diff options
Diffstat (limited to 'SRC/sgesvx.f')
| -rw-r--r-- | SRC/sgesvx.f | 479 |
1 files changed, 479 insertions, 0 deletions
diff --git a/SRC/sgesvx.f b/SRC/sgesvx.f new file mode 100644 index 00000000..24dc987b --- /dev/null +++ b/SRC/sgesvx.f @@ -0,0 +1,479 @@ + SUBROUTINE SGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, + $ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, + $ WORK, IWORK, INFO ) +* +* -- LAPACK driver routine (version 3.1) -- +* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. +* November 2006 +* +* .. Scalar Arguments .. + CHARACTER EQUED, FACT, TRANS + INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS + REAL RCOND +* .. +* .. Array Arguments .. + INTEGER IPIV( * ), IWORK( * ) + REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), + $ BERR( * ), C( * ), FERR( * ), R( * ), + $ WORK( * ), X( LDX, * ) +* .. +* +* Purpose +* ======= +* +* SGESVX uses the LU factorization to compute the solution to a real +* system of linear equations +* A * X = B, +* where A is an N-by-N matrix and X and B are N-by-NRHS matrices. +* +* Error bounds on the solution and a condition estimate are also +* provided. +* +* Description +* =========== +* +* The following steps are performed: +* +* 1. If FACT = 'E', real scaling factors are computed to equilibrate +* the system: +* TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B +* TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B +* TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B +* Whether or not the system will be equilibrated depends on the +* scaling of the matrix A, but if equilibration is used, A is +* overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') +* or diag(C)*B (if TRANS = 'T' or 'C'). +* +* 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the +* matrix A (after equilibration if FACT = 'E') as +* A = P * L * U, +* where P is a permutation matrix, L is a unit lower triangular +* matrix, and U is upper triangular. +* +* 3. If some U(i,i)=0, so that U is exactly singular, then the routine +* returns with INFO = i. Otherwise, the factored form of A is used +* to estimate the condition number of the matrix A. If the +* reciprocal of the condition number is less than machine precision, +* INFO = N+1 is returned as a warning, but the routine still goes on +* to solve for X and compute error bounds as described below. +* +* 4. The system of equations is solved for X using the factored form +* of A. +* +* 5. Iterative refinement is applied to improve the computed solution +* matrix and calculate error bounds and backward error estimates +* for it. +* +* 6. If equilibration was used, the matrix X is premultiplied by +* diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so +* that it solves the original system before equilibration. +* +* Arguments +* ========= +* +* FACT (input) CHARACTER*1 +* Specifies whether or not the factored form of the matrix A is +* supplied on entry, and if not, whether the matrix A should be +* equilibrated before it is factored. +* = 'F': On entry, AF and IPIV contain the factored form of A. +* If EQUED is not 'N', the matrix A has been +* equilibrated with scaling factors given by R and C. +* A, AF, and IPIV are not modified. +* = 'N': The matrix A will be copied to AF and factored. +* = 'E': The matrix A will be equilibrated if necessary, then +* copied to AF and factored. +* +* TRANS (input) CHARACTER*1 +* Specifies the form of the system of equations: +* = 'N': A * X = B (No transpose) +* = 'T': A**T * X = B (Transpose) +* = 'C': A**H * X = B (Transpose) +* +* N (input) INTEGER +* The number of linear equations, i.e., the order of the +* matrix A. N >= 0. +* +* NRHS (input) INTEGER +* The number of right hand sides, i.e., the number of columns +* of the matrices B and X. NRHS >= 0. +* +* A (input/output) REAL array, dimension (LDA,N) +* On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is +* not 'N', then A must have been equilibrated by the scaling +* factors in R and/or C. A is not modified if FACT = 'F' or +* 'N', or if FACT = 'E' and EQUED = 'N' on exit. +* +* On exit, if EQUED .ne. 'N', A is scaled as follows: +* EQUED = 'R': A := diag(R) * A +* EQUED = 'C': A := A * diag(C) +* EQUED = 'B': A := diag(R) * A * diag(C). +* +* LDA (input) INTEGER +* The leading dimension of the array A. LDA >= max(1,N). +* +* AF (input or output) REAL array, dimension (LDAF,N) +* If FACT = 'F', then AF is an input argument and on entry +* contains the factors L and U from the factorization +* A = P*L*U as computed by SGETRF. If EQUED .ne. 'N', then +* AF is the factored form of the equilibrated matrix A. +* +* If FACT = 'N', then AF is an output argument and on exit +* returns the factors L and U from the factorization A = P*L*U +* of the original matrix A. +* +* If FACT = 'E', then AF is an output argument and on exit +* returns the factors L and U from the factorization A = P*L*U +* of the equilibrated matrix A (see the description of A for +* the form of the equilibrated matrix). +* +* LDAF (input) INTEGER +* The leading dimension of the array AF. LDAF >= max(1,N). +* +* IPIV (input or output) INTEGER array, dimension (N) +* If FACT = 'F', then IPIV is an input argument and on entry +* contains the pivot indices from the factorization A = P*L*U +* as computed by SGETRF; row i of the matrix was interchanged +* with row IPIV(i). +* +* If FACT = 'N', then IPIV is an output argument and on exit +* contains the pivot indices from the factorization A = P*L*U +* of the original matrix A. +* +* If FACT = 'E', then IPIV is an output argument and on exit +* contains the pivot indices from the factorization A = P*L*U +* of the equilibrated matrix A. +* +* EQUED (input or output) CHARACTER*1 +* Specifies the form of equilibration that was done. +* = 'N': No equilibration (always true if FACT = 'N'). +* = 'R': Row equilibration, i.e., A has been premultiplied by +* diag(R). +* = 'C': Column equilibration, i.e., A has been postmultiplied +* by diag(C). +* = 'B': Both row and column equilibration, i.e., A has been +* replaced by diag(R) * A * diag(C). +* EQUED is an input argument if FACT = 'F'; otherwise, it is an +* output argument. +* +* R (input or output) REAL array, dimension (N) +* The row scale factors for A. If EQUED = 'R' or 'B', A is +* multiplied on the left by diag(R); if EQUED = 'N' or 'C', R +* is not accessed. R is an input argument if FACT = 'F'; +* otherwise, R is an output argument. If FACT = 'F' and +* EQUED = 'R' or 'B', each element of R must be positive. +* +* C (input or output) REAL array, dimension (N) +* The column scale factors for A. If EQUED = 'C' or 'B', A is +* multiplied on the right by diag(C); if EQUED = 'N' or 'R', C +* is not accessed. C is an input argument if FACT = 'F'; +* otherwise, C is an output argument. If FACT = 'F' and +* EQUED = 'C' or 'B', each element of C must be positive. +* +* B (input/output) REAL array, dimension (LDB,NRHS) +* On entry, the N-by-NRHS right hand side matrix B. +* On exit, +* if EQUED = 'N', B is not modified; +* if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by +* diag(R)*B; +* if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is +* overwritten by diag(C)*B. +* +* LDB (input) INTEGER +* The leading dimension of the array B. LDB >= max(1,N). +* +* X (output) REAL array, dimension (LDX,NRHS) +* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X +* to the original system of equations. Note that A and B are +* modified on exit if EQUED .ne. 'N', and the solution to the +* equilibrated system is inv(diag(C))*X if TRANS = 'N' and +* EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' +* and EQUED = 'R' or 'B'. +* +* LDX (input) INTEGER +* The leading dimension of the array X. LDX >= max(1,N). +* +* RCOND (output) REAL +* The estimate of the reciprocal condition number of the matrix +* A after equilibration (if done). If RCOND is less than the +* machine precision (in particular, if RCOND = 0), the matrix +* is singular to working precision. This condition is +* indicated by a return code of INFO > 0. +* +* FERR (output) REAL array, dimension (NRHS) +* The estimated forward error bound for each solution vector +* X(j) (the j-th column of the solution matrix X). +* If XTRUE is the true solution corresponding to X(j), FERR(j) +* is an estimated upper bound for the magnitude of the largest +* element in (X(j) - XTRUE) divided by the magnitude of the +* largest element in X(j). The estimate is as reliable as +* the estimate for RCOND, and is almost always a slight +* overestimate of the true error. +* +* BERR (output) REAL array, dimension (NRHS) +* The componentwise relative backward error of each solution +* vector X(j) (i.e., the smallest relative change in +* any element of A or B that makes X(j) an exact solution). +* +* WORK (workspace/output) REAL array, dimension (4*N) +* On exit, WORK(1) contains the reciprocal pivot growth +* factor norm(A)/norm(U). The "max absolute element" norm is +* used. If WORK(1) is much less than 1, then the stability +* of the LU factorization of the (equilibrated) matrix A +* could be poor. This also means that the solution X, condition +* estimator RCOND, and forward error bound FERR could be +* unreliable. If factorization fails with 0<INFO<=N, then +* WORK(1) contains the reciprocal pivot growth factor for the +* leading INFO columns of A. +* +* IWORK (workspace) INTEGER array, dimension (N) +* +* INFO (output) INTEGER +* = 0: successful exit +* < 0: if INFO = -i, the i-th argument had an illegal value +* > 0: if INFO = i, and i is +* <= N: U(i,i) is exactly zero. The factorization has +* been completed, but the factor U is exactly +* singular, so the solution and error bounds +* could not be computed. RCOND = 0 is returned. +* = N+1: U is nonsingular, but RCOND is less than machine +* precision, meaning that the matrix is singular +* to working precision. Nevertheless, the +* solution and error bounds are computed because +* there are a number of situations where the +* computed solution can be more accurate than the +* value of RCOND would suggest. +* +* ===================================================================== +* +* .. Parameters .. + REAL ZERO, ONE + PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) +* .. +* .. Local Scalars .. + LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU + CHARACTER NORM + INTEGER I, INFEQU, J + REAL AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN, + $ ROWCND, RPVGRW, SMLNUM +* .. +* .. External Functions .. + LOGICAL LSAME + REAL SLAMCH, SLANGE, SLANTR + EXTERNAL LSAME, SLAMCH, SLANGE, SLANTR +* .. +* .. External Subroutines .. + EXTERNAL SGECON, SGEEQU, SGERFS, SGETRF, SGETRS, SLACPY, + $ SLAQGE, XERBLA +* .. +* .. Intrinsic Functions .. + INTRINSIC MAX, MIN +* .. +* .. Executable Statements .. +* + 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 = SLAMCH( 'Safe minimum' ) + BIGNUM = ONE / SMLNUM + END IF +* +* Test the input parameters. +* + 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 +* + IF( INFO.NE.0 ) THEN + CALL XERBLA( 'SGESVX', -INFO ) + RETURN + END IF +* + IF( EQUIL ) THEN +* +* Compute row and column scalings to equilibrate the matrix A. +* + CALL SGEEQU( N, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFEQU ) + IF( INFEQU.EQ.0 ) THEN +* +* Equilibrate the matrix. +* + CALL SLAQGE( 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 +* +* Scale the right hand side. +* + 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 +* + IF( NOFACT .OR. EQUIL ) THEN +* +* Compute the LU factorization of A. +* + CALL SLACPY( 'Full', N, N, A, LDA, AF, LDAF ) + CALL SGETRF( N, N, AF, LDAF, IPIV, INFO ) +* +* Return if INFO is non-zero. +* + IF( INFO.GT.0 ) THEN +* +* Compute the reciprocal pivot growth factor of the +* leading rank-deficient INFO columns of A. +* + RPVGRW = SLANTR( 'M', 'U', 'N', INFO, INFO, AF, LDAF, + $ WORK ) + IF( RPVGRW.EQ.ZERO ) THEN + RPVGRW = ONE + ELSE + RPVGRW = SLANGE( 'M', N, INFO, A, LDA, WORK ) / RPVGRW + END IF + WORK( 1 ) = RPVGRW + RCOND = ZERO + RETURN + END IF + END IF +* +* Compute the norm of the matrix A and the +* reciprocal pivot growth factor RPVGRW. +* + IF( NOTRAN ) THEN + NORM = '1' + ELSE + NORM = 'I' + END IF + ANORM = SLANGE( NORM, N, N, A, LDA, WORK ) + RPVGRW = SLANTR( 'M', 'U', 'N', N, N, AF, LDAF, WORK ) + IF( RPVGRW.EQ.ZERO ) THEN + RPVGRW = ONE + ELSE + RPVGRW = SLANGE( 'M', N, N, A, LDA, WORK ) / RPVGRW + END IF +* +* Compute the reciprocal of the condition number of A. +* + CALL SGECON( NORM, N, AF, LDAF, ANORM, RCOND, WORK, IWORK, INFO ) +* +* Compute the solution matrix X. +* + CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) + CALL SGETRS( TRANS, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO ) +* +* Use iterative refinement to improve the computed solution and +* compute error bounds and backward error estimates for it. +* + CALL SGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, + $ LDX, FERR, BERR, WORK, IWORK, INFO ) +* +* Transform the solution matrix X to a solution of the original +* system. +* + 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 +* +* Set INFO = N+1 if the matrix is singular to working precision. +* + IF( RCOND.LT.SLAMCH( 'Epsilon' ) ) + $ INFO = N + 1 +* + WORK( 1 ) = RPVGRW + RETURN +* +* End of SGESVX +* + END |