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authorjason <jason@8a072113-8704-0410-8d35-dd094bca7971>2008-10-28 01:38:50 +0000
committerjason <jason@8a072113-8704-0410-8d35-dd094bca7971>2008-10-28 01:38:50 +0000
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+ 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