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%------------------------------------------------------------------------------- % The QuickStart.tex file. %------------------------------------------------------------------------------- \documentclass[11pt]{article} \newcommand{\m}[1]{{\bf{#1}}} % for matrices and vectors \newcommand{\tr}{^{\sf T}} % transpose \topmargin 0in \textheight 9in \oddsidemargin 0pt \evensidemargin 0pt \textwidth 6.5in \begin{document} \author{Timothy A. Davis \\ Dept. of Computer and Information Science and Engineering \\ Univ. of Florida, Gainesville, FL} \title{UMFPACK Version 4.4 Quick Start Guide} \date{Jan. 28, 2005} \maketitle %------------------------------------------------------------------------------- \begin{abstract} UMFPACK is a set of routines for solving unsymmetric sparse linear systems, $\m{Ax}=\m{b}$, 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 start'' guide for Unix users of the C interface. \end{abstract} %------------------------------------------------------------------------------- UMFPACK Version 4.4 (Jan. 28, 2005), Copyright\copyright 2005 by Timothy A. Davis. All Rights Reserved. Refer to the UMFPACK User Guide for the License. See \newline http://www.cise.ufl.edu/research/sparse/umfpack for the code and full documentation. %------------------------------------------------------------------------------- \section{Overview} %------------------------------------------------------------------------------- UMFPACK is a set of routines for solving systems of linear equations, $\m{Ax}=\m{b}$, when $\m{A}$ is sparse and unsymmetric. The sparse matrix $\m{A}$ can be square or rectangular, singular or non-singular, and real or complex (or any combination). Only square matrices $\m{A}$ can be used to solve $\m{Ax}=\m{b}$ or related systems. Rectangular matrices can only be factorized. UMFPACK Version 4.3 is a built-in routine in MATLAB 7.1, used by the forward and backslash operator, and the {\tt lu} routine. MATLAB 6.5 used Version 4.0. The following is a short introduction to Unix users of the C interface of UMFPACK. %------------------------------------------------------------------------------- The C-callable UMFPACK library consists of 32 user-callable routines and one include file. Twenty-eight 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 {\tt int}'s as integers (not {\tt long}'s). Refer to the User Guide for information about the complex and long integer versions. The include file {\tt umfpack.h} must be included in any C program that uses UMFPACK. For more details, see: {\em A column pre-ordering strategy for the unsymmetric-pattern multifrontal method}, Davis, T. A., ACM Trans. Math. Software, vol 30. no 2, 2004, pp. 165-195, and {\em Algorithm 832: {UMFPACK}, an unsymmetric-pattern multifrontal method}, same issue, pp. 196-199. %------------------------------------------------------------------------------- \section{Primary routines, and a simple example} %------------------------------------------------------------------------------- Five primary UMFPACK routines are required to factorize $\m{A}$ or solve $\m{Ax}=\m{b}$. An overview of the primary features of the routines is given in Section~\ref{Primary}. 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, computing the determinant, and reporting results. See the User Guide for more information. \begin{itemize} \item {\tt umfpack\_di\_symbolic}: Pre-orders the columns of $\m{A}$ to reduce fill-in and performs a symbolic analysis. Returns an opaque {\tt Symbolic} object as a {\tt void *} pointer. The object contains the symbolic analysis and is needed for the numerical factorization. \item {\tt umfpack\_di\_numeric}: Numerically scales and then factorizes a sparse matrix $\m{PAQ}$, $\m{PRAQ}$, or $\m{PR}^{-1}\m{AQ}$ into the product $\m{LU}$, where $\m{P}$ and $\m{Q}$ are permutation matrices, $\m{R}$ is a diagonal matrix of scale factors, $\m{L}$ is lower triangular with unit diagonal, and $\m{U}$ is upper triangular. Requires the symbolic ordering and analysis computed by {\tt umfpack\_di\_symbolic}. Returns an opaque {\tt Numeric} object as a {\tt void *} pointer. The object contains the numerical factorization and is used by {\tt umfpack\_di\_solve}. \item {\tt umfpack\_di\_solve}: Solves a sparse linear system ($\m{Ax}=\m{b}$, $\m{A}\tr\m{x}=\m{b}$, or systems involving just $\m{L}$ or $\m{U}$), using the numeric factorization computed by {\tt umfpack\_di\_numeric}. \item {\tt umfpack\_di\_free\_symbolic}: Frees the {\tt Symbolic} object created by {\tt umfpack\_di\_symbolic}. \item {\tt umfpack\_di\_free\_numeric}: Frees the {\tt Numeric} object created by {\tt umfpack\_di\_numeric}. \end{itemize} The matrix $\m{A}$ 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 {\tt m}-by-{\tt n}, with {\tt nz} entries: {\footnotesize \begin{verbatim} int Ap [n+1] ; int Ai [nz] ; double Ax [nz] ; \end{verbatim} } All nonzeros are entries, but an entry may be numerically zero. The row indices of entries in column {\tt j} are stored in {\tt Ai[Ap[j]} ... {\tt Ap[j+1]-1]}. The corresponding numerical values are stored in {\tt Ax[Ap[j]} ... {\tt 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 {\tt Ap[0]} must be zero. The total number of entries in the matrix is thus {\tt nz = Ap[n]}. Except for the fact that extra zero entries can be included, there is thus a unique compressed column representation of any given matrix $\m{A}$. Here is a simple main program, {\tt umfpack\_simple.c}, that illustrates the basic usage of UMFPACK. {\footnotesize \begin{verbatim} #include <stdio.h> #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) ; umfpack_di_free_symbolic (&Symbolic) ; (void) umfpack_di_solve (UMFPACK_A, Ap, Ai, Ax, x, b, Numeric, null, null) ; umfpack_di_free_numeric (&Numeric) ; for (i = 0 ; i < n ; i++) printf ("x [%d] = %g\n", i, x [i]) ; return (0) ; } \end{verbatim} } The {\tt Ap}, {\tt Ai}, and {\tt Ax} arrays represent the matrix \[ \m{A} = \left[ \begin{array}{rrrrr} 2 & 3 & 0 & 0 & 0 \\ 3 & 0 & 4 & 0 & 6 \\ 0 & -1 & -3 & 2 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 4 & 2 & 0 & 1 \\ \end{array} \right]. \] and the solution is $\m{x} = [1 \, 2 \, 3 \, 4 \, 5]\tr$. The program uses default control settings and does not return any statistics about the ordering, factorization, or solution ({\tt Control} and {\tt Info} are both {\tt (double *) NULL}). For routines to manipulate a simpler ``triplet-form'' data structure for your sparse matrix $\m{A}$, refer to the UMFPACK User Guide. %------------------------------------------------------------------------------- \section{Synopsis of primary C-callable routines} \label{Synopsis} %------------------------------------------------------------------------------- The matrix $\m{A}$ is {\tt m}-by-{\tt n} with {\tt nz} entries. The optional {\tt umfpack\_di\_defaults} routine loads the default control parameters into the {\tt Control} array. The settings can then be modified before passing the array to the other routines. Refer to Section~\ref{Primary} for more details. {\footnotesize \begin{verbatim} #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 = umfpack_di_symbolic (m, n, Ap, Ai, Ax, &Symbolic, Control, Info) ; status = umfpack_di_numeric (Ap, Ai, Ax, Symbolic, &Numeric, Control, Info) ; status = umfpack_di_solve (sys, Ap, Ai, Ax, X, B, Numeric, Control, Info) ; umfpack_di_free_symbolic (&Symbolic) ; umfpack_di_free_numeric (&Numeric) ; umfpack_di_defaults (Control) ; \end{verbatim} } %------------------------------------------------------------------------------- \section{Installation} \label{Install} %------------------------------------------------------------------------------- You will need to install both UMFPACK v4.4 and AMD v1.1 (or AMD v1.0) to use UMFPACK. The {\tt UMFPACK} and {\tt 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 {\tt AMD/Make} and {\tt UMFPACK/Make} directories (Note that {\tt UMFPACK/Make} is a symbolic link to {\tt AMD/Make}). You can edit the {\tt Make.include} file 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 {\tt make}. This will compile the libraries ({\tt AMD/Lib/libamd.a} and {\tt UMFPACK/Lib/libumfpack.a}). A demo of the AMD ordering routine will be compiled and tested in the {\tt AMD/Demo} directory, and five demo programs will then be compiled and tested in the {\tt 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 {\tt make lib} instead. If you compile UMFPACK and AMD and then later change the {\tt Make.include} file or your system-specific configuration file such as {\tt Make.linux}, then you should type {\tt make purge} and then {\tt make} to recompile. Here are the various parameters that you can control in your {\tt Make.include} file: \begin{itemize} \item {\tt CC = } your C compiler, such as {\tt cc}. \item {\tt RANLIB = } your system's {\tt ranlib} program, if needed. \item {\tt CFLAGS = } optimization flags, such as {\tt -O}. \item {\tt CONFIG = } configuration settings, for the BLAS, memory allocation routines, and timing routines. \item {\tt LIB = } your libraries, such as {\tt -lm} or {\tt -lblas}. \item {\tt RM =} the command to delete a file. \item {\tt MV =} the command to rename a file. \item {\tt MEX =} the command to compile a MATLAB mexFunction. \item {\tt F77 =} the command to compile a Fortran program (optional). \item {\tt F77FLAGS =} the Fortran compiler flags (optional). \item {\tt F77LIB =} the Fortran libraries (optional). \end{itemize} The {\tt CONFIG} string can include combinations of the following; most deal with how the BLAS are called: \begin{itemize} \item {\tt -DNBLAS} if you do not have any BLAS at all. \item {\tt -DCBLAS} if you have the C-BLAS. \item {\tt -DNSUNPERF} if you are on Solaris but do not have the Sun Performance Library. \item {\tt -DNSCSL} if you on SGI IRIX but do not have the SCSL BLAS library. \item {\tt -DLONGBLAS} if your BLAS can take {\tt long} integer input arguments. \item Options for controlling how C calls the Fortran BLAS: {\tt -DBLAS\_BY\_VALUE}, {\tt -DBLAS\_NO\_UNDERSCORE}, and {\tt -DBLAS\_CHAR\_ARG}. These are set automatically for Windows, Sun Solaris, SGI Irix, Red Hat Linux, Compaq Alpha, and AIX (the IBM RS 6000). \item {\tt -DGETRUSAGE} if you have the {\tt getrusage} function. \item {\tt -DLP64} if you are compiling in the LP64 model (32 bit {\tt int}'s, 64 bit {\tt long}'s, and 64 bit pointers). \item {\tt -DNUTIL} if you wish to compile the MATLAB-callable UMFPACK mexFunction with the {\tt mxMalloc}, {\tt mxRealloc} and {\tt mxFree} routines, instead of the undocumented (but superior) {\tt utMalloc}, {\tt utRealloc}, and {\tt utFree} routines. \item {\tt -DNPOSIX} if you do not have the POSIX-compliant {\tt sysconf} and {\tt times} routines. \item {\tt -DNRECIPROCAL} controls a trade-off between speed and accuracy. This is off by default (speed preferred over accuracy) except when compiling for MATLAB. \end{itemize} When you compile your program that uses the C-callable UMFPACK library, you need to add the both {\tt UMFPACK/Lib/libumfpack.a} and {\tt AMD/Lib/libamd.a} libraries, and you need to tell your compiler to look in the directories {\tt UMFPACK/Include} and {\tt AMD/Include} for include files. See {\tt 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 \begin{verbatim} #include "umfpack.h" \end{verbatim} statement, as described in Section~\ref{Synopsis}. %------------------------------------------------------------------------------- \newpage \section{The primary UMFPACK routines} \label{Primary} %------------------------------------------------------------------------------- \subsection{umfpack\_di\_symbolic} {\footnotesize \begin{verbatim} int umfpack_di_symbolic ( int n_row, int n_col, const int Ap [ ], const int Ai [ ], const double Ax [ ], void **Symbolic, const double Control [UMFPACK_CONTROL], double Info [UMFPACK_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-n1-nempty_row) -by- (n_col-n1-nempty_col), where n1 = 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 n_row ; Input argument, not modified. Int n_col ; 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+1] ; 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-1. 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 0-based). double Ax [nz] ; Optional input argument, not modified. 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 umfpack_di_numeric. 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_ERROR_out_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_SIZE_OF_UNIT]: 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 umfpack_di_numeric 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 umfpack_di_numeric 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 umfpack_di_numeric 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. \end{verbatim} } %------------------------------------------------------------------------------- \newpage \subsection{umfpack\_di\_numeric} {\footnotesize \begin{verbatim} int umfpack_di_numeric ( const int Ap [ ], const int Ai [ ], const double Ax [ ], void *Symbolic, void **Numeric, const double Control [UMFPACK_CONTROL], double Info [UMFPACK_INFO] ) ; Purpose: Given a sparse matrix A in column-oriented form, and a symbolic analysis computed by umfpack_di_symbolic, the umfpack_di_numeric 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 umfpack_di_numeric. 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 i j 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 j i i: x 0 j: 0 x In both cases, the symmetry of the Schur complement is preserved. Control [UMFPACK_SCALE]: 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_ERROR_out_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 umfpack_di_numeric. 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. \end{verbatim} } %------------------------------------------------------------------------------- \newpage \subsection{umfpack\_di\_solve} {\footnotesize \begin{verbatim} 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 [UMFPACK_CONTROL], double Info [UMFPACK_INFO] ) ; Purpose: Given LU factors computed by umfpack_di_numeric (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 system solved UMFPACK_A Ax=b UMFPACK_At A'x=b UMFPACK_Pt_L P'Lx=b UMFPACK_L Lx=b UMFPACK_Lt_P L'Px=b UMFPACK_Lt L'x=b UMFPACK_U_Qt UQ'x=b UMFPACK_U Ux=b UMFPACK_Q_Ut QU'x=b UMFPACK_Ut U'x=b Iterative refinement can be optionally performed when sys is any of the following: 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 [UMFPACK_IRSTEP] >= 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 umfpack_di_numeric. The umfpack_di_solve routine does not check the contents of these arguments, so the results are undefined if Ap, Ai, Ax, are modified between the calls the umfpack_di_numeric and umfpack_di_solve. These three arrays do not need to be present (NULL pointers can be passed) if Control [UMFPACK_IRSTEP] 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 umfpack_di_numeric. double Control [UMFPACK_CONTROL] ; Input argument, not modified. If a (double *) NULL pointer is passed, then the default control settings are used. Control [UMFPACK_IRSTEP]: 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 [UMFPACK_INFO] ; 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 umfpack_di_solve (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 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 0] b = [ 1 ] returns x = [ 0.5 ] [0 0] [ 0 ] [ Inf ] UMFPACK_ERROR_out_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. \end{verbatim} } %------------------------------------------------------------------------------- \newpage \subsection{umfpack\_di\_free\_symbolic} {\footnotesize \begin{verbatim} void umfpack_di_free_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. \end{verbatim} } %------------------------------------------------------------------------------- \subsection{umfpack\_di\_free\_numeric} {\footnotesize \begin{verbatim} void umfpack_di_free_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. \end{verbatim} } %------------------------------------------------------------------------------- \subsection{umfpack\_di\_defaults} {\footnotesize \begin{verbatim} void umfpack_di_defaults ( 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. \end{verbatim} } \end{document}