SUBROUTINE DSYSVX( 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
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
$ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
* ..
*
* Purpose
* =======
*
* DSYSVX 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 DSYTRF.
*
* 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 DSYTRF.
* 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 DSYTRF.
*
* B (input) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION
* 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) 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/output) DOUBLE PRECISION 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 DSYTRF.
*
* 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 ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, NOFACT
INTEGER LWKOPT, NB
DOUBLE PRECISION ANORM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANSY
EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY
* ..
* .. External Subroutines ..
EXTERNAL DLACPY, DSYCON, DSYRFS, DSYTRF, DSYTRS, 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, 'DSYTRF', UPLO, N, -1, -1, -1 )
LWKOPT = MAX( LWKOPT, N*NB )
END IF
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYSVX', -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 DLACPY( UPLO, N, N, A, LDA, AF, LDAF )
CALL DSYTRF( 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 = DLANSY( 'I', UPLO, N, A, LDA, WORK )
*
* Compute the reciprocal of the condition number of A.
*
CALL DSYCON( UPLO, N, AF, LDAF, IPIV, ANORM, RCOND, WORK, IWORK,
$ INFO )
*
* Compute the solution vectors X.
*
CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL DSYTRS( 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 DSYRFS( 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.DLAMCH( 'Epsilon' ) )
$ INFO = N + 1
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DSYSVX
*
END