SUBROUTINE ZGBSVXX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, $ LDAFB, IPIV, EQUED, R, C, 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, TRANS INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS, $ N_ERR_BNDS DOUBLE PRECISION RCOND, RPVGRW * .. * .. Array Arguments .. INTEGER IPIV( * ) COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), $ X( LDX , * ),WORK( * ) DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ), $ ERR_BNDS_NORM( NRHS, * ), $ ERR_BNDS_COMP( NRHS, * ), RWORK( * ) * .. * * Purpose * ======= * * ZGBSVXX uses the LU factorization to compute the solution to a * complex*16 system of linear equations A * X = B, where A is an * N-by-N matrix and X and B are N-by-NRHS matrices. * * If requested, both normwise and maximum componentwise error bounds * are returned. ZGBSVXX 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. * * ZGBSVXX 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 * ZGBSVXX 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 ZGBSVXX would itself produce. * * Description * =========== * * The following steps are performed: * * 1. If FACT = 'E', double precision scaling factors are computed to equilibrate * the system: * * TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B * TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B * TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') * or diag(C)*B (if TRANS = 'T' or 'C'). * * 2. If FACT = 'N' or 'E', the LU decomposition is used to factor * the matrix A (after equilibration if FACT = 'E') as * * A = P * L * U, * * where P is a permutation matrix, L is a unit lower triangular * matrix, and U is upper triangular. * * 3. If some U(i,i)=0, so that U 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 (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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') 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 and IPIV contain the factored form of A. * If EQUED is not 'N', the matrix A has been * equilibrated with scaling factors given by R and C. * A, AF, and IPIV 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. * * TRANS (input) CHARACTER*1 * Specifies the form of the system of equations: * = 'N': A * X = B (No transpose) * = 'T': A**T * X = B (Transpose) * = 'C': A**H * X = B (Conjugate Transpose = Transpose) * * N (input) INTEGER * The number of linear equations, i.e., the order of the * matrix A. N >= 0. * * KL (input) INTEGER * The number of subdiagonals within the band of A. KL >= 0. * * KU (input) INTEGER * The number of superdiagonals within the band of A. KU >= 0. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of columns * of the matrices B and X. NRHS >= 0. * * AB (input/output) COMPLEX*16 array, dimension (LDAB,N) * On entry, the matrix A in band storage, in rows 1 to KL+KU+1. * The j-th column of A is stored in the j-th column of the * array AB as follows: * AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl) * * If FACT = 'F' and EQUED is not 'N', then AB must have been * equilibrated by the scaling factors in R and/or C. AB is not * modified if FACT = 'F' or 'N', or if FACT = 'E' and * EQUED = 'N' on exit. * * On exit, if EQUED .ne. 'N', A is scaled as follows: * EQUED = 'R': A := diag(R) * A * EQUED = 'C': A := A * diag(C) * EQUED = 'B': A := diag(R) * A * diag(C). * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KL+KU+1. * * AFB (input or output) COMPLEX*16 array, dimension (LDAFB,N) * If FACT = 'F', then AFB is an input argument and on entry * contains details of the LU factorization of the band matrix * A, as computed by ZGBTRF. U is stored as an upper triangular * band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, * and the multipliers used during the factorization are stored * in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is * the factored form of the equilibrated matrix A. * * If FACT = 'N', then AF is an output argument and on exit * returns the factors L and U from the factorization A = P*L*U * of the original matrix A. * * If FACT = 'E', then AF is an output argument and on exit * returns the factors L and U from the factorization A = P*L*U * of the equilibrated matrix A (see the description of A for * the form of the equilibrated matrix). * * LDAFB (input) INTEGER * The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. * * IPIV (input or output) INTEGER array, dimension (N) * If FACT = 'F', then IPIV is an input argument and on entry * contains the pivot indices from the factorization A = P*L*U * as computed by DGETRF; row i of the matrix was interchanged * with row IPIV(i). * * If FACT = 'N', then IPIV is an output argument and on exit * contains the pivot indices from the factorization A = P*L*U * of the original matrix A. * * If FACT = 'E', then IPIV is an output argument and on exit * contains the pivot indices from the factorization A = P*L*U * of the equilibrated matrix A. * * EQUED (input or output) CHARACTER*1 * Specifies the form of equilibration that was done. * = 'N': No equilibration (always true if FACT = 'N'). * = 'R': Row equilibration, i.e., A has been premultiplied by * diag(R). * = 'C': Column equilibration, i.e., A has been postmultiplied * by diag(C). * = 'B': Both row and column equilibration, i.e., A has been * replaced by diag(R) * A * diag(C). * EQUED is an input argument if FACT = 'F'; otherwise, it is an * output argument. * * R (input or output) DOUBLE PRECISION array, dimension (N) * The row scale factors for A. If EQUED = 'R' or 'B', A is * multiplied on the left by diag(R); if EQUED = 'N' or 'C', R * is not accessed. R is an input argument if FACT = 'F'; * otherwise, R is an output argument. If FACT = 'F' and * EQUED = 'R' or 'B', each element of R must be positive. * If R is output, each element of R is a power of the radix. * If R is input, each element of R 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. * * C (input or output) DOUBLE PRECISION array, dimension (N) * The column scale factors for A. If EQUED = 'C' or 'B', A is * multiplied on the right by diag(C); if EQUED = 'N' or 'R', C * is not accessed. C is an input argument if FACT = 'F'; * otherwise, C is an output argument. If FACT = 'F' and * EQUED = 'C' or 'B', each element of C must be positive. * If C is output, each element of C is a power of the radix. * If C is input, each element of C 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 TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by * diag(R)*B; * if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is * overwritten by diag(C)*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(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or * inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. * * 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 COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU INTEGER INFEQU, I, J, KL, KU DOUBLE PRECISION AMAX, BIGNUM, COLCND, RCMAX, RCMIN, $ ROWCND, SMLNUM * .. * .. External Functions .. EXTERNAL LSAME, DLAMCH, ZLA_GBRPVGRW LOGICAL LSAME DOUBLE PRECISION DLAMCH, ZLA_GBRPVGRW * .. * .. External Subroutines .. EXTERNAL ZGBEQUB, ZGBTRF, ZGBTRS, ZLACPY, ZLAQGB, $ XERBLA, ZLASCL2, ZGBRFSX * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. Executable Statements .. * INFO = 0 NOFACT = LSAME( FACT, 'N' ) EQUIL = LSAME( FACT, 'E' ) NOTRAN = LSAME( TRANS, 'N' ) SMLNUM = DLAMCH( 'Safe minimum' ) BIGNUM = ONE / SMLNUM IF( NOFACT .OR. EQUIL ) THEN EQUED = 'N' ROWEQU = .FALSE. COLEQU = .FALSE. ELSE ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) END IF * * 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 ZGBRFSX. * RPVGRW = ZERO * * Test the input parameters. PARAMS is not tested until DGERFSX. * IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT. $ LSAME( FACT, 'F' ) ) THEN INFO = -1 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. $ LSAME( TRANS, 'C' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KL.LT.0 ) THEN INFO = -4 ELSE IF( KU.LT.0 ) THEN INFO = -5 ELSE IF( NRHS.LT.0 ) THEN INFO = -6 ELSE IF( LDAB.LT.KL+KU+1 ) THEN INFO = -8 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN INFO = -10 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN INFO = -12 ELSE IF( ROWEQU ) THEN RCMIN = BIGNUM RCMAX = ZERO DO 10 J = 1, N RCMIN = MIN( RCMIN, R( J ) ) RCMAX = MAX( RCMAX, R( J ) ) 10 CONTINUE IF( RCMIN.LE.ZERO ) THEN INFO = -13 ELSE IF( N.GT.0 ) THEN ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) ELSE ROWCND = ONE END IF END IF IF( COLEQU .AND. INFO.EQ.0 ) THEN RCMIN = BIGNUM RCMAX = ZERO DO 20 J = 1, N RCMIN = MIN( RCMIN, C( J ) ) RCMAX = MAX( RCMAX, C( J ) ) 20 CONTINUE IF( RCMIN.LE.ZERO ) THEN INFO = -14 ELSE IF( N.GT.0 ) THEN COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) ELSE COLCND = ONE END IF END IF IF( INFO.EQ.0 ) THEN IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -15 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -16 END IF END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZGBSVXX', -INFO ) RETURN END IF * IF( EQUIL ) THEN * * Compute row and column scalings to equilibrate the matrix A. * CALL ZGBEQUB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, $ AMAX, INFEQU ) IF( INFEQU.EQ.0 ) THEN * * Equilibrate the matrix. * CALL ZLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, $ AMAX, EQUED ) ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) END IF * * If the scaling factors are not applied, set them to 1.0. * IF ( .NOT.ROWEQU ) THEN DO J = 1, N R( J ) = 1.0D+0 END DO END IF IF ( .NOT.COLEQU ) THEN DO J = 1, N C( J ) = 1.0D+0 END DO END IF END IF * * Scale the right-hand side. * IF( NOTRAN ) THEN IF( ROWEQU ) CALL ZLASCL2( N, NRHS, R, B, LDB ) ELSE IF( COLEQU ) CALL ZLASCL2( N, NRHS, C, B, LDB ) END IF * IF( NOFACT .OR. EQUIL ) THEN * * Compute the LU factorization of A. * DO 40, J = 1, N DO 30, I = KL+1, 2*KL+KU+1 AFB( I, J ) = AB( I-KL, J ) 30 CONTINUE 40 CONTINUE CALL ZGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, 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_GBRPVGRW( N, KL, KU, INFO, AB, LDAB, AFB, $ LDAFB ) RETURN END IF END IF * * Compute the reciprocal pivot growth factor RPVGRW. * RPVGRW = ZLA_GBRPVGRW( N, KL, KU, N, AB, LDAB, AFB, LDAFB ) * * Compute the solution matrix X. * CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL ZGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX, $ INFO ) * * Use iterative refinement to improve the computed solution and * compute error bounds and backward error estimates for it. * CALL ZGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, $ IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, $ WORK, RWORK, INFO ) * * Scale solutions. * IF ( COLEQU .AND. NOTRAN ) THEN CALL ZLASCL2( N, NRHS, C, X, LDX ) ELSE IF ( ROWEQU .AND. .NOT.NOTRAN ) THEN CALL ZLASCL2( N, NRHS, R, X, LDX ) END IF * RETURN * * End of ZGBSVXX * END