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SUBROUTINE DSPOSV( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK,
$ SWORK, ITER, INFO )
*
* -- LAPACK PROTOTYPE driver routine (version 3.3.1) --
* Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..
* -- April 2011 --
*
* ..
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
REAL SWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( N, * ),
$ X( LDX, * )
* ..
*
* Purpose
* =======
*
* DSPOSV computes the solution to a real system of linear equations
* A * X = B,
* where A is an N-by-N symmetric positive definite matrix and X and B
* are N-by-NRHS matrices.
*
* DSPOSV first attempts to factorize the matrix in SINGLE PRECISION
* and use this factorization within an iterative refinement procedure
* to produce a solution with DOUBLE PRECISION normwise backward error
* quality (see below). If the approach fails the method switches to a
* DOUBLE PRECISION factorization and solve.
*
* The iterative refinement is not going to be a winning strategy if
* the ratio SINGLE PRECISION performance over DOUBLE PRECISION
* performance is too small. A reasonable strategy should take the
* number of right-hand sides and the size of the matrix into account.
* This might be done with a call to ILAENV in the future. Up to now, we
* always try iterative refinement.
*
* The iterative refinement process is stopped if
* ITER > ITERMAX
* or for all the RHS we have:
* RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
* where
* o ITER is the number of the current iteration in the iterative
* refinement process
* o RNRM is the infinity-norm of the residual
* o XNRM is the infinity-norm of the solution
* o ANRM is the infinity-operator-norm of the matrix A
* o EPS is the machine epsilon returned by DLAMCH('Epsilon')
* The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00
* respectively.
*
* Arguments
* =========
*
* 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 matrix B. NRHS >= 0.
*
* A (input/output) DOUBLE PRECISION array,
* dimension (LDA,N)
* On entry, 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.
* On exit, if iterative refinement has been successfully used
* (INFO.EQ.0 and ITER.GE.0, see description below), then A is
* unchanged, if double precision factorization has been used
* (INFO.EQ.0 and ITER.LT.0, see description below), then the
* array A contains the factor U or L from the Cholesky
* factorization A = U**T*U or A = L*L**T.
*
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* 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, the N-by-NRHS solution matrix X.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* WORK (workspace) DOUBLE PRECISION array, dimension (N,NRHS)
* This array is used to hold the residual vectors.
*
* SWORK (workspace) REAL array, dimension (N*(N+NRHS))
* This array is used to use the single precision matrix and the
* right-hand sides or solutions in single precision.
*
* ITER (output) INTEGER
* < 0: iterative refinement has failed, double precision
* factorization has been performed
* -1 : the routine fell back to full precision for
* implementation- or machine-specific reasons
* -2 : narrowing the precision induced an overflow,
* the routine fell back to full precision
* -3 : failure of SPOTRF
* -31: stop the iterative refinement after the 30th
* iterations
* > 0: iterative refinement has been sucessfully used.
* Returns the number of iterations
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, the leading minor of order i of (DOUBLE
* PRECISION) A is not positive definite, so the
* factorization could not be completed, and the solution
* has not been computed.
*
* =====================================================================
*
* .. Parameters ..
LOGICAL DOITREF
PARAMETER ( DOITREF = .TRUE. )
*
INTEGER ITERMAX
PARAMETER ( ITERMAX = 30 )
*
DOUBLE PRECISION BWDMAX
PARAMETER ( BWDMAX = 1.0E+00 )
*
DOUBLE PRECISION NEGONE, ONE
PARAMETER ( NEGONE = -1.0D+0, ONE = 1.0D+0 )
*
* .. Local Scalars ..
INTEGER I, IITER, PTSA, PTSX
DOUBLE PRECISION ANRM, CTE, EPS, RNRM, XNRM
*
* .. External Subroutines ..
EXTERNAL DAXPY, DSYMM, DLACPY, DLAT2S, DLAG2S, SLAG2D,
$ SPOTRF, SPOTRS, XERBLA
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH, DLANSY
LOGICAL LSAME
EXTERNAL IDAMAX, DLAMCH, DLANSY, LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
ITER = 0
*
* Test the input parameters.
*
IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPOSV', -INFO )
RETURN
END IF
*
* Quick return if (N.EQ.0).
*
IF( N.EQ.0 )
$ RETURN
*
* Skip single precision iterative refinement if a priori slower
* than double precision factorization.
*
IF( .NOT.DOITREF ) THEN
ITER = -1
GO TO 40
END IF
*
* Compute some constants.
*
ANRM = DLANSY( 'I', UPLO, N, A, LDA, WORK )
EPS = DLAMCH( 'Epsilon' )
CTE = ANRM*EPS*SQRT( DBLE( N ) )*BWDMAX
*
* Set the indices PTSA, PTSX for referencing SA and SX in SWORK.
*
PTSA = 1
PTSX = PTSA + N*N
*
* Convert B from double precision to single precision and store the
* result in SX.
*
CALL DLAG2S( N, NRHS, B, LDB, SWORK( PTSX ), N, INFO )
*
IF( INFO.NE.0 ) THEN
ITER = -2
GO TO 40
END IF
*
* Convert A from double precision to single precision and store the
* result in SA.
*
CALL DLAT2S( UPLO, N, A, LDA, SWORK( PTSA ), N, INFO )
*
IF( INFO.NE.0 ) THEN
ITER = -2
GO TO 40
END IF
*
* Compute the Cholesky factorization of SA.
*
CALL SPOTRF( UPLO, N, SWORK( PTSA ), N, INFO )
*
IF( INFO.NE.0 ) THEN
ITER = -3
GO TO 40
END IF
*
* Solve the system SA*SX = SB.
*
CALL SPOTRS( UPLO, N, NRHS, SWORK( PTSA ), N, SWORK( PTSX ), N,
$ INFO )
*
* Convert SX back to double precision
*
CALL SLAG2D( N, NRHS, SWORK( PTSX ), N, X, LDX, INFO )
*
* Compute R = B - AX (R is WORK).
*
CALL DLACPY( 'All', N, NRHS, B, LDB, WORK, N )
*
CALL DSYMM( 'Left', UPLO, N, NRHS, NEGONE, A, LDA, X, LDX, ONE,
$ WORK, N )
*
* Check whether the NRHS normwise backward errors satisfy the
* stopping criterion. If yes, set ITER=0 and return.
*
DO I = 1, NRHS
XNRM = ABS( X( IDAMAX( N, X( 1, I ), 1 ), I ) )
RNRM = ABS( WORK( IDAMAX( N, WORK( 1, I ), 1 ), I ) )
IF( RNRM.GT.XNRM*CTE )
$ GO TO 10
END DO
*
* If we are here, the NRHS normwise backward errors satisfy the
* stopping criterion. We are good to exit.
*
ITER = 0
RETURN
*
10 CONTINUE
*
DO 30 IITER = 1, ITERMAX
*
* Convert R (in WORK) from double precision to single precision
* and store the result in SX.
*
CALL DLAG2S( N, NRHS, WORK, N, SWORK( PTSX ), N, INFO )
*
IF( INFO.NE.0 ) THEN
ITER = -2
GO TO 40
END IF
*
* Solve the system SA*SX = SR.
*
CALL SPOTRS( UPLO, N, NRHS, SWORK( PTSA ), N, SWORK( PTSX ), N,
$ INFO )
*
* Convert SX back to double precision and update the current
* iterate.
*
CALL SLAG2D( N, NRHS, SWORK( PTSX ), N, WORK, N, INFO )
*
DO I = 1, NRHS
CALL DAXPY( N, ONE, WORK( 1, I ), 1, X( 1, I ), 1 )
END DO
*
* Compute R = B - AX (R is WORK).
*
CALL DLACPY( 'All', N, NRHS, B, LDB, WORK, N )
*
CALL DSYMM( 'L', UPLO, N, NRHS, NEGONE, A, LDA, X, LDX, ONE,
$ WORK, N )
*
* Check whether the NRHS normwise backward errors satisfy the
* stopping criterion. If yes, set ITER=IITER>0 and return.
*
DO I = 1, NRHS
XNRM = ABS( X( IDAMAX( N, X( 1, I ), 1 ), I ) )
RNRM = ABS( WORK( IDAMAX( N, WORK( 1, I ), 1 ), I ) )
IF( RNRM.GT.XNRM*CTE )
$ GO TO 20
END DO
*
* If we are here, the NRHS normwise backward errors satisfy the
* stopping criterion, we are good to exit.
*
ITER = IITER
*
RETURN
*
20 CONTINUE
*
30 CONTINUE
*
* If we are at this place of the code, this is because we have
* performed ITER=ITERMAX iterations and never satisified the
* stopping criterion, set up the ITER flag accordingly and follow
* up on double precision routine.
*
ITER = -ITERMAX - 1
*
40 CONTINUE
*
* Single-precision iterative refinement failed to converge to a
* satisfactory solution, so we resort to double precision.
*
CALL DPOTRF( UPLO, N, A, LDA, INFO )
*
IF( INFO.NE.0 )
$ RETURN
*
CALL DLACPY( 'All', N, NRHS, B, LDB, X, LDX )
CALL DPOTRS( UPLO, N, NRHS, A, LDA, X, LDX, INFO )
*
RETURN
*
* End of DSPOSV.
*
END
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