SUBROUTINE CPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
$ S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
$ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
$ NPARAMS, PARAMS, WORK, RWORK, INFO )
*
* -- LAPACK driver routine (version 3.2.2) --
* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
* -- Jason Riedy of Univ. of California Berkeley. --
* -- June 2010 --
*
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley and NAG Ltd. --
*
IMPLICIT NONE
* ..
* .. Scalar Arguments ..
CHARACTER EQUED, FACT, UPLO
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
$ N_ERR_BNDS
REAL RCOND, RPVGRW
* ..
* .. Array Arguments ..
COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
$ WORK( * ), X( LDX, * )
REAL S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
$ ERR_BNDS_NORM( NRHS, * ),
$ ERR_BNDS_COMP( NRHS, * )
* ..
*
* Purpose
* =======
*
* CPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
* to compute the solution to a complex 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.
*
* If requested, both normwise and maximum componentwise error bounds
* are returned. CPOSVXX will return a solution with a tiny
* guaranteed error (O(eps) where eps is the working machine
* precision) unless the matrix is very ill-conditioned, in which
* case a warning is returned. Relevant condition numbers also are
* calculated and returned.
*
* CPOSVXX accepts user-provided factorizations and equilibration
* factors; see the definitions of the FACT and EQUED options.
* Solving with refinement and using a factorization from a previous
* CPOSVXX call will also produce a solution with either O(eps)
* errors or warnings, but we cannot make that claim for general
* user-provided factorizations and equilibration factors if they
* differ from what CPOSVXX would itself produce.
*
* 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**T* U, if UPLO = 'U', or
* A = L * L**T, 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 (see argument RCOND). If the reciprocal of the condition number
* is less than machine precision, 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. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
* the routine will use iterative refinement to try to get a small
* error and error bounds. Refinement calculates the residual to at
* least twice the working precision.
*
* 6. If equilibration was used, the matrix X is premultiplied by
* diag(S) so that it solves the original system before
* equilibration.
*
* Arguments
* =========
*
* Some optional parameters are bundled in the PARAMS array. These
* settings determine how refinement is performed, but often the
* defaults are acceptable. If the defaults are acceptable, users
* can pass NPARAMS = 0 which prevents the source code from accessing
* the PARAMS argument.
*
* 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 is not 'N', the matrix A has been
* equilibrated with scaling factors given by S.
* A and AF are not 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 array, dimension (LDA,N)
* On entry, the symmetric 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 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**T*U or A = L*L**T, 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**T*U or A = L*L**T 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**T*U or A = L*L**T 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': Both row and column equilibration, 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) REAL array, dimension (N)
* The row scale factors for A. If EQUED = 'Y', A is multiplied on
* the left and right by diag(S). 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. If S is output, each
* element of S is a power of the radix. If S is input, each element
* of S should be a power of the radix to ensure a reliable solution
* and error estimates. Scaling by powers of the radix does not cause
* rounding errors unless the result underflows or overflows.
* Rounding errors during scaling lead to refining with a matrix that
* is not equivalent to the input matrix, producing error estimates
* that may not be reliable.
*
* B (input/output) COMPLEX array, dimension (LDB,NRHS)
* On entry, the N-by-NRHS right hand 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 array, dimension (LDX,NRHS)
* If INFO = 0, the N-by-NRHS solution matrix X to the original
* system of equations. Note that A and B are modified on exit if
* EQUED .ne. 'N', 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) REAL
* Reciprocal scaled condition number. This is an estimate of the
* reciprocal Skeel condition number of the matrix A after
* equilibration (if done). If this is less than the machine
* precision (in particular, if it is zero), the matrix is singular
* to working precision. Note that the error may still be small even
* if this number is very small and the matrix appears ill-
* conditioned.
*
* RPVGRW (output) REAL
* Reciprocal pivot growth. On exit, this contains the reciprocal
* pivot growth factor norm(A)/norm(U). The "max absolute element"
* norm is used. If this is much less than 1, then the stability of
* the LU factorization of the (equilibrated) matrix A could be poor.
* This also means that the solution X, estimated condition numbers,
* and error bounds could be unreliable. If factorization fails with
* 0 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
* has been completed, but the factor U is exactly singular, so
* the solution and error bounds could not be computed. RCOND = 0
* is returned.
* = N+J: The solution corresponding to the Jth right-hand side is
* not guaranteed. The solutions corresponding to other right-
* hand sides K with K > J may not be guaranteed as well, but
* only the first such right-hand side is reported. If a small
* componentwise error is not requested (PARAMS(3) = 0.0) then
* the Jth right-hand side is the first with a normwise error
* bound that is not guaranteed (the smallest J such
* that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
* the Jth right-hand side is the first with either a normwise or
* componentwise error bound that is not guaranteed (the smallest
* J such that either ERR_BNDS_NORM(J,1) = 0.0 or
* ERR_BNDS_COMP(J,1) = 0.0). See the definition of
* ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
* about all of the right-hand sides check ERR_BNDS_NORM or
* ERR_BNDS_COMP.
*
* ==================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
INTEGER CMP_ERR_I, PIV_GROWTH_I
PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
$ BERR_I = 3 )
PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
$ PIV_GROWTH_I = 9 )
* ..
* .. Local Scalars ..
LOGICAL EQUIL, NOFACT, RCEQU
INTEGER INFEQU, J
REAL AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
* ..
* .. External Functions ..
EXTERNAL LSAME, SLAMCH, CLA_PORPVGRW
LOGICAL LSAME
REAL SLAMCH, CLA_PORPVGRW
* ..
* .. External Subroutines ..
EXTERNAL CPOCON, CPOEQUB, CPOTRF, CPOTRS, CLACPY,
$ CLAQHE, XERBLA, CLASCL2, CPORFSX
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
EQUIL = LSAME( FACT, 'E' )
SMLNUM = SLAMCH( 'Safe minimum' )
BIGNUM = ONE / SMLNUM
IF( NOFACT .OR. EQUIL ) THEN
EQUED = 'N'
RCEQU = .FALSE.
ELSE
RCEQU = LSAME( EQUED, 'Y' )
ENDIF
*
* Default is failure. If an input parameter is wrong or
* factorization fails, make everything look horrible. Only the
* pivot growth is set here, the rest is initialized in CPORFSX.
*
RPVGRW = ZERO
*
* Test the input parameters. PARAMS is not tested until CPORFSX.
*
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( 'CPOSVXX', -INFO )
RETURN
END IF
*
IF( EQUIL ) THEN
*
* Compute row and column scalings to equilibrate the matrix A.
*
CALL CPOEQUB( N, A, LDA, S, SCOND, AMAX, INFEQU )
IF( INFEQU.EQ.0 ) THEN
*
* Equilibrate the matrix.
*
CALL CLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
RCEQU = LSAME( EQUED, 'Y' )
END IF
END IF
*
* Scale the right-hand side.
*
IF( RCEQU ) CALL CLASCL2( N, NRHS, S, B, LDB )
*
IF( NOFACT .OR. EQUIL ) THEN
*
* Compute the Cholesky factorization of A.
*
CALL CLACPY( UPLO, N, N, A, LDA, AF, LDAF )
CALL CPOTRF( UPLO, N, AF, LDAF, INFO )
*
* Return if INFO is non-zero.
*
IF( INFO.GT.0 ) THEN
*
* Pivot in column INFO is exactly 0
* Compute the reciprocal pivot growth factor of the
* leading rank-deficient INFO columns of A.
*
RPVGRW = CLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, RWORK )
RETURN
END IF
END IF
*
* Compute the reciprocal pivot growth factor RPVGRW.
*
RPVGRW = CLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, RWORK )
*
* Compute the solution matrix X.
*
CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL CPOTRS( 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 CPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF,
$ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
$ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO )
*
* Scale solutions.
*
IF ( RCEQU ) THEN
CALL CLASCL2( N, NRHS, S, X, LDX )
END IF
*
RETURN
*
* End of CPOSVXX
*
END