/* * SPARSE FORTRAN MODULE * * Author: Advising professor: * Kenneth S. Kundert Alberto Sangiovanni-Vincentelli * UC Berkeley * * This module contains routines that interface Sparse1.3 to a calling * program written in fortran. Almost every externally available Sparse1.3 * routine has a counterpart defined in this file, with the name the * same except the `sp' prefix is changed to `sf'. The spADD_ELEMENT * and spADD_QUAD macros are also replaced with the sfAdd1 and sfAdd4 * functions defined in this file. * * To ease porting this file to different operating systems, the names of * the functions can be easily redefined (search for `Routine Renaming'). * A simple example of a FORTRAN program that calls Sparse is included in * this file (search for Example). When interfacing to a FORTRAN program, * the ARRAY_OFFSET option should be set to NO (see spConfig.h). * * DISCLAIMER: * These interface routines were written by a C programmer who has little * experience with FORTRAN. The routines have had minimal testing. * Any interface between two languages is going to have portability * problems, this one is no exception. * * * >>> User accessible functions contained in this file: * sfCreate() * sfDestroy() * sfStripFills() * sfClear() * sfGetElement() * sfGetAdmittance() * sfGetQuad() * sfGetOnes() * sfAdd1Real() * sfAdd1Imag() * sfAdd1Complex() * sfAdd4Real() * sfAdd4Imag() * sfAdd4Complex() * sfOrderAndFactor() * sfFactor() * sfPartition() * sfSolve() * sfSolveTransposed() * sfPrint() * sfFileMatrix() * sfFileVector() * sfFileStats() * sfMNA_Preorder() * sfScale() * sfMultiply() * sfTransMultiply() * sfDeterminant() * sfError() * sfWhereSingular() * sfGetSize() * sfSetReal() * sfSetComplex() * sfFillinCount() * sfElementCount() * sfDeleteRowAndCol() * sfPseudoCondition() * sfCondition() * sfNorm() * sfLargestElement() * sfRoundoff() */ /* * FORTRAN -- C COMPATIBILITY * * Fortran and C data types correspond in the following way: * -- C -- -- FORTRAN -- * int INTEGER*4 or INTEGER*2 (machine dependent, usually int*4) * long INTEGER*4 * float REAL * double DOUBLE PRECISION (used by default in preference to float) * * The complex number format used by Sparse is compatible with that * used by FORTRAN. C pointers are passed to FORTRAN as longs, they should * not be used in any way in FORTRAN. */ /* * Revision and copyright information. * * Copyright (c) 1985,86,87,88 * by Kenneth S. Kundert and the University of California. * * Permission to use, copy, modify, and distribute this software and its * documentation for any purpose and without fee is hereby granted, provided * that the above copyright notice appear in all copies and supporting * documentation and that the authors and the University of California * are properly credited. The authors and the University of California * make no representations as to the suitability of this software for * any purpose. It is provided `as is', without express or implied warranty. */ /* * IMPORTS * * >>> Import descriptions: * spConfig.h * Macros that customize the sparse matrix routines. * spmatrix.h * Macros and declarations to be imported by the user. * spDefs.h * Matrix type and macro definitions for the sparse matrix routines. * spFortran.h * Definition of profiles */ #define spINSIDE_SPARSE #include "spConfig.h" #include "spmatrix.h" #include "spDefs.h" #include "spFortran.h" #ifdef FORTRAN /* * Example of a FORTRAN Program Calling Sparse * int matrix, error, sfCreate, sfGetElement, spFactor int element(10) double precision rhs(4), solution(4) c matrix = sfCreate(4,0,error) element(1) = sfGetElement(matrix,1,1) element(2) = sfGetElement(matrix,1,2) element(3) = sfGetElement(matrix,2,1) element(4) = sfGetElement(matrix,2,2) element(5) = sfGetElement(matrix,2,3) element(6) = sfGetElement(matrix,3,2) element(7) = sfGetElement(matrix,3,3) element(8) = sfGetElement(matrix,3,4) element(9) = sfGetElement(matrix,4,3) element(10) = sfGetElement(matrix,4,4) call sfClear(matrix) call sfAdd1Real(element(1), 2d0) call sfAdd1Real(element(2), -1d0) call sfAdd1Real(element(3), -1d0) call sfAdd1Real(element(4), 3d0) call sfAdd1Real(element(5), -1d0) call sfAdd1Real(element(6), -1d0) call sfAdd1Real(element(7), 3d0) call sfAdd1Real(element(8), -1d0) call sfAdd1Real(element(9), -1d0) call sfAdd1Real(element(10), 3d0) call sfprint(matrix, .false., .false.) rhs(1) = 34d0 rhs(2) = 0d0 rhs(3) = 0d0 rhs(4) = 0d0 error = sfFactor(matrix) call sfSolve(matrix, rhs, solution) write (6, 10) rhs(1), rhs(2), rhs(3), rhs(4) 10 format (f 10.2) end * */ long sfCreate(int *Size, int *Complex, int *Error ) { /* Begin `sfCreate'. */ return (long)spCreate(*Size, *Complex, Error ); } void sfDestroy( long *Matrix ) { /* Begin `sfDestroy'. */ spDestroy((char *)*Matrix); return; } #ifdef STRIP void sfStripFills( long *Matrix ) { /* Begin `sfStripFills'. */ spStripFills((char *)*Matrix); return; } #endif /* * CLEAR MATRIX * * Sets every element of the matrix to zero and clears the error flag. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to matrix that is to be cleared. */ void sfClear( long *Matrix ) { /* Begin `sfClear'. */ spClear((char *)*Matrix); return; } /* * SINGLE ELEMENT ADDITION TO MATRIX BY INDEX * * Finds element [Row,Col] and returns a pointer to it. If element is * not found then it is created and spliced into matrix. This routine * is only to be used after spCreate() and before spMNA_Preorder(), * spFactor() or spOrderAndFactor(). Returns a pointer to the * Real portion of a MatrixElement. This pointer is later used by * sfAddxxxxx() to directly access element. * * >>> Returns: [INTEGER] * Returns a pointer to the element. This pointer is then used to directly * access the element during successive builds. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to the matrix that the element is to be added to. * Row <input> (int *) [INTEGER or INTEGER*2] * Row index for element. Must be in the range of [0..Size] unless * the options EXPANDABLE or TRANSLATE are used. Elements placed in * row zero are discarded. In no case may Row be less than zero. * Col <input> (int *) [INTEGER or INTEGER*2] * Column index for element. Must be in the range of [0..Size] unless * the options EXPANDABLE or TRANSLATE are used. Elements placed in * column zero are discarded. In no case may Col be less than zero. * * >>> Possible errors: * spNO_MEMORY * Error is not cleared in this routine. */ long sfGetElement( long *Matrix, int *Row, int *Col ) { /* Begin `sfGetElement'. */ return (long)spGetElement((char *)*Matrix, *Row, *Col); } #ifdef QUAD_ELEMENT /* * ADDITION OF ADMITTANCE TO MATRIX BY INDEX * * Performs same function as sfGetElement except rather than one * element, all four Matrix elements for a floating component are * added. This routine also works if component is grounded. Positive * elements are placed at [Node1,Node2] and [Node2,Node1]. This * routine is only to be used after sfCreate() and before * sfMNA_Preorder(), sfFactor() or sfOrderAndFactor(). * * >>> Returns: [INTEGER or INTEGER*2] * Error code. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to the matrix that component is to be entered in. * Node1 <input> (int *) [INTEGER or INTEGER*2] * Row and column indices for elements. Must be in the range of [0..Size] * unless the options EXPANDABLE or TRANSLATE are used. Node zero is the * ground node. In no case may Node1 be less than zero. * Node2 <input> (int *) [INTEGER or INTEGER*2] * Row and column indices for elements. Must be in the range of [0..Size] * unless the options EXPANDABLE or TRANSLATE are used. Node zero is the * ground node. In no case may Node2 be less than zero. * Template <output> (long[4]) [INTEGER (4)] * Collection of pointers to four elements that are later used to directly * address elements. User must supply the template, this routine will * fill it. * * Possible errors: * spNO_MEMORY * Error is not cleared in this routine. */ int sfGetAdmittance( long *Matrix, int *Node1, int *Node2, long Template[4] ) { /* Begin `spGetAdmittance'. */ return ( spGetAdmittance((char *)*Matrix, *Node1, *Node2, (struct spTemplate *)Template ) ); } #endif /* QUAD_ELEMENT */ #ifdef QUAD_ELEMENT /* * ADDITION OF FOUR ELEMENTS TO MATRIX BY INDEX * * Similar to sfGetAdmittance, except that sfGetAdmittance only * handles 2-terminal components, whereas sfGetQuad handles simple * 4-terminals as well. These 4-terminals are simply generalized * 2-terminals with the option of having the sense terminals different * from the source and sink terminals. sfGetQuad adds four * elements to the matrix. Positive elements occur at Row1,Col1 * Row2,Col2 while negative elements occur at Row1,Col2 and Row2,Col1. * The routine works fine if any of the rows and columns are zero. * This routine is only to be used after sfCreate() and before * sfMNA_Preorder(), sfFactor() or sfOrderAndFactor() * unless TRANSLATE is set true. * * >>> Returns: [INTEGER or INTEGER*2] * Error code. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to the matrix that component is to be entered in. * Row1 <input> (int *) [INTEGER or INTEGER*2] * First row index for elements. Must be in the range of [0..Size] * unless the options EXPANDABLE or TRANSLATE are used. Zero is the * ground row. In no case may Row1 be less than zero. * Row2 <input> (int *) [INTEGER or INTEGER*2] * Second row index for elements. Must be in the range of [0..Size] * unless the options EXPANDABLE or TRANSLATE are used. Zero is the * ground row. In no case may Row2 be less than zero. * Col1 <input> (int *) [INTEGER or INTEGER*2] * First column index for elements. Must be in the range of [0..Size] * unless the options EXPANDABLE or TRANSLATE are used. Zero is the * ground column. In no case may Col1 be less than zero. * Col2 <input> (int *) [INTEGER or INTEGER*2] * Second column index for elements. Must be in the range of [0..Size] * unless the options EXPANDABLE or TRANSLATE are used. Zero is the * ground column. In no case may Col2 be less than zero. * Template <output> (long[4]) [INTEGER (4)] * Collection of pointers to four elements that are later used to directly * address elements. User must supply the template, this routine will * fill it. * * Possible errors: * spNO_MEMORY * Error is not cleared in this routine. */ int sfGetQuad( long *Matrix, int *Row1, int *Row2, int *Col1, int *Col2, long Template[4] ) { /* Begin `spGetQuad'. */ return ( spGetQuad( (char *)*Matrix, *Row1, *Row2, *Col1, *Col2, (struct spTemplate *)Template ) ); } #endif /* QUAD_ELEMENT */ #ifdef QUAD_ELEMENT /* * ADDITION OF FOUR STRUCTURAL ONES TO MATRIX BY INDEX * * Performs similar function to sfGetQuad() except this routine is * meant for components that do not have an admittance representation. * * The following stamp is used: * Pos Neg Eqn * Pos [ . . 1 ] * Neg [ . . -1 ] * Eqn [ 1 -1 . ] * * >>> Returns: [INTEGER or INTEGER*2] * Error code. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to the matrix that component is to be entered in. * Pos <input> (int *) [INTEGER or INTEGER*2] * See stamp above. Must be in the range of [0..Size] * unless the options EXPANDABLE or TRANSLATE are used. Zero is the * ground row. In no case may Pos be less than zero. * Neg <input> (int *) [INTEGER or INTEGER*2] * See stamp above. Must be in the range of [0..Size] * unless the options EXPANDABLE or TRANSLATE are used. Zero is the * ground row. In no case may Neg be less than zero. * Eqn <input> (int *) [INTEGER or INTEGER*2] * See stamp above. Must be in the range of [0..Size] * unless the options EXPANDABLE or TRANSLATE are used. Zero is the * ground row. In no case may Eqn be less than zero. * Template <output> (long[4]) [INTEGER (4)] * Collection of pointers to four elements that are later used to directly * address elements. User must supply the template, this routine will * fill it. * * Possible errors: * spNO_MEMORY * Error is not cleared in this routine. */ int sfGetOnes(long *Matrix, int *Pos, int *Neg, int *Eqn, long Template[4]) { /* Begin `sfGetOnes'. */ return ( spGetOnes( (char *)*Matrix, *Pos, *Neg, *Eqn, (struct spTemplate *)Template ) ); } #endif /* QUAD_ELEMENT */ /* * ADD ELEMENT(S) DIRECTLY TO MATRIX * * Adds a value to an element or a set of four element in a matrix. * These elements are referenced by pointer, and so must already have * been created by spGetElement(), spGetAdmittance(), spGetQuad(), or * spGetOnes(). * * >>> Arguments: * Element <input> (long *) [INTEGER] * Pointer to the element that is to be added to. * Template <input> (long[4]) [INTEGER (4)] * Pointer to the element that is to be added to. * Real <input> (spREAL *) [REAL or DOUBLE PRECISION] * Real portion of the number to be added to the element. * Imag <input> (spREAL *) [REAL or DOUBLE PRECISION] * Imaginary portion of the number to be added to the element. */ void sfAdd1Real( long *Element, RealNumber *Real ) { /* Begin `sfAdd1Real'. */ *((RealNumber *)*Element) += *Real; } #ifdef spCOMPLEX void sfAdd1Imag( long *Element, RealNumber *Imag ) { /* Begin `sfAdd1Imag'. */ *(((RealNumber *)*Element)+1) += *Imag; } void sfAdd1Complex( long *Element, RealNumber *Real, RealNumber *Imag ) { /* Begin `sfAdd1Complex'. */ *((RealNumber *)*Element) += *Real; *(((RealNumber *)*Element)+1) += *Imag; } #endif /* spCOMPLEX */ #ifdef QUAD_ELEMENT void sfAdd4Real( long Template[4], RealNumber *Real ) { /* Begin `sfAdd4Real'. */ *((RealNumber *)Template[0]) += *Real; *((RealNumber *)Template[1]) += *Real; *((RealNumber *)Template[2]) -= *Real; *((RealNumber *)Template[3]) -= *Real; } #ifdef spCOMPLEX void sfAdd4Imag( long Template[4], RealNumber *Imag ) { /* Begin `sfAdd4Imag'. */ *(((RealNumber *)Template[0])+1) += *Imag; *(((RealNumber *)Template[1])+1) += *Imag; *(((RealNumber *)Template[2])+1) -= *Imag; *(((RealNumber *)Template[3])+1) -= *Imag; } void sfAdd4Complex( long Template[4], RealNumber *Real, RealNumber *Imag ) { /* Begin `sfAdd4Complex'. */ *((RealNumber *)Template[0]) += *Real; *((RealNumber *)Template[1]) += *Real; *((RealNumber *)Template[2]) -= *Real; *((RealNumber *)Template[3]) -= *Real; *(((RealNumber *)Template[0])+1) += *Imag; *(((RealNumber *)Template[1])+1) += *Imag; *(((RealNumber *)Template[2])+1) -= *Imag; *(((RealNumber *)Template[3])+1) -= *Imag; } #endif /* spCOMPLEX */ #endif /* QUAD_ELEMENT */ /* * ORDER AND FACTOR MATRIX * * This routine chooses a pivot order for the matrix and factors it * into LU form. It handles both the initial factorization and subsequent * factorizations when a reordering is desired. This is handled in a manner * that is transparent to the user. The routine uses a variation of * Gauss's method where the pivots are associated with L and the * diagonal terms of U are one. * * >>> Returned: [INTEGER of INTEGER*2] * The error code is returned. Possible errors are listed below. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to matrix. * RHS <input> (RealVector) [REAL (1) or DOUBLE PRECISION (1)] * Representative right-hand side vector that is used to determine * pivoting order when the right hand side vector is sparse. If * RHS is a NULL pointer then the RHS vector is assumed to * be full and it is not used when determining the pivoting * order. * RelThreshold <input> (RealNumber *) [REAL or DOUBLE PRECISION] * This number determines what the pivot relative threshold will * be. It should be between zero and one. If it is one then the * pivoting method becomes complete pivoting, which is very slow * and tends to fill up the matrix. If it is set close to zero * the pivoting method becomes strict Markowitz with no * threshold. The pivot threshold is used to eliminate pivot * candidates that would cause excessive element growth if they * were used. Element growth is the cause of roundoff error. * Element growth occurs even in well-conditioned matrices. * Setting the RelThreshold large will reduce element growth and * roundoff error, but setting it too large will cause execution * time to be excessive and will result in a large number of * fill-ins. If this occurs, accuracy can actually be degraded * because of the large number of operations required on the * matrix due to the large number of fill-ins. A good value seems * to be 0.001. The default is chosen by giving a value larger * than one or less than or equal to zero. This value should be * increased and the matrix resolved if growth is found to be * excessive. Changing the pivot threshold does not improve * performance on matrices where growth is low, as is often the * case with ill-conditioned matrices. Once a valid threshold is * given, it becomes the new default. The default value of * RelThreshold was choosen for use with nearly diagonally * dominant matrices such as node- and modified-node admittance * matrices. For these matrices it is usually best to use * diagonal pivoting. For matrices without a strong diagonal, it * is usually best to use a larger threshold, such as 0.01 or * 0.1. * AbsThreshold <input> (RealNumber *) [REAL or DOUBLE PRECISION] * The absolute magnitude an element must have to be considered * as a pivot candidate, except as a last resort. This number * should be set significantly smaller than the smallest diagonal * element that is is expected to be placed in the matrix. If * there is no reasonable prediction for the lower bound on these * elements, then AbsThreshold should be set to zero. * AbsThreshold is used to reduce the possibility of choosing as a * pivot an element that has suffered heavy cancellation and as a * result mainly consists of roundoff error. Once a valid * threshold is given, it becomes the new default. * DiagPivoting <input> (long *) [LOGICAL] * A flag indicating that pivot selection should be confined to the * diagonal if possible. If DiagPivoting is nonzero and if * DIAGONAL_PIVOTING is enabled pivots will be chosen only from * the diagonal unless there are no diagonal elements that satisfy * the threshold criteria. Otherwise, the entire reduced * submatrix is searched when looking for a pivot. The diagonal * pivoting in Sparse is efficient and well refined, while the * off-diagonal pivoting is not. For symmetric and near symmetric * matrices, it is best to use diagonal pivoting because it * results in the best performance when reordering the matrix and * when factoring the matrix without ordering. If there is a * considerable amount of nonsymmetry in the matrix, then * off-diagonal pivoting may result in a better equation ordering * simply because there are more pivot candidates to choose from. * A better ordering results in faster subsequent factorizations. * However, the initial pivot selection process takes considerably * longer for off-diagonal pivoting. * * >>> Possible errors: * spNO_MEMORY * spSINGULAR * spSMALL_PIVOT * Error is cleared in this function. */ int sfOrderAndFactor( long *Matrix, RealNumber RHS[], RealNumber *RelThreshold, RealNumber* AbsThreshold, long *DiagPivoting ) { /* Begin `sfOrderAndFactor'. */ return spOrderAndFactor( (char *)*Matrix, RHS, *RelThreshold, *AbsThreshold, (SPBOOLEAN)*DiagPivoting ); } /* * FACTOR MATRIX * * This routine is the companion routine to spOrderAndFactor(). * Unlike sfOrderAndFactor(), sfFactor() cannot change the ordering. * It is also faster than sfOrderAndFactor(). The standard way of * using these two routines is to first use sfOrderAndFactor() for the * initial factorization. For subsequent factorizations, sfFactor() * is used if there is some assurance that little growth will occur * (say for example, that the matrix is diagonally dominant). If * sfFactor() is called for the initial factorization of the matrix, * then sfOrderAndFactor() is automatically called with the default * threshold. This routine uses "row at a time" LU factorization. * Pivots are associated with the lower triangular matrix and the * diagonals of the upper triangular matrix are ones. * * >>> Returned: [INTEGER or INTEGER*2] * The error code is returned. Possible errors are listed below. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to matrix. * * >>> Possible errors: * spNO_MEMORY * spSINGULAR * spZERO_DIAG * spSMALL_PIVOT * Error is cleared in this function. */ int sfFactor( long *Matrix ) { /* Begin `sfFactor'. */ return spFactor((char *)*Matrix); } /* * PARTITION MATRIX * * This routine determines the cost to factor each row using both * direct and indirect addressing and decides, on a row-by-row basis, * which addressing mode is fastest. This information is used in * sfFactor() to speed the factorization. * * When factoring a previously ordered matrix using sfFactor(), Sparse * operates on a row-at-a-time basis. For speed, on each step, the * row being updated is copied into a full vector and the operations * are performed on that vector. This can be done one of two ways, * either using direct addressing or indirect addressing. Direct * addressing is fastest when the matrix is relatively dense and * indirect addressing is best when the matrix is quite sparse. The * user selects the type of partition used with Mode. If Mode is set * to spDIRECT_PARTITION, then the all rows are placed in the direct * addressing partition. Similarly, if Mode is set to * spINDIRECT_PARTITION, then the all rows are placed in the indirect * addressing partition. By setting Mode to spAUTO_PARTITION, the * user allows Sparse to select the partition for each row * individually. sfFactor() generally runs faster if Sparse is * allowed to choose its own partitioning, however choosing a * partition is expensive. The time required to choose a partition is * of the same order of the cost to factor the matrix. If you plan to * factor a large number of matrices with the same structure, it is * best to let Sparse choose the partition. Otherwise, you should * choose the partition based on the predicted density of the matrix. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to matrix. * Mode <input> (int *) [INTEGER or INTEGER*2] * Mode must be one of three special codes: spDIRECT_PARTITION, * spINDIRECT_PARTITION, or spAUTO_PARTITION. */ void sfPartition( long *Matrix, int *Mode ) { /* Begin `sfPartition'. */ spPartition((char *)*Matrix, *Mode); } /* * SOLVE MATRIX EQUATION * * Performs forward elimination and back substitution to find the * unknown vector from the RHS vector and factored matrix. This * routine assumes that the pivots are associated with the lower * triangular (L) matrix and that the diagonal of the upper triangular * (U) matrix consists of ones. This routine arranges the computation * in different way than is traditionally used in order to exploit the * sparsity of the right-hand side. See the reference in spRevision. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to matrix. * RHS <input> (RealVector) [REAL (1) or DOUBLE PRECISION (1)] * RHS is the input data array, the right hand side. This data is * undisturbed and may be reused for other solves. * Solution <output> (RealVector) [REAL (1) or DOUBLE PRECISION (1)] * Solution is the output data array. This routine is constructed such that * RHS and Solution can be the same array. * iRHS <input> (RealVector) [REAL (1) or DOUBLE PRECISION (1)] * iRHS is the imaginary portion of the input data array, the right * hand side. This data is undisturbed and may be reused for other solves. * This argument is only necessary if matrix is complex and if * spSEPARATED_COMPLEX_VECTOR is set true. * iSolution <output> (RealVector) [REAL (1) or DOUBLE PRECISION (1)] * iSolution is the imaginary portion of the output data array. This * routine is constructed such that iRHS and iSolution can be * the same array. This argument is only necessary if matrix is complex * and if spSEPARATED_COMPLEX_VECTOR is set true. * * >>> Obscure Macros * IMAG_VECTORS * Replaces itself with `, iRHS, iSolution' if the options spCOMPLEX and * spSEPARATED_COMPLEX_VECTORS are set, otherwise it disappears * without a trace. */ /*VARARGS3*/ void sfSolve( long *Matrix, RealVector RHS, RealVector Solution IMAG_VECTORS ) { /* Begin `sfSolve'. */ spSolve( (char *)*Matrix, RHS, Solution IMAG_VECTORS ); } #ifdef TRANSPOSE /* * SOLVE TRANSPOSED MATRIX EQUATION * * Performs forward elimination and back substitution to find the * unknown vector from the RHS vector and transposed factored * matrix. This routine is useful when performing sensitivity analysis * on a circuit using the adjoint method. This routine assumes that * the pivots are associated with the untransposed lower triangular * (L) matrix and that the diagonal of the untransposed upper * triangular (U) matrix consists of ones. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to matrix. * RHS <input> (RealVector) [REAL (1) or DOUBLE PRECISION (1)] * RHS is the input data array, the right hand side. This data is * undisturbed and may be reused for other solves. * Solution <output> (RealVector) [REAL (1) or DOUBLE PRECISION (1)] * Solution is the output data array. This routine is constructed such that * RHS and Solution can be the same array. * iRHS <input> (RealVector) [REAL (1) or DOUBLE PRECISION (1)] * iRHS is the imaginary portion of the input data array, the right * hand side. This data is undisturbed and may be reused for other solves. * If spSEPARATED_COMPLEX_VECTOR is set false, or if matrix is real, there * is no need to supply this array. * iSolution <output> (RealVector) [REAL (1) or DOUBLE PRECISION (1)] * iSolution is the imaginary portion of the output data array. This * routine is constructed such that iRHS and iSolution can be * the same array. If spSEPARATED_COMPLEX_VECTOR is set false, or if * matrix is real, there is no need to supply this array. * * >>> Obscure Macros * IMAG_VECTORS * Replaces itself with `, iRHS, iSolution' if the options spCOMPLEX and * spSEPARATED_COMPLEX_VECTORS are set, otherwise it disappears * without a trace. */ /*VARARGS3*/ void sfSolveTransposed( long *Matrix, RealVector RHS, RealVector Solution IMAG_VECTORS ) { /* Begin `sfSolveTransposed'. */ spSolveTransposed( (char *)*Matrix, RHS, Solution IMAG_VECTORS ); } #endif /* TRANSPOSE */ #ifdef DOCUMENTATION /* * PRINT MATRIX * * Formats and send the matrix to standard output. Some elementary * statistics are also output. The matrix is output in a format that is * readable by people. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to matrix. * PrintReordered <input> (long *) [LOGICAL] * Indicates whether the matrix should be printed out in its original * form, as input by the user, or whether it should be printed in its * reordered form, as used by the matrix routines. A zero indicates that * the matrix should be printed as inputed, a one indicates that it * should be printed reordered. * Data <input> (long *) [LOGICAL] * Boolean flag that when false indicates that output should be * compressed such that only the existence of an element should be * indicated rather than giving the actual value. Thus 10 times as many * can be printed on a row. A zero signifies that the matrix should * be printed compressed. A one indicates that the matrix should be * printed in all its glory. * Header <input> (long *) [LOGICAL] * Flag indicating that extra information such as the row and column * numbers should be printed. */ void sfPrint( long *Matrix, long *Data, long *PrintReordered, long *Header ) { /* Begin `sfPrint'. */ spPrint( (char *)*Matrix, (int)*PrintReordered, (int)*Data, (int)*Header ); } #endif /* DOCUMENTATION */ #ifdef DOCUMENTATION /* * OUTPUT MATRIX TO FILE * * Writes matrix to file in format suitable to be read back in by the * matrix test program. Data is sent to a file with a fixed name because * it is impossible to pass strings from FORTRAN to C in a manner that is * portable. * * >>> Returns: * One is returned if routine was successful, otherwise zero is returned. * The calling function can query errno (the system global error variable) * as to the reason why this routine failed. * * >>> Arguments: [LOGICAL] * Matrix <input> (long *) [INTEGER] * Pointer to matrix. * Reordered <input> (long *) [LOGICAL] * Specifies whether matrix should be output in reordered form, * or in original order. * Data <input> (long *) [LOGICAL] * Indicates that the element values should be output along with * the indices for each element. This parameter must be true if * matrix is to be read by the sparse test program. * Header <input> (long *) [LOGICAL] * Indicates that header is desired. This parameter must be true if * matrix is to be read by the sparse test program. */ #define MATRIX_FILE_NAME "spMatrix" #define STATS_FILE_NAME "spStats" long sfFileMatrix( long *Matrix, long *Reordered, long *Data, long *Header ) { /* Begin `sfFileMatrix'. */ return spFileMatrix( (char *)*Matrix, MATRIX_FILE_NAME, "", (int)*Reordered, (int)*Data, (int)*Header ); } #endif /* DOCUMENTATION */ #ifdef DOCUMENTATION /* * OUTPUT SOURCE VECTOR TO FILE * * Writes vector to file in format suitable to be read back in by the * matrix test program. This routine should be executed after the function * sfFileMatrix. * * >>> Returns: * One is returned if routine was successful, otherwise zero is returned. * The calling function can query errno (the system global error variable) * as to the reason why this routine failed. * * >>> Arguments: * Matrix <input> (long *) * Pointer to matrix. * RHS <input> (RealNumber []) [REAL (1) or DOUBLE PRECISION (1)] * Right-hand side vector. This is only the real portion if * spSEPARATED_COMPLEX_VECTORS is true. * iRHS <input> (RealNumber []) [REAL (1) or DOUBLE PRECISION (1)] * Right-hand side vector, imaginary portion. Not necessary if matrix * is real or if spSEPARATED_COMPLEX_VECTORS is set false. */ int sfFileVector( long *Matrix, RealVector RHS IMAG_RHS ) { /* Begin `sfFileVector'. */ return spFileVector( (char *)*Matrix, MATRIX_FILE_NAME, RHS IMAG_RHS ); } #endif /* DOCUMENTATION */ #ifdef DOCUMENTATION /* * OUTPUT STATISTICS TO FILE * * Writes useful information concerning the matrix to a file. Should be * executed after the matrix is factored. * * >>> Returns: [LOGICAL] * One is returned if routine was successful, otherwise zero is returned. * The calling function can query errno (the system global error variable) * as to the reason why this routine failed. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to matrix. */ int sfFileStats( long *Matrix ) { /* Begin `sfFileStats'. */ return spFileStats( (char *)*Matrix, STATS_FILE_NAME, "" ); } #endif /* DOCUMENTATION */ #ifdef MODIFIED_NODAL /* * PREORDER MODIFIED NODE ADMITTANCE MATRIX TO REMOVE ZEROS FROM DIAGONAL * * This routine massages modified node admittance matrices to remove * zeros from the diagonal. It takes advantage of the fact that the * row and column associated with a zero diagonal usually have * structural ones placed symmetricly. This routine should be used * only on modified node admittance matrices and should be executed * after the matrix has been built but before the factorization * begins. It should be executed for the initial factorization only * and should be executed before the rows have been linked. Thus it * should be run before using spScale(), spMultiply(), * spDeleteRowAndCol(), or spNorm(). * * This routine exploits the fact that the structural one are placed * in the matrix in symmetric twins. For example, the stamps for * grounded and a floating voltage sources are * grounded: floating: * [ x x 1 ] [ x x 1 ] * [ x x ] [ x x -1 ] * [ 1 ] [ 1 -1 ] * Notice for the grounded source, there is one set of twins, and for * the grounded, there are two sets. We remove the zero from the diagonal * by swapping the rows associated with a set of twins. For example: * grounded: floating 1: floating 2: * [ 1 ] [ 1 -1 ] [ x x 1 ] * [ x x ] [ x x -1 ] [ 1 -1 ] * [ x x 1 ] [ x x 1 ] [ x x -1 ] * * It is important to deal with any zero diagonals that only have one * set of twins before dealing with those that have more than one because * swapping row destroys the symmetry of any twins in the rows being * swapped, which may limit future moves. Consider * [ x x 1 ] * [ x x -1 1 ] * [ 1 -1 ] * [ 1 ] * There is one set of twins for diagonal 4 and two for diagonal3. * Dealing with diagonal for first requires swapping rows 2 and 4. * [ x x 1 ] * [ 1 ] * [ 1 -1 ] * [ x x -1 1 ] * We can now deal with diagonal 3 by swapping rows 1 and 3. * [ 1 -1 ] * [ 1 ] * [ x x 1 ] * [ x x -1 1 ] * And we are done, there are no zeros left on the diagonal. However, if * we originally dealt with diagonal 3 first, we could swap rows 2 and 3 * [ x x 1 ] * [ 1 -1 ] * [ x x -1 1 ] * [ 1 ] * Diagonal 4 no longer has a symmetric twin and we cannot continue. * * So we always take care of lone twins first. When none remain, we * choose arbitrarily a set of twins for a diagonal with more than one set * and swap the rows corresponding to that twin. We then deal with any * lone twins that were created and repeat the procedure until no * zero diagonals with symmetric twins remain. * * In this particular implementation, columns are swapped rather than rows. * The algorithm used in this function was developed by Ken Kundert and * Tom Quarles. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to the matrix to be preordered. */ void sfMNA_Preorder( long *Matrix ) { /* Begin `sfMNA_Preorder'. */ spMNA_Preorder( (char *)*Matrix ); } #endif /* MODIFIED_NODAL */ #ifdef SCALING /* * SCALE MATRIX * * This function scales the matrix to enhance the possibility of * finding a good pivoting order. Note that scaling enhances accuracy * of the solution only if it affects the pivoting order, so it makes * no sense to scale the matrix before spFactor(). If scaling is * desired it should be done before spOrderAndFactor(). There * are several things to take into account when choosing the scale * factors. First, the scale factors are directly multiplied against * the elements in the matrix. To prevent roundoff, each scale factor * should be equal to an int power of the number base of the * machine. Since most machines operate in base two, scale factors * should be a power of two. Second, the matrix should be scaled such * that the matrix of element uncertainties is equilibrated. Third, * this function multiplies the scale factors by the elements, so if * one row tends to have uncertainties 1000 times smaller than the * other rows, then its scale factor should be 1024, not 1/1024. * Fourth, to save time, this function does not scale rows or columns * if their scale factors are equal to one. Thus, the scale factors * should be normalized to the most common scale factor. Rows and * columns should be normalized separately. For example, if the size * of the matrix is 100 and 10 rows tend to have uncertainties near * 1e-6 and the remaining 90 have uncertainties near 1e-12, then the * scale factor for the 10 should be 1/1,048,576 and the scale factors * for the remaining 90 should be 1. Fifth, since this routine * directly operates on the matrix, it is necessary to apply the scale * factors to the RHS and Solution vectors. It may be easier to * simply use spOrderAndFactor() on a scaled matrix to choose the * pivoting order, and then throw away the matrix. Subsequent * factorizations, performed with spFactor(), will not need to have * the RHS and Solution vectors descaled. Lastly, this function * should not be executed before the function spMNA_Preorder. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to the matrix to be scaled. * SolutionScaleFactors <input> (RealVector) [REAL(1) or DOUBLE PRECISION(1)] * The array of Solution scale factors. These factors scale the columns. * All scale factors are real valued. * RHS_ScaleFactors <input> (RealVector) [REAL(1) or DOUBLE PRECISION(1)] * The array of RHS scale factors. These factors scale the rows. * All scale factors are real valued. */ void sfScale( long *Matrix, RealVector RHS_ScaleFactors, RealVector SolutionScaleFactors ) { /* Begin `sfScale'. */ spScale( (char *)*Matrix, RHS_ScaleFactors, SolutionScaleFactors ); } #endif /* SCALING */ #ifdef MULTIPLICATION /* * MATRIX MULTIPLICATION * * Multiplies matrix by solution vector to find source vector. * Assumes matrix has not been factored. This routine can be used * as a test to see if solutions are correct. It should not be used * before PreorderFoModifiedNodal(). * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to the matrix. * RHS <output> (RealVector) [REAL(1) or DOUBLE PRECISION(1)] * RHS is the right hand side. This is what is being solved for. * Solution <input> (RealVector) [REAL(1) or DOUBLE PRECISION(1)] * Solution is the vector being multiplied by the matrix. * iRHS <output> (RealVector) [REAL(1) or DOUBLE PRECISION(1)] * iRHS is the imaginary portion of the right hand side. This is * what is being solved for. This is only necessary if the matrix is * complex and spSEPARATED_COMPLEX_VECTORS is true. * iSolution <input> (RealVector) [REAL(1) or DOUBLE PRECISION(1)] * iSolution is the imaginary portion of the vector being multiplied * by the matrix. This is only necessary if the matrix is * complex and spSEPARATED_COMPLEX_VECTORS is true. * * >>> Obscure Macros * IMAG_VECTORS * Replaces itself with `, iRHS, iSolution' if the options spCOMPLEX and * spSEPARATED_COMPLEX_VECTORS are set, otherwise it disappears * without a trace. */ void sfMultiply( long *Matrix, RealVector RHS, RealVector Solution IMAG_VECTORS ) { /* Begin `sfMultiply'. */ spMultiply( (char *)*Matrix, RHS, Solution IMAG_VECTORS ); } #endif /* MULTIPLICATION */ #if MULTIPLICATION AND TRANSPOSE /* * TRANSPOSED MATRIX MULTIPLICATION * * Multiplies transposed matrix by solution vector to find source vector. * Assumes matrix has not been factored. This routine can be used * as a test to see if solutions are correct. It should not be used * before PreorderFoModifiedNodal(). * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to the matrix. * RHS <output> (RealVector) [REAL(1) or DOUBLE PRECISION(1)] * RHS is the right hand side. This is what is being solved for. * Solution <input> (RealVector) [REAL(1) or DOUBLE PRECISION(1)] * Solution is the vector being multiplied by the matrix. * iRHS <output> (RealVector) [REAL(1) or DOUBLE PRECISION(1)] * iRHS is the imaginary portion of the right hand side. This is * what is being solved for. This is only necessary if the matrix is * complex and spSEPARATED_COMPLEX_VECTORS is true. * iSolution <input> (RealVector) [REAL(1) or DOUBLE PRECISION(1)] * iSolution is the imaginary portion of the vector being multiplied * by the matrix. This is only necessary if the matrix is * complex and spSEPARATED_COMPLEX_VECTORS is true. * * >>> Obscure Macros * IMAG_VECTORS * Replaces itself with `, iRHS, iSolution' if the options spCOMPLEX and * spSEPARATED_COMPLEX_VECTORS are set, otherwise it disappears * without a trace. */ void sfMultTransposed( long *Matrix, RealVector RHS, RealVector Solution IMAG_VECTORS ) { /* Begin `sfMultTransposed'. */ spMultTransposed( (char *)*Matrix, RHS, Solution IMAG_VECTORS ); } #endif /* MULTIPLICATION AND TRANSPOSE */ #ifdef DETERMINANT /* * CALCULATE DETERMINANT * * This routine in capable of calculating the determinant of the * matrix once the LU factorization has been performed. Hence, only * use this routine after spFactor() and before spClear(). * The determinant equals the product of all the diagonal elements of * the lower triangular matrix L, except that this product may need * negating. Whether the product or the negative product equals the * determinant is determined by the number of row and column * interchanges performed. Note that the determinants of matrices can * be very large or very small. On large matrices, the determinant * can be far larger or smaller than can be represented by a floating * point number. For this reason the determinant is scaled to a * reasonable value and the logarithm of the scale factor is returned. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * A pointer to the matrix for which the determinant is desired. * pExponent <output> (int *) [INTEGER or INTEGER*2] * The logarithm base 10 of the scale factor for the determinant. To * find * the actual determinant, Exponent should be added to the exponent of * DeterminantReal. * pDeterminant <output> (RealNumber *) [REAL or DOUBLE PRECISION] * The real portion of the determinant. This number is scaled to be * greater than or equal to 1.0 and less than 10.0. * piDeterminant <output> (RealNumber *) [REAL or DOUBLE PRECISION] * The imaginary portion of the determinant. When the matrix is real * this pointer need not be supplied, nothing will be returned. This * number is scaled to be greater than or equal to 1.0 and less than 10.0. */ #ifdef spCOMPLEX void sfDeterminant( long *Matrix, int *pExponent, RealNumber *pDeterminant, RealNumber *piDeterminant ) { /* Begin `sfDeterminant'. */ spDeterminant( (char *)*Matrix, pExponent, pDeterminant, piDeterminant ); } #else /* spCOMPLEX */ void sfDeterminant( long *Matrix, int *pExponent, RealNumber *pDeterminant ) { /* Begin `sfDeterminant'. */ spDeterminant( (char *)*Matrix, pExponent, pDeterminant ); } #endif /* spCOMPLEX */ #endif /* DETERMINANT */ /* * RETURN MATRIX ERROR STATUS * * This function is used to determine the error status of the given matrix. * * >>> Returned: [INTEGER or INTEGER*2] * The error status of the given matrix. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * The matrix for which the error status is desired. */ int sfError( long *Matrix ) { /* Begin `sfError'. */ return spError( (char *)*Matrix ); } /* * WHERE IS MATRIX SINGULAR * * This function returns the row and column number where the matrix was * detected as singular or where a zero was detected on the diagonal. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * The matrix for which the error status is desired. * pRow <output> (int *) [INTEGER or INTEGER*2] * The row number. * pCol <output> (int *) [INTEGER or INTEGER*2] * The column number. */ void sfWhereSingular( long *Matrix, int *Row, int *Col ) { /* Begin `sfWhereSingular'. */ spWhereSingular( (char *)*Matrix, Row, Col ); } /* * MATRIX SIZE * * Returns the size of the matrix. Either the internal or external size of * the matrix is returned. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to matrix. * External <input> (SPBOOLEAN) [LOGICAL] * If External is set true, the external size , i.e., the value of the * largest external row or column number encountered is returned. * Otherwise the true size of the matrix is returned. These two sizes * may differ if the TRANSLATE option is set true. */ int sfGetSize( long *Matrix, long *External ) { /* Begin `sfGetSize'. */ return spGetSize( (char *)*Matrix, (SPBOOLEAN)*External ); } /* * SET MATRIX COMPLEX OR REAL * * Forces matrix to be either real or complex. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to matrix. */ void sfSetReal( long *Matrix ) { /* Begin `sfSetReal'. */ spSetReal( (char *)*Matrix ); } void sfSetComplex( long *Matrix ) { /* Begin `sfSetComplex'. */ spSetComplex( (char *)*Matrix ); } /* * ELEMENT OR FILL-IN COUNT * * Two functions used to return simple statistics. Either the number * of total elements, or the number of fill-ins can be returned. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to matrix. */ int sfFillinCount( long *Matrix ) { /* Begin `sfFillinCount'. */ return spFillinCount( (char *)*Matrix ); } int sfElementCount( long *Matrix ) { /* Begin `sfElementCount'. */ return spElementCount( (char *)*Matrix ); } #if TRANSLATE AND DELETE /* * DELETE A ROW AND COLUMN FROM THE MATRIX * * Deletes a row and a column from a matrix. * * Sparse will abort if an attempt is made to delete a row or column that * doesn't exist. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to the matrix in which the row and column are to be deleted. * Row <input> (int) [INTEGER or INTEGER*2] * Row to be deleted. * Col <input> (int) [INTEGER or INTEGER*2] * Column to be deleted. */ void sfDeleteRowAndCol( long *Matrix, int *Row, int *Col ) { /* Begin `sfDeleteRowAndCol'. */ spDeleteRowAndCol( (char *)*Matrix, *Row, *Col ); } #endif #ifdef PSEUDOCONDITION /* * CALCULATE PSEUDOCONDITION * * Computes the magnitude of the ratio of the largest to the smallest * pivots. This quantity is an indicator of ill-conditioning in the * matrix. If this ratio is large, and if the matrix is scaled such * that uncertainties in the RHS and the matrix entries are * equilibrated, then the matrix is ill-conditioned. However, a small * ratio does not necessarily imply that the matrix is * well-conditioned. This routine must only be used after a matrix * has been factored by sfOrderAndFactor() or sfFactor() and before it * is cleared by sfClear() or spInitialize(). The pseudocondition is faster * to compute than the condition number calculated by sfCondition(), but * is not as informative. * * >>> Returns: [REAL or DOUBLE PRECISION] * The magnitude of the ratio of the largest to smallest pivot used during * previous factorization. If the matrix was singular, zero is returned. * * >>> Arguments: * Matrix <input> (long *) * Pointer to the matrix. */ RealNumber sfPseudoCondition( long *Matrix ) { /* Begin `sfPseudoCondition'. */ return spPseudoCondition( (char *)Matrix ); } #endif #ifdef CONDITION /* * ESTIMATE CONDITION NUMBER * * Computes an estimate of the condition number using a variation on * the LINPACK condition number estimation algorithm. This quantity is * an indicator of ill-conditioning in the matrix. To avoid problems * with overflow, the reciprocal of the condition number is returned. * If this number is small, and if the matrix is scaled such that * uncertainties in the RHS and the matrix entries are equilibrated, * then the matrix is ill-conditioned. If the this number is near * one, the matrix is well conditioned. This routine must only be * used after a matrix has been factored by sfOrderAndFactor() or * sfFactor() and before it is cleared by sfClear() or spInitialize(). * * Unlike the LINPACK condition number estimator, this routines * returns the L infinity condition number. This is an artifact of * Sparse placing ones on the diagonal of the upper triangular matrix * rather than the lower. This difference should be of no importance. * * References: * A.K. Cline, C.B. Moler, G.W. Stewart, J.H. Wilkinson. An estimate * for the condition number of a matrix. SIAM Journal on Numerical * Analysis. Vol. 16, No. 2, pages 368-375, April 1979. * * J.J. Dongarra, C.B. Moler, J.R. Bunch, G.W. Stewart. LINPACK * User's Guide. SIAM, 1979. * * Roger G. Grimes, John G. Lewis. Condition number estimation for * sparse matrices. SIAM Journal on Scientific and Statistical * Computing. Vol. 2, No. 4, pages 384-388, December 1981. * * Dianne Prost O'Leary. Estimating matrix condition numbers. SIAM * Journal on Scientific and Statistical Computing. Vol. 1, No. 2, * pages 205-209, June 1980. * * >>> Returns: [REAL or DOUBLE PRECISION] * The reciprocal of the condition number. If the matrix was singular, * zero is returned. * * >>> Arguments: * eMatrix <input> (long *) * Pointer to the matrix. * NormOfMatrix <input> (RealNumber *) [REAL or DOUBLE PRECISION] * The L-infinity norm of the unfactored matrix as computed by * spNorm(). * pError <output> (int *) [INTEGER or INTEGER*2] * Used to return error code. * * >>> Possible errors: * spSINGULAR * spNO_MEMORY */ RealNumber sfCondition( long * Matrix, RealNumber *NormOfMatrix, int *pError ) { /* Begin `sfCondition'. */ return spCondition( (char *)*Matrix, *NormOfMatrix, pError ); } /* * L-INFINITY MATRIX NORM * * Computes the L-infinity norm of an unfactored matrix. It is a fatal * error to pass this routine a factored matrix. * * One difficulty is that the rows may not be linked. * * >>> Returns: [REAL or DOUBLE PRECISION] * The largest absolute row sum of matrix. * * >>> Arguments: * Matrix <input> (long *) * Pointer to the matrix. */ RealNumber sfNorm( long *Matrix ) { /* Begin `sfNorm'. */ return spNorm( (char *)*Matrix ); } #endif /* CONDITION */ #ifdef STABILITY /* * STABILITY OF FACTORIZATION * * The following routines are used to gauge the stability of a * factorization. If the factorization is determined to be too unstable, * then the matrix should be reordered. The routines compute quantities * that are needed in the computation of a bound on the error attributed * to any one element in the matrix during the factorization. In other * words, there is a matrix E = [e_ij] of error terms such that A+E = LU. * This routine finds a bound on |e_ij|. Erisman & Reid [1] showed that * |e_ij| < 3.01 u rho m_ij, where u is the machine rounding unit, * rho = max a_ij where the max is taken over every row i, column j, and * step k, and m_ij is the number of multiplications required in the * computation of l_ij if i > j or u_ij otherwise. Barlow [2] showed that * rho < max_i || l_i ||_p max_j || u_j ||_q where 1/p + 1/q = 1. * * The first routine finds the magnitude on the largest element in the * matrix. If the matrix has not yet been factored, the largest * element is found by direct search. If the matrix is factored, a * bound on the largest element in any of the reduced submatrices is * computed using Barlow with p = oo and q = 1. The ratio of these * two numbers is the growth, which can be used to determine if the * pivoting order is adequate. A large growth implies that * considerable error has been made in the factorization and that it * is probably a good idea to reorder the matrix. If a large growth * in encountered after using spFactor(), reconstruct the matrix and * refactor using spOrderAndFactor(). If a large growth is * encountered after using spOrderAndFactor(), refactor using * spOrderAndFactor() with the pivot threshold increased, say to 0.1. * * Using only the size of the matrix as an upper bound on m_ij and * Barlow's bound, the user can estimate the size of the matrix error * terms e_ij using the bound of Erisman and Reid. The second routine * computes a tighter bound (with more work) based on work by Gear * [3], |e_ij| < 1.01 u rho (t c^3 + (1 + t)c^2) where t is the * threshold and c is the maximum number of off-diagonal elements in * any row of L. The expensive part of computing this bound is * determining the maximum number of off-diagonals in L, which changes * only when the order of the matrix changes. This number is computed * and saved, and only recomputed if the matrix is reordered. * * [1] A. M. Erisman, J. K. Reid. Monitoring the stability of the * triangular factorization of a sparse matrix. Numerische * Mathematik. Vol. 22, No. 3, 1974, pp 183-186. * * [2] J. L. Barlow. A note on monitoring the stability of triangular * decomposition of sparse matrices. "SIAM Journal of Scientific * and Statistical Computing." Vol. 7, No. 1, January 1986, pp 166-168. * * [3] I. S. Duff, A. M. Erisman, J. K. Reid. "Direct Methods for Sparse * Matrices." Oxford 1986. pp 99. */ /* * LARGEST ELEMENT IN MATRIX * * >>> Returns: [REAL or DOUBLE PRECISION] * If matrix is not factored, returns the magnitude of the largest element in * the matrix. If the matrix is factored, a bound on the magnitude of the * largest element in any of the reduced submatrices is returned. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to the matrix. */ RealNumber sfLargestElement( long *Matrix ) { /* Begin `sfLargestElement'. */ return spLargestElement( (char *)Matrix ); } /* * MATRIX ROUNDOFF ERROR * * >>> Returns: [REAL or DOUBLE PRECISION] * Returns a bound on the magnitude of the largest element in E = A - LU. * * >>> Arguments: * Matrix <input> (long *) [INTEGER] * Pointer to the matrix. * Rho <input> (RealNumber *) [REAL or DOUBLE PRECISION] * The bound on the magnitude of the largest element in any of the * reduced submatrices. This is the number computed by the function * spLargestElement() when given a factored matrix. If this number is * negative, the bound will be computed automatically. */ RealNumber sfRoundoff( long *Matrix, RealNumber *Rho ) { /* Begin `sfRoundoff'. */ return spRoundoff( (char *)*Matrix, *Rho ); } #endif #endif /* FORTRAN */

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