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author | jason <jason@8a072113-8704-0410-8d35-dd094bca7971> | 2008-10-28 01:38:50 +0000 |
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committer | jason <jason@8a072113-8704-0410-8d35-dd094bca7971> | 2008-10-28 01:38:50 +0000 |
commit | baba851215b44ac3b60b9248eb02bcce7eb76247 (patch) | |
tree | 8c0f5c006875532a30d4409f5e94b0f310ff00a7 /SRC/zposvx.f | |
download | lapack-baba851215b44ac3b60b9248eb02bcce7eb76247.tar.gz lapack-baba851215b44ac3b60b9248eb02bcce7eb76247.tar.bz2 lapack-baba851215b44ac3b60b9248eb02bcce7eb76247.zip |
Move LAPACK trunk into position.
Diffstat (limited to 'SRC/zposvx.f')
-rw-r--r-- | SRC/zposvx.f | 376 |
1 files changed, 376 insertions, 0 deletions
diff --git a/SRC/zposvx.f b/SRC/zposvx.f new file mode 100644 index 00000000..ddf0524c --- /dev/null +++ b/SRC/zposvx.f @@ -0,0 +1,376 @@ + SUBROUTINE ZPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, + $ S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, + $ RWORK, INFO ) +* +* -- LAPACK driver routine (version 3.1) -- +* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. +* November 2006 +* +* .. Scalar Arguments .. + CHARACTER EQUED, FACT, UPLO + INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS + DOUBLE PRECISION RCOND +* .. +* .. Array Arguments .. + DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * ) + COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), + $ WORK( * ), X( LDX, * ) +* .. +* +* Purpose +* ======= +* +* ZPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to +* compute the solution to a complex system of linear equations +* A * X = B, +* where A is an N-by-N Hermitian positive definite 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: +* diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * 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(S)*A*diag(S) and B by diag(S)*B. +* +* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to +* factor the matrix A (after equilibration if FACT = 'E') as +* A = U**H* U, if UPLO = 'U', or +* A = L * L**H, if UPLO = 'L', +* where U is an upper triangular matrix and L is a lower triangular +* matrix. +* +* 3. If the leading i-by-i principal minor is not positive definite, +* 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(S) 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 contains the factored form of A. +* If EQUED = 'Y', the matrix A has been equilibrated +* with scaling factors given by S. A and AF will not +* be 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. +* +* UPLO (input) CHARACTER*1 +* = 'U': Upper triangle of A is stored; +* = 'L': Lower triangle of A is stored. +* +* 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) COMPLEX*16 array, dimension (LDA,N) +* On entry, the Hermitian matrix A, except if FACT = 'F' and +* EQUED = 'Y', then A must contain the equilibrated matrix +* diag(S)*A*diag(S). If UPLO = 'U', the leading +* N-by-N upper triangular part of A contains the upper +* triangular part of the matrix A, and the strictly lower +* triangular part of A is not referenced. If UPLO = 'L', the +* leading N-by-N lower triangular part of A contains the lower +* triangular part of the matrix A, and the strictly upper +* triangular part of A is not referenced. A is not modified if +* FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. +* +* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by +* diag(S)*A*diag(S). +* +* LDA (input) INTEGER +* The leading dimension of the array A. LDA >= max(1,N). +* +* AF (input or output) COMPLEX*16 array, dimension (LDAF,N) +* If FACT = 'F', then AF is an input argument and on entry +* contains the triangular factor U or L from the Cholesky +* factorization A = U**H*U or A = L*L**H, in the same storage +* format as A. If EQUED .ne. 'N', then AF is the factored form +* of the equilibrated matrix diag(S)*A*diag(S). +* +* If FACT = 'N', then AF is an output argument and on exit +* returns the triangular factor U or L from the Cholesky +* factorization A = U**H*U or A = L*L**H of the original +* matrix A. +* +* If FACT = 'E', then AF is an output argument and on exit +* returns the triangular factor U or L from the Cholesky +* factorization A = U**H*U or A = L*L**H 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). +* +* EQUED (input or output) CHARACTER*1 +* Specifies the form of equilibration that was done. +* = 'N': No equilibration (always true if FACT = 'N'). +* = 'Y': Equilibration was done, i.e., A has been replaced by +* diag(S) * A * diag(S). +* EQUED is an input argument if FACT = 'F'; otherwise, it is an +* output argument. +* +* S (input or output) DOUBLE PRECISION array, dimension (N) +* The scale factors for A; not accessed if EQUED = 'N'. S is +* an input argument if FACT = 'F'; otherwise, S is an output +* argument. If FACT = 'F' and EQUED = 'Y', each element of S +* must be positive. +* +* B (input/output) COMPLEX*16 array, dimension (LDB,NRHS) +* On entry, the N-by-NRHS righthand side matrix B. +* On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', +* B is overwritten by diag(S) * B. +* +* LDB (input) INTEGER +* The leading dimension of the array B. LDB >= max(1,N). +* +* X (output) COMPLEX*16 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 if EQUED = 'Y', +* A and B are modified on exit, and the solution to the +* equilibrated system is inv(diag(S))*X. +* +* LDX (input) INTEGER +* The leading dimension of the array X. LDX >= max(1,N). +* +* RCOND (output) DOUBLE PRECISION +* 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) COMPLEX*16 array, dimension (2*N) +* +* RWORK (workspace) DOUBLE PRECISION 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: the leading minor of order i of A is +* not positive definite, so the factorization +* could not be completed, and the solution has not +* been 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 .. + DOUBLE PRECISION ZERO, ONE + PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) +* .. +* .. Local Scalars .. + LOGICAL EQUIL, NOFACT, RCEQU + INTEGER I, INFEQU, J + DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM +* .. +* .. External Functions .. + LOGICAL LSAME + DOUBLE PRECISION DLAMCH, ZLANHE + EXTERNAL LSAME, DLAMCH, ZLANHE +* .. +* .. External Subroutines .. + EXTERNAL XERBLA, ZLACPY, ZLAQHE, ZPOCON, ZPOEQU, ZPORFS, + $ ZPOTRF, ZPOTRS +* .. +* .. Intrinsic Functions .. + INTRINSIC MAX, MIN +* .. +* .. Executable Statements .. +* + INFO = 0 + NOFACT = LSAME( FACT, 'N' ) + EQUIL = LSAME( FACT, 'E' ) + IF( NOFACT .OR. EQUIL ) THEN + EQUED = 'N' + RCEQU = .FALSE. + ELSE + RCEQU = LSAME( EQUED, 'Y' ) + SMLNUM = DLAMCH( '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.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) + $ 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. + $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN + INFO = -9 + ELSE + IF( RCEQU ) THEN + SMIN = BIGNUM + SMAX = ZERO + DO 10 J = 1, N + SMIN = MIN( SMIN, S( J ) ) + SMAX = MAX( SMAX, S( J ) ) + 10 CONTINUE + IF( SMIN.LE.ZERO ) THEN + INFO = -10 + ELSE IF( N.GT.0 ) THEN + SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) + ELSE + SCOND = ONE + END IF + END IF + IF( INFO.EQ.0 ) THEN + IF( LDB.LT.MAX( 1, N ) ) THEN + INFO = -12 + ELSE IF( LDX.LT.MAX( 1, N ) ) THEN + INFO = -14 + END IF + END IF + END IF +* + IF( INFO.NE.0 ) THEN + CALL XERBLA( 'ZPOSVX', -INFO ) + RETURN + END IF +* + IF( EQUIL ) THEN +* +* Compute row and column scalings to equilibrate the matrix A. +* + CALL ZPOEQU( N, A, LDA, S, SCOND, AMAX, INFEQU ) + IF( INFEQU.EQ.0 ) THEN +* +* Equilibrate the matrix. +* + CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) + RCEQU = LSAME( EQUED, 'Y' ) + END IF + END IF +* +* Scale the right hand side. +* + IF( RCEQU ) THEN + DO 30 J = 1, NRHS + DO 20 I = 1, N + B( I, J ) = S( I )*B( I, J ) + 20 CONTINUE + 30 CONTINUE + END IF +* + IF( NOFACT .OR. EQUIL ) THEN +* +* Compute the Cholesky factorization A = U'*U or A = L*L'. +* + CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF ) + CALL ZPOTRF( UPLO, N, AF, LDAF, INFO ) +* +* Return if INFO is non-zero. +* + IF( INFO.GT.0 )THEN + RCOND = ZERO + RETURN + END IF + END IF +* +* Compute the norm of the matrix A. +* + ANORM = ZLANHE( '1', UPLO, N, A, LDA, RWORK ) +* +* Compute the reciprocal of the condition number of A. +* + CALL ZPOCON( UPLO, N, AF, LDAF, ANORM, RCOND, WORK, RWORK, INFO ) +* +* Compute the solution matrix X. +* + CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) + CALL ZPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO ) +* +* Use iterative refinement to improve the computed solution and +* compute error bounds and backward error estimates for it. +* + CALL ZPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, + $ FERR, BERR, WORK, RWORK, INFO ) +* +* Transform the solution matrix X to a solution of the original +* system. +* + IF( RCEQU ) THEN + DO 50 J = 1, NRHS + DO 40 I = 1, N + X( I, J ) = S( I )*X( I, J ) + 40 CONTINUE + 50 CONTINUE + DO 60 J = 1, NRHS + FERR( J ) = FERR( J ) / SCOND + 60 CONTINUE + END IF +* +* Set INFO = N+1 if the matrix is singular to working precision. +* + IF( RCOND.LT.DLAMCH( 'Epsilon' ) ) + $ INFO = N + 1 +* + RETURN +* +* End of ZPOSVX +* + END |