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Diffstat (limited to 'SRC/ssysvx.f')
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diff --git a/SRC/ssysvx.f b/SRC/ssysvx.f new file mode 100644 index 00000000..04123b9b --- /dev/null +++ b/SRC/ssysvx.f @@ -0,0 +1,300 @@ + SUBROUTINE SSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, + $ LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK, + $ IWORK, INFO ) +* +* -- LAPACK driver routine (version 3.1) -- +* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. +* November 2006 +* +* .. Scalar Arguments .. + CHARACTER FACT, UPLO + INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS + REAL RCOND +* .. +* .. Array Arguments .. + INTEGER IPIV( * ), IWORK( * ) + REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), + $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * ) +* .. +* +* Purpose +* ======= +* +* SSYSVX uses the diagonal pivoting factorization to compute the +* solution to a real system of linear equations A * X = B, +* where A is an N-by-N symmetric 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 = 'N', the diagonal pivoting method is used to factor A. +* The form of the factorization is +* A = U * D * U**T, if UPLO = 'U', or +* A = L * D * L**T, if UPLO = 'L', +* where U (or L) is a product of permutation and unit upper (lower) +* triangular matrices, and D is symmetric and block diagonal with +* 1-by-1 and 2-by-2 diagonal blocks. +* +* 2. If some D(i,i)=0, so that D 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. +* +* 3. The system of equations is solved for X using the factored form +* of A. +* +* 4. Iterative refinement is applied to improve the computed solution +* matrix and calculate error bounds and backward error estimates +* for it. +* +* Arguments +* ========= +* +* FACT (input) CHARACTER*1 +* Specifies whether or not the factored form of A has been +* supplied on entry. +* = 'F': On entry, AF and IPIV contain the factored form of +* A. AF and IPIV will not be modified. +* = 'N': The matrix A will be 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) REAL array, dimension (LDA,N) +* The symmetric matrix A. 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. +* +* 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 block diagonal matrix D and the multipliers used +* to obtain the factor U or L from the factorization +* A = U*D*U**T or A = L*D*L**T as computed by SSYTRF. +* +* If FACT = 'N', then AF is an output argument and on exit +* returns the block diagonal matrix D and the multipliers used +* to obtain the factor U or L from the factorization +* A = U*D*U**T or A = L*D*L**T. +* +* 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 details of the interchanges and the block structure +* of D, as determined by SSYTRF. +* If IPIV(k) > 0, then rows and columns k and IPIV(k) were +* interchanged and D(k,k) is a 1-by-1 diagonal block. +* If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and +* columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) +* is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = +* IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were +* interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. +* +* If FACT = 'N', then IPIV is an output argument and on exit +* contains details of the interchanges and the block structure +* of D, as determined by SSYTRF. +* +* B (input) REAL array, dimension (LDB,NRHS) +* The N-by-NRHS right hand side matrix 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. +* +* 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. 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 (MAX(1,LWORK)) +* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. +* +* LWORK (input) INTEGER +* The length of WORK. LWORK >= max(1,3*N), and for best +* performance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where +* NB is the optimal blocksize for SSYTRF. +* +* If LWORK = -1, then a workspace query is assumed; the routine +* only calculates the optimal size of the WORK array, returns +* this value as the first entry of the WORK array, and no error +* message related to LWORK is issued by XERBLA. +* +* 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: D(i,i) is exactly zero. The factorization +* has been completed but the factor D is exactly +* singular, so the solution and error bounds could +* not be computed. RCOND = 0 is returned. +* = N+1: D 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 + PARAMETER ( ZERO = 0.0E+0 ) +* .. +* .. Local Scalars .. + LOGICAL LQUERY, NOFACT + INTEGER LWKOPT, NB + REAL ANORM +* .. +* .. External Functions .. + LOGICAL LSAME + INTEGER ILAENV + REAL SLAMCH, SLANSY + EXTERNAL ILAENV, LSAME, SLAMCH, SLANSY +* .. +* .. External Subroutines .. + EXTERNAL SLACPY, SSYCON, SSYRFS, SSYTRF, SSYTRS, XERBLA +* .. +* .. Intrinsic Functions .. + INTRINSIC MAX +* .. +* .. Executable Statements .. +* +* Test the input parameters. +* + INFO = 0 + NOFACT = LSAME( FACT, 'N' ) + LQUERY = ( LWORK.EQ.-1 ) + IF( .NOT.NOFACT .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( LDB.LT.MAX( 1, N ) ) THEN + INFO = -11 + ELSE IF( LDX.LT.MAX( 1, N ) ) THEN + INFO = -13 + ELSE IF( LWORK.LT.MAX( 1, 3*N ) .AND. .NOT.LQUERY ) THEN + INFO = -18 + END IF +* + IF( INFO.EQ.0 ) THEN + LWKOPT = MAX( 1, 3*N ) + IF( NOFACT ) THEN + NB = ILAENV( 1, 'SSYTRF', UPLO, N, -1, -1, -1 ) + LWKOPT = MAX( LWKOPT, N*NB ) + END IF + WORK( 1 ) = LWKOPT + END IF +* + IF( INFO.NE.0 ) THEN + CALL XERBLA( 'SSYSVX', -INFO ) + RETURN + ELSE IF( LQUERY ) THEN + RETURN + END IF +* + IF( NOFACT ) THEN +* +* Compute the factorization A = U*D*U' or A = L*D*L'. +* + CALL SLACPY( UPLO, N, N, A, LDA, AF, LDAF ) + CALL SSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, LWORK, 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 = SLANSY( 'I', UPLO, N, A, LDA, WORK ) +* +* Compute the reciprocal of the condition number of A. +* + CALL SSYCON( UPLO, N, AF, LDAF, IPIV, ANORM, RCOND, WORK, IWORK, + $ INFO ) +* +* Compute the solution vectors X. +* + CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) + CALL SSYTRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO ) +* +* Use iterative refinement to improve the computed solutions and +* compute error bounds and backward error estimates for them. +* + CALL SSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, + $ LDX, FERR, BERR, WORK, IWORK, INFO ) +* +* Set INFO = N+1 if the matrix is singular to working precision. +* + IF( RCOND.LT.SLAMCH( 'Epsilon' ) ) + $ INFO = N + 1 +* + WORK( 1 ) = LWKOPT +* + RETURN +* +* End of SSYSVX +* + END |