontal matrices together. If L and U are the upper bound on the
pattern of the factors, then this flop count estimate can be
represented in MATLAB (for real matrices, not complex) as:

Lnz = full (sum (spones (L))) 1 ; % nz in each col of L
Unz = full (sum (spones (U')))' 1 ; % nz in each row of U
flops = 2*Lnz*Unz + sum (Lnz) ;

The actual "true flop" count found by umfpackdinumeric will be
less than this estimate.

Info [UMFPACK_LNZ_ESTIMATE]: an estimate of the number of nonzeros in
L, including the diagonal. Since L is unit-diagonal, the diagonal
of L is not stored. This estimate is a strict upper bound on the
actual nonzeros in L to be computed by umfpack_di_numeric.

Info [UMFPACK_UNZ_ESTIMATE]: an estimate of the number of nonzeros in
U, including the diagonal. This estimate is a strict upper bound on
the actual nonzeros in U to be computed by umfpack_di_numeric.

Info [UMFPACK_SYMBOLIC_TIME]: The CPU time taken, in seconds.

Info [UMFPACK_STRATEGY_USED]: The ordering strategy used:
UMFPACK_STRATEGY_SYMMETRIC, UMFPACK_STRATEGY_UNSYMMETRIC, or
UMFPACK STRATEGY 2BY2.
















5.2 umfpack_di_numeric

int umfpack_di_numeric
(


const int Ap [ ],
const int Ai [ ],
const double Ax [ ],
void *Symbolic,
void **Numeric,
const double Control
double Info [UMFPACK


[UMFPACK_CONTROL],
INFO]


Purpose:

Given a sparse matrix A in column-oriented form, and a symbolic analysis
computed by umfpack_di_symbolic, the umfpackdinumeric routine performs the
numerical factorization, PAQ=LU, PRAQ=LU, or P(R\A)Q=LU, where P and Q are
permutation matrices (represented as permutation vectors), R is the row
scaling, L is unit-lower triangular, and U is upper triangular. This is
required before the system Ax=b (or other related linear systems) can be
solved. umfpack_di_numeric can be called multiple times for each call to
umfpack_di_symbolic, to factorize a sequence of matrices with identical
nonzero pattern. Simply compute the Symbolic object once, with
umfpack_di_symbolic, and reuse it for subsequent matrices.
umfpack_di_numeric safely detects if the pattern changes, and sets an
appropriate error code.

Returns:

The status code is returned. See Info [UMFPACK_STATUS], below.

Arguments:

Int Ap [n_col+1] ; Input argument, not modified.

This must be identical to the Ap array passed to umfpack_di_symbolic.
The value of n_col is what was passed to umfpack_di_symbolic (this is
held in the Symbolic object).


Int Ai [nz] ;


Input argument, not modified, of size nz = Ap [n_col].


This must be identical to the Ai array passed to umfpack_di_symbolic.


double Ax [nz] ;


Input argument, not modified, of size nz = Ap [n_col].


The numerical values of the sparse matrix A. The nonzero pattern (row
indices) for column j is stored in Ai [(Ap [j]) ... (Ap [j+1]-1)], and
the corresponding numerical values are stored in
Ax [(Ap [j]) ... (Ap [j+1]-1)].


void *Symbolic ;


Input argument, not modified.


The Symbolic object, which holds the symbolic factorization computed by
umfpack_di_symbolic. The Symbolic object is not modified by
umfpackdinumeric.


void **Numeric ;


Output argument.

















**Numeric is the address of a (void *) pointer variable in the user's
calling routine (see Syntax, above). On input, the contents of this
variable are not defined. On output, this variable holds a (void *)
pointer to the Numeric object (if successful), or (void *) NULL if
a failure occurred.

double Control [UMFPACK_CONTROL] ; Input argument, not modified.

If a (double *) NULL pointer is passed, then the default control
settings are used. Only the primary parameters are listed below:

Control [UMFPACK_PIVOT_TOLERANCE]: relative pivot tolerance for
threshold partial pivoting with row interchanges. In any given
column, an entry is numerically acceptable if its absolute value is
greater than or equal to Control [UMFPACK_PIVOT_TOLERANCE] times
the largest absolute value in the column. A value of 1.0 gives true
partial pivoting. If less than or equal to zero, then any nonzero
entry is numerically acceptable as a pivot (this is changed from
Version 4.0). Default: 0.1.

Smaller values tend to lead to sparser LU factors, but the solution
to the linear system can become inaccurate. Larger values can lead
to a more accurate solution (but not always), and usually an
increase in the total work.

Control [UMFPACK_SYM_PIVOT_TOLERANCE]: This parameter is new to V4.1.
If diagonal pivoting is attempted (the symmetric or symmetric-2by2
strategies are used) then this parameter is used to control when the
diagonal entry is selected in a given pivot column. The absolute
value of the entry must be >= Control [UMFPACK_SYM_PIVOT_TOLERANCE]
times the largest absolute value in the column. A value of zero
will ensure that no off-diagonal pivoting is performed, except that
zero diagonal entries are not selected if there are any off-diagonal
nonzero entries.

If an off-diagonal pivot is selected, an attempt is made to restore
symmetry later on. Suppose A (i,j) is selected, where i != j.
If column i has not yet been selected as a pivot column, then
the entry A (j,i) is redefined as a "diagonal" entry, except that
the tighter tolerance (Control [UMFPACK_PIVOT_TOLERANCE]) is
applied. This strategy has an effect similar to 2-by-2 pivoting
for symmetric indefinite matrices. If a 2-by-2 block pivot with
nonzero structure

ij
i: 0 x
j: x 0

is selected in a symmetric indefinite factorization method, the
2-by-2 block is inverted and a rank-2 update is applied. In
UMFPACK, this 2-by-2 block would be reordered as

ji
i: x 0
j: 0 x

In both cases, the symmetry of the Schur complement is preserved.

Control [UMFPACKSCALE]: This parameter is new to V4.1. Version 4.0

















did not scale the matrix. Note that the user's input matrix is
never modified, only an internal copy is scaled.

There are three valid settings for this parameter. If any other
value is provided, the default is used.

UMFPACK_SCALE_NONE: no scaling is performed.

UMFPACK_SCALE_SUM: each row of the input matrix A is divided by
the sum of the absolute values of the entries in that row.
The scaled matrix has an infinity norm of 1.

UMFPACK_SCALE_MAX: each row of the input matrix A is divided by
the maximum the absolute values of the entries in that row.
In the scaled matrix the largest entry in each row has
a magnitude exactly equal to 1.

Scaling is very important for the "symmetric" strategy when
diagonal pivoting is attempted. It also improves the performance
of the "unsymmetric" strategy.

Default: UMFPACK SCALE SUM.

double Info [UMFPACK_INFO] ; Output argument.

Contains statistics about the numeric factorization. If a
(double *) NULL pointer is passed, then no statistics are returned in
Info (this is not an error condition). The following statistics are
computed in umfpack_di_numeric (only the primary statistics are listed):

Info [UMFPACK_STATUS]: status code. This is also the return value,
whether or not Info is present.

UMFPACK OK

Numeric factorization was successful. umfpack_di_numeric
computed a valid numeric factorization.

UMFPACK_WARNING_singular_matrix

Numeric factorization was successful, but the matrix is
singular. umfpack_di_numeric computed a valid numeric
factorization, but you will get a divide by zero in
umfpack_di_solve. For the other cases below, no Numeric object
is created (*Numeric is (void *) NULL).

UMFPACK_ERRORout of memory

Insufficient memory to complete the numeric factorization.

UMFPACK_ERROR_argument_missing

One or more required arguments are missing.

UMFPACK_ERROR_invalid_Symbolic_object

Symbolic object provided as input is invalid.

UMFPACK_ERROR_different_pattern


















The pattern (Ap and/or Ai) has changed since the call to
umfpack_di_symbolic which produced the Symbolic object.

Info [UMFPACK_NUMERIC_SIZE]: the actual final size (in Units) of the
entire Numeric object, including the final size of the variable
part of the object. Info [UMFPACK_NUMERIC_SIZE_ESTIMATE],
an estimate, was computed by umfpack_di_symbolic. The estimate is
normally an upper bound on the actual final size, but this is not
guaranteed.

Info [UMFPACK_PEAK_MEMORY]: the actual peak memory usage (in Units) of
both umfpack_di_symbolic and umfpackdinumeric. An estimate,
Info [UMFPACK_PEAK_MEMORY_ESTIMATE], was computed by
umfpack_di_symbolic. The estimate is normally an upper bound on the
actual peak usage, but this is not guaranteed. With testing on
hundreds of matrix arising in real applications, I have never
observed a matrix where this estimate or the Numeric size estimate
was less than the actual result, but this is theoretically possible.
Please send me one if you find such a matrix.

Info [UMFPACK_FLOPS]: the actual count of the (useful) floating-point
operations performed. An estimate, Info [UMFPACK_FLOPS_ESTIMATE],
was computed by umfpack_di_symbolic. The estimate is guaranteed to
be an upper bound on this flop count. The flop count excludes
"useless" flops on zero values, flops performed during the pivot
search (for tentative updates and assembly of candidate columns),
and flops performed to add frontal matrices together.

Info [UMFPACK_LNZ]: the actual nonzero entries in final factor L,
including the diagonal. This excludes any zero entries in L,
although some of these are stored in the Numeric object. The
Info [UMFPACK_LU_ENTRIES] statistic does account for all
explicitly stored zeros, however. Info [UMFPACK_LNZ_ESTIMATE],
an estimate, was computed by umfpack_di_symbolic. The estimate is
guaranteed to be an upper bound on Info [UMFPACK_LNZ].

Info [UMFPACK_UNZ]: the actual nonzero entries in final factor U,
including the diagonal. This excludes any zero entries in U,
although some of these are stored in the Numeric object. The
Info [UMFPACK_LU_ENTRIES] statistic does account for all
explicitly stored zeros, however. Info [UMFPACK_UNZ_ESTIMATE],
an estimate, was computed by umfpack_di_symbolic. The estimate is
guaranteed to be an upper bound on Info [UMFPACK_UNZ].

Info [UMFPACK_NUMERIC_TIME]: The CPU time taken, in seconds.

















5.3 umfpack_di_solve

int umfpack_di_solve
(


int sys,
const int Ap [ ],
const int Ai [ ],
const double Ax [ ],
double X [ ],
const double B [ ],
void *Numeric,
const double Control
double Info [UMFPACK_


[UMFPACK_CONTROL],
INFO]


Purpose:

Given LU factors computed by umfpackdinumeric (PAQ=LU, PRAQ=LU, or
P(R\A)Q=LU) and the right-hand-side, B, solve a linear system for the
solution X. Iterative refinement is optionally performed. Only square
systems are handled. Singular matrices result in a divide-by-zero for all
systems except those involving just the matrix L. Iterative refinement is
not performed for singular matrices.

In the discussion below, n is equal to n_row and n_col, because only
square systems are handled.

Returns:

The status code is returned. See Info [UMFPACK_STATUS], below.

Arguments:

Int sys ; Input argument, not modified.

Defines which system to solve. (') is the linear algebraic transpose.


sys value

UMFPACK A
UMFPACK At
UMFPACK Pt L
UMFPACK L
UMFPACK Lt P
UMFPACK Lt
UMFPACK_U_Qt
UMFPACK U
UMFPACK_Q_Ut
UMFPACK Ut


system solved

Ax=b
A'x=b
P'Lx=b
Lx=b
L'Px=b
L'x=b
UQ'x=b
Ux=b
QU'x=b
U'x=b


Iterative refinement can be
the following:


optionally performed when sys is any of


UMFPACK A Ax=b
UMFPACK At A'x=b

For the other values of the sys argument, iterative refinement is not
performed (Control [UMFPACK_IRSTEP], Ap, Ai, and Ax are ignored).

















Int Ap [n+1] ; Input argument, not modified.
Int Ai [nz] ; Input argument, not modified.
double Ax [nz] ; Input argument, not modified.

If iterative refinement is requested (Control [UMFPACKIRSTEP] >= 1,
Ax=b or A'x=b is being solved, and A is nonsingular), then
these arrays must be identical to the same ones passed to
umfpackdinumeric. The umfpackdisolve routine does not check the
contents of these arguments, so the results are undefined if Ap, Ai, Ax,
are modified between the calls the umfpackdinumeric and
umfpackdisolve. These three arrays do not need to be present (NULL
pointers can be passed) if Control [UMFPACKIRSTEP] is zero, or if a
system other than Ax=b or A'x=b is being solved, or if A is
singular, since in each of these cases A is not accessed.

double X [n] ; Output argument.

The solution to the linear system, where n = n_row = n_col is the
dimension of the matrices A, L, and U.

double B [n] ; Input argument, not modified.

The right-hand side vector, b, stored as a conventional array of size n
(or two arrays of size n for complex versions). This routine does not
solve for multiple right-hand-sides, nor does it allow b to be stored in
a sparse-column form.

void *Numeric ; Input argument, not modified.

Numeric must point to a valid Numeric object, computed by
umfpackdinumeric.

double Control [UMFPACK_CONTROL] ; Input argument, not modified.

If a (double *) NULL pointer is passed, then the default control
settings are used.

Control [UMFPACKIRSTEP]: The maximum number of iterative refinement
steps to attempt. A value less than zero is treated as zero. If
less than 1, or if Ax=b or A'x=b is not being solved, or
if A is singular, then the Ap, Ai, and Ax arguments are not
accessed. Default: 2.

double Info [UMFPACKINFO] ; Output argument.

Contains statistics about the solution factorization. If a
(double *) NULL pointer is passed, then no statistics are returned in
Info (this is not an error condition). The following statistics are
computed in umfpackdisolve (only the primary statistics are listed):

Info [UMFPACKSTATUS]: status code. This is also the return value,
whether or not Info is present.

UMFPACK OK

The linear system was successfully solved.

UMFPACK_WARNING_singular_matrix

















A divide-by-zero occurred. Your solution will contain Inf's
and/or NaN's. Some parts of the solution may be valid. For
example, solving Ax=b with

A = [2 01 b = [ 1] returns x = [ 0.5 ]
[0 0] [ 0 ] [ Inf ]

UMFPACK_ERRORout of memory

Insufficient memory to solve the linear system.

UMFPACK_ERROR_argument_missing

One or more required arguments are missing. The B and X
arguments are always required. Info and Control are not
required. Ap, Ai and Ax are required if Ax=b or
A'x=b is to be solved, the (default) iterative
refinement is requested, and the matrix A is nonsingular.

UMFPACK_ERROR_invalid_system

The sys argument is not valid, or the matrix A is not square.

UMFPACK_ERROR_invalid_Numeric_object

The Numeric object is not valid.

Info [UMFPACK_SOLVE_FLOPS]: the number of floating point operations
performed to solve the linear system. This includes the work
taken for all iterative refinement steps, including the backtrack
(if any).

Info [UMFPACK_SOLVE_TIME]: The time taken, in seconds.
















5.4 umfpack_di_free_symbolic

void umfpackdifree_symbolic
(
void **Symbolic
);

Purpose:

Deallocates the Symbolic object and sets the Symbolic handle to NULL.

Arguments:

void **Symbolic ; Input argument, deallocated and Symbolic is
set to (void *) NULL on output.


5.5 umfpack_di_free_numeric

void umfpackdifree_numeric
(
void **Numeric
);

Purpose:

Deallocates the Numeric object and sets the Numeric handle to NULL.

Arguments:

void **Numeric ; Input argument, deallocated and Numeric is
set to (void *) NULL on output.


5.6 umfpack_di_defaults

void umfpackdidefaults
(
double Control [UMFPACK_CONTROL]
);

Purpose:

Sets the default control parameter settings.

Arguments:

double Control [UMFPACK_CONTROL] ; Output argument.

Control is set to the default control parameter settings.




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