SUBROUTINE ZPOSVXX( 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 3.2.1) -- * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- * -- Jason Riedy of Univ. of California Berkeley. -- * -- April 2009 -- * * -- 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 DOUBLE PRECISION RCOND, RPVGRW * .. * .. Array Arguments .. COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), $ WORK( * ), X( LDX, * ) DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ), $ ERR_BNDS_NORM( NRHS, * ), $ ERR_BNDS_COMP( NRHS, * ) * .. * * Purpose * ======= * * ZPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T * to compute the solution to a complex*16 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. ZPOSVXX 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. * * ZPOSVXX 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 * ZPOSVXX 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 ZPOSVXX would itself produce. * * Description * =========== * * The following steps are performed: * * 1. If FACT = 'E', double precision 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*16 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*16 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) DOUBLE PRECISION 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*16 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*16 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) DOUBLE PRECISION * 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) DOUBLE PRECISION * 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 .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+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 DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM * .. * .. External Functions .. EXTERNAL LSAME, DLAMCH, ZLA_PORPVGRW LOGICAL LSAME DOUBLE PRECISION DLAMCH, ZLA_PORPVGRW * .. * .. External Subroutines .. EXTERNAL ZPOCON, ZPOEQUB, ZPOTRF, ZPOTRS, ZLACPY, $ ZLAQHE, XERBLA, ZLASCL2, ZPORFSX * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. Executable Statements .. * INFO = 0 NOFACT = LSAME( FACT, 'N' ) EQUIL = LSAME( FACT, 'E' ) SMLNUM = DLAMCH( '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 ZPORFSX. * RPVGRW = ZERO * * Test the input parameters. PARAMS is not tested until ZPORFSX. * 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( 'ZPOSVXX', -INFO ) RETURN END IF * IF( EQUIL ) THEN * * Compute row and column scalings to equilibrate the matrix A. * CALL ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFEQU ) IF( INFEQU.EQ.0 ) THEN * * Equilibrate the matrix. * CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) RCEQU = LSAME( EQUED, 'Y' ) END IF END IF * * Scale the right-hand side. * IF( RCEQU ) CALL ZLASCL2( N, NRHS, S, B, LDB ) * IF( NOFACT .OR. EQUIL ) THEN * * Compute the LU factorization of A. * CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF ) CALL ZPOTRF( 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 = ZLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, RWORK ) RETURN END IF END IF * * Compute the reciprocal pivot growth factor RPVGRW. * RPVGRW = ZLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, RWORK ) * * Compute the solution matrix X. * CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL ZPOTRS( 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 ZPORFSX( 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 ZLASCL2( N, NRHS, S, X, LDX ) END IF * RETURN * * End of ZPOSVXX * END