SUBROUTINE MB02PD( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, $ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, $ IWORK, DWORK, INFO ) C C SLICOT RELEASE 5.0. C C Copyright (c) 2002-2009 NICONET e.V. C C This program is free software: you can redistribute it and/or C modify it under the terms of the GNU General Public License as C published by the Free Software Foundation, either version 2 of C the License, or (at your option) any later version. C C This program is distributed in the hope that it will be useful, C but WITHOUT ANY WARRANTY; without even the implied warranty of C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the C GNU General Public License for more details. C C You should have received a copy of the GNU General Public License C along with this program. If not, see C . C C PURPOSE C C To solve (if well-conditioned) the matrix equations C C op( A )*X = B, C C where X and B are N-by-NRHS matrices, A is an N-by-N matrix and C op( A ) is one of C C op( A ) = A or op( A ) = A'. C C Error bounds on the solution and a condition estimate are also C provided. C C ARGUMENTS C C Mode Parameters C C FACT CHARACTER*1 C Specifies whether or not the factored form of the matrix A C is supplied on entry, and if not, whether the matrix A C should be equilibrated before it is factored. C = 'F': On entry, AF and IPIV contain the factored form C of A. If EQUED is not 'N', the matrix A has been C equilibrated with scaling factors given by R C and C. A, AF, and IPIV are not modified. C = 'N': The matrix A will be copied to AF and factored. C = 'E': The matrix A will be equilibrated if necessary, C then copied to AF and factored. C C TRANS CHARACTER*1 C Specifies the form of the system of equations as follows: C = 'N': A * X = B (No transpose); C = 'T': A**T * X = B (Transpose); C = 'C': A**H * X = B (Transpose). C C Input/Output Parameters C C N (input) INTEGER C The number of linear equations, i.e., the order of the C matrix A. N >= 0. C C NRHS (input) INTEGER C The number of right hand sides, i.e., the number of C columns of the matrices B and X. NRHS >= 0. C C A (input/output) DOUBLE PRECISION array, dimension (LDA,N) C On entry, the leading N-by-N part of this array must C contain the matrix A. If FACT = 'F' and EQUED is not 'N', C then A must have been equilibrated by the scaling factors C in R and/or C. A is not modified if FACT = 'F' or 'N', C or if FACT = 'E' and EQUED = 'N' on exit. C On exit, if EQUED .NE. 'N', the leading N-by-N part of C this array contains the matrix A scaled as follows: C EQUED = 'R': A := diag(R) * A; C EQUED = 'C': A := A * diag(C); C EQUED = 'B': A := diag(R) * A * diag(C). C C LDA INTEGER C The leading dimension of the array A. LDA >= max(1,N). C C AF (input or output) DOUBLE PRECISION array, dimension C (LDAF,N) C If FACT = 'F', then AF is an input argument and on entry C the leading N-by-N part of this array must contain the C factors L and U from the factorization A = P*L*U as C computed by DGETRF. If EQUED .NE. 'N', then AF is the C factored form of the equilibrated matrix A. C If FACT = 'N', then AF is an output argument and on exit C the leading N-by-N part of this array contains the factors C L and U from the factorization A = P*L*U of the original C matrix A. C If FACT = 'E', then AF is an output argument and on exit C the leading N-by-N part of this array contains the factors C L and U from the factorization A = P*L*U of the C equilibrated matrix A (see the description of A for the C form of the equilibrated matrix). C C LDAF (input) INTEGER C The leading dimension of the array AF. LDAF >= max(1,N). C C IPIV (input or output) INTEGER array, dimension (N) C If FACT = 'F', then IPIV is an input argument and on entry C it must contain the pivot indices from the factorization C A = P*L*U as computed by DGETRF; row i of the matrix was C interchanged with row IPIV(i). C If FACT = 'N', then IPIV is an output argument and on exit C it contains the pivot indices from the factorization C A = P*L*U of the original matrix A. C If FACT = 'E', then IPIV is an output argument and on exit C it contains the pivot indices from the factorization C A = P*L*U of the equilibrated matrix A. C C EQUED (input or output) CHARACTER*1 C Specifies the form of equilibration that was done as C follows: C = 'N': No equilibration (always true if FACT = 'N'); C = 'R': Row equilibration, i.e., A has been premultiplied C by diag(R); C = 'C': Column equilibration, i.e., A has been C postmultiplied by diag(C); C = 'B': Both row and column equilibration, i.e., A has C been replaced by diag(R) * A * diag(C). C EQUED is an input argument if FACT = 'F'; otherwise, it is C an output argument. C C R (input or output) DOUBLE PRECISION array, dimension (N) C The row scale factors for A. If EQUED = 'R' or 'B', A is C multiplied on the left by diag(R); if EQUED = 'N' or 'C', C R is not accessed. R is an input argument if FACT = 'F'; C otherwise, R is an output argument. If FACT = 'F' and C EQUED = 'R' or 'B', each element of R must be positive. C C C (input or output) DOUBLE PRECISION array, dimension (N) C The column scale factors for A. If EQUED = 'C' or 'B', C A is multiplied on the right by diag(C); if EQUED = 'N' C or 'R', C is not accessed. C is an input argument if C FACT = 'F'; otherwise, C is an output argument. If C FACT = 'F' and EQUED = 'C' or 'B', each element of C must C be positive. C C B (input/output) DOUBLE PRECISION array, dimension C (LDB,NRHS) C On entry, the leading N-by-NRHS part of this array must C contain the right-hand side matrix B. C On exit, C if EQUED = 'N', B is not modified; C if TRANS = 'N' and EQUED = 'R' or 'B', the leading C N-by-NRHS part of this array contains diag(R)*B; C if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', the leading C N-by-NRHS part of this array contains diag(C)*B. C C LDB INTEGER C The leading dimension of the array B. LDB >= max(1,N). C C X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) C If INFO = 0 or INFO = N+1, the leading N-by-NRHS part of C this array contains the solution matrix X to the original C system of equations. Note that A and B are modified on C exit if EQUED .NE. 'N', and the solution to the C equilibrated system is inv(diag(C))*X if TRANS = 'N' and C EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or C 'C' and EQUED = 'R' or 'B'. C C LDX (input) INTEGER C The leading dimension of the array X. LDX >= max(1,N). C C RCOND (output) DOUBLE PRECISION C The estimate of the reciprocal condition number of the C matrix A after equilibration (if done). If RCOND is less C than the machine precision (in particular, if RCOND = 0), C the matrix is singular to working precision. This C condition is indicated by a return code of INFO > 0. C For efficiency reasons, RCOND is computed only when the C matrix A is factored, i.e., for FACT = 'N' or 'E'. For C FACT = 'F', RCOND is not used, but it is assumed that it C has been computed and checked before the routine call. C C FERR (output) DOUBLE PRECISION array, dimension (NRHS) C The estimated forward error bound for each solution vector C X(j) (the j-th column of the solution matrix X). C If XTRUE is the true solution corresponding to X(j), C FERR(j) is an estimated upper bound for the magnitude of C the largest element in (X(j) - XTRUE) divided by the C magnitude of the largest element in X(j). The estimate C is as reliable as the estimate for RCOND, and is almost C always a slight overestimate of the true error. C C BERR (output) DOUBLE PRECISION array, dimension (NRHS) C The componentwise relative backward error of each solution C vector X(j) (i.e., the smallest relative change in C any element of A or B that makes X(j) an exact solution). C C Workspace C C IWORK INTEGER array, dimension (N) C C DWORK DOUBLE PRECISION array, dimension (4*N) C On exit, DWORK(1) contains the reciprocal pivot growth C factor norm(A)/norm(U). The "max absolute element" norm is C used. If DWORK(1) is much less than 1, then the stability C of the LU factorization of the (equilibrated) matrix A C could be poor. This also means that the solution X, C condition estimator RCOND, and forward error bound FERR C could be unreliable. If factorization fails with C 0 < INFO <= N, then DWORK(1) contains the reciprocal pivot C growth factor for the leading INFO columns of A. C C Error Indicator C C INFO INTEGER C = 0: successful exit; C < 0: if INFO = -i, the i-th argument had an illegal C value; C > 0: if INFO = i, and i is C <= N: U(i,i) is exactly zero. The factorization C has been completed, but the factor U is C exactly singular, so the solution and error C bounds could not be computed. RCOND = 0 is C returned. C = N+1: U is nonsingular, but RCOND is less than C machine precision, meaning that the matrix is C singular to working precision. Nevertheless, C the solution and error bounds are computed C because there are a number of situations C where the computed solution can be more C accurate than the value of RCOND would C suggest. C The positive values for INFO are set only when the C matrix A is factored, i.e., for FACT = 'N' or 'E'. C C METHOD C C The following steps are performed: C C 1. If FACT = 'E', real scaling factors are computed to equilibrate C the system: C C TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B C TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B C TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B C C Whether or not the system will be equilibrated depends on the C scaling of the matrix A, but if equilibration is used, A is C overwritten by diag(R)*A*diag(C) and B by diag(R)*B C (if TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C'). C C 2. If FACT = 'N' or 'E', the LU decomposition is used to factor C the matrix A (after equilibration if FACT = 'E') as C A = P * L * U, C where P is a permutation matrix, L is a unit lower triangular C matrix, and U is upper triangular. C C 3. If some U(i,i)=0, so that U is exactly singular, then the C routine returns with INFO = i. Otherwise, the factored form C of A is used to estimate the condition number of the matrix A. C If the reciprocal of the condition number is less than machine C precision, INFO = N+1 is returned as a warning, but the routine C still goes on to solve for X and compute error bounds as C described below. C C 4. The system of equations is solved for X using the factored form C of A. C C 5. Iterative refinement is applied to improve the computed C solution matrix and calculate error bounds and backward error C estimates for it. C C 6. If equilibration was used, the matrix X is premultiplied by C diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so C that it solves the original system before equilibration. C C REFERENCES C C [1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J., C Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., C Ostrouchov, S., Sorensen, D. C LAPACK Users' Guide: Second Edition, SIAM, Philadelphia, 1995. C C FURTHER COMMENTS C C This is a simplified version of the LAPACK Library routine DGESVX, C useful when several sets of matrix equations with the same C coefficient matrix A and/or A' should be solved. C C NUMERICAL ASPECTS C 3 C The algorithm requires 0(N ) operations. C C CONTRIBUTORS C C V. Sima, Research Institute for Informatics, Bucharest, Apr. 1999. C C REVISIONS C C - C C KEYWORDS C C Condition number, matrix algebra, matrix operations. C C ****************************************************************** C C .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) C .. Scalar Arguments .. CHARACTER EQUED, FACT, TRANS INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS DOUBLE PRECISION RCOND C .. C .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), $ BERR( * ), C( * ), DWORK( * ), FERR( * ), $ R( * ), X( LDX, * ) C .. C .. Local Scalars .. LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU CHARACTER NORM INTEGER I, INFEQU, J DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN, $ ROWCND, RPVGRW, SMLNUM C .. C .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANGE, DLANTR EXTERNAL LSAME, DLAMCH, DLANGE, DLANTR C .. C .. External Subroutines .. EXTERNAL DGECON, DGEEQU, DGERFS, DGETRF, DGETRS, DLACPY, $ DLAQGE, XERBLA C .. C .. Intrinsic Functions .. INTRINSIC MAX, MIN C .. C .. Save Statement .. SAVE RPVGRW C .. C .. Executable Statements .. C INFO = 0 NOFACT = LSAME( FACT, 'N' ) EQUIL = LSAME( FACT, 'E' ) NOTRAN = LSAME( TRANS, 'N' ) IF( NOFACT .OR. EQUIL ) THEN EQUED = 'N' ROWEQU = .FALSE. COLEQU = .FALSE. ELSE ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) SMLNUM = DLAMCH( 'Safe minimum' ) BIGNUM = ONE / SMLNUM END IF C C Test the input parameters. C IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) ) $ THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. $ LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN INFO = -10 ELSE IF( ROWEQU ) THEN RCMIN = BIGNUM RCMAX = ZERO DO 10 J = 1, N RCMIN = MIN( RCMIN, R( J ) ) RCMAX = MAX( RCMAX, R( J ) ) 10 CONTINUE IF( RCMIN.LE.ZERO ) THEN INFO = -11 ELSE IF( N.GT.0 ) THEN ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) ELSE ROWCND = ONE END IF END IF IF( COLEQU .AND. INFO.EQ.0 ) THEN RCMIN = BIGNUM RCMAX = ZERO DO 20 J = 1, N RCMIN = MIN( RCMIN, C( J ) ) RCMAX = MAX( RCMAX, C( J ) ) 20 CONTINUE IF( RCMIN.LE.ZERO ) THEN INFO = -12 ELSE IF( N.GT.0 ) THEN COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) ELSE COLCND = ONE END IF END IF IF( INFO.EQ.0 ) THEN IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -14 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -16 END IF END IF END IF C IF( INFO.NE.0 ) THEN CALL XERBLA( 'MB02PD', -INFO ) RETURN END IF C IF( EQUIL ) THEN C C Compute row and column scalings to equilibrate the matrix A. C CALL DGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFEQU ) IF( INFEQU.EQ.0 ) THEN C C Equilibrate the matrix. C CALL DLAQGE( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, $ EQUED ) ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) END IF END IF C C Scale the right hand side. C IF( NOTRAN ) THEN IF( ROWEQU ) THEN DO 40 J = 1, NRHS DO 30 I = 1, N B( I, J ) = R( I )*B( I, J ) 30 CONTINUE 40 CONTINUE END IF ELSE IF( COLEQU ) THEN DO 60 J = 1, NRHS DO 50 I = 1, N B( I, J ) = C( I )*B( I, J ) 50 CONTINUE 60 CONTINUE END IF C IF( NOFACT .OR. EQUIL ) THEN C C Compute the LU factorization of A. C CALL DLACPY( 'Full', N, N, A, LDA, AF, LDAF ) CALL DGETRF( N, N, AF, LDAF, IPIV, INFO ) C C Return if INFO is non-zero. C IF( INFO.NE.0 ) THEN IF( INFO.GT.0 ) THEN C C Compute the reciprocal pivot growth factor of the C leading rank-deficient INFO columns of A. C RPVGRW = DLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF, $ DWORK ) IF( RPVGRW.EQ.ZERO ) THEN RPVGRW = ONE ELSE RPVGRW = DLANGE( 'M', N, INFO, A, LDA, DWORK ) / $ RPVGRW END IF DWORK( 1 ) = RPVGRW RCOND = ZERO END IF RETURN END IF C C Compute the norm of the matrix A and the C reciprocal pivot growth factor RPVGRW. C IF( NOTRAN ) THEN NORM = '1' ELSE NORM = 'I' END IF ANORM = DLANGE( NORM, N, N, A, LDA, DWORK ) RPVGRW = DLANTR( 'M', 'U', 'N', N, N, AF, LDAF, DWORK ) IF( RPVGRW.EQ.ZERO ) THEN RPVGRW = ONE ELSE RPVGRW = DLANGE( 'M', N, N, A, LDA, DWORK ) / RPVGRW END IF C C Compute the reciprocal of the condition number of A. C CALL DGECON( NORM, N, AF, LDAF, ANORM, RCOND, DWORK, IWORK, $ INFO ) C C Set INFO = N+1 if the matrix is singular to working precision. C IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) $ INFO = N + 1 END IF C C Compute the solution matrix X. C CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL DGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO ) C C Use iterative refinement to improve the computed solution and C compute error bounds and backward error estimates for it. C CALL DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, $ LDX, FERR, BERR, DWORK, IWORK, INFO ) C C Transform the solution matrix X to a solution of the original C system. C IF( NOTRAN ) THEN IF( COLEQU ) THEN DO 80 J = 1, NRHS DO 70 I = 1, N X( I, J ) = C( I )*X( I, J ) 70 CONTINUE 80 CONTINUE DO 90 J = 1, NRHS FERR( J ) = FERR( J ) / COLCND 90 CONTINUE END IF ELSE IF( ROWEQU ) THEN DO 110 J = 1, NRHS DO 100 I = 1, N X( I, J ) = R( I )*X( I, J ) 100 CONTINUE 110 CONTINUE DO 120 J = 1, NRHS FERR( J ) = FERR( J ) / ROWCND 120 CONTINUE END IF C DWORK( 1 ) = RPVGRW RETURN C C *** Last line of MB02PD *** END