/* zgesvxx.f -- translated by f2c (version 20061008). You must link the resulting object file with libf2c: on Microsoft Windows system, link with libf2c.lib; on Linux or Unix systems, link with .../path/to/libf2c.a -lm or, if you install libf2c.a in a standard place, with -lf2c -lm -- in that order, at the end of the command line, as in cc *.o -lf2c -lm Source for libf2c is in /netlib/f2c/libf2c.zip, e.g., http://www.netlib.org/f2c/libf2c.zip */ #include "f2c.h" #include "blaswrap.h" /* Subroutine */ int zgesvxx_(char *fact, char *trans, integer *n, integer * nrhs, doublecomplex *a, integer *lda, doublecomplex *af, integer * ldaf, integer *ipiv, char *equed, doublereal *r__, doublereal *c__, doublecomplex *b, integer *ldb, doublecomplex *x, integer *ldx, doublereal *rcond, doublereal *rpvgrw, doublereal *berr, integer * n_err_bnds__, doublereal *err_bnds_norm__, doublereal * err_bnds_comp__, integer *nparams, doublereal *params, doublecomplex * work, doublereal *rwork, integer *info) { /* System generated locals */ integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1, x_offset, err_bnds_norm_dim1, err_bnds_norm_offset, err_bnds_comp_dim1, err_bnds_comp_offset, i__1; doublereal d__1, d__2; /* Local variables */ integer j; extern doublereal zla_rpvgrw__(integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *); doublereal amax; extern logical lsame_(char *, char *); doublereal rcmin, rcmax; logical equil; extern doublereal dlamch_(char *); doublereal colcnd; logical nofact; extern /* Subroutine */ int xerbla_(char *, integer *); doublereal bignum; extern /* Subroutine */ int zlaqge_(integer *, integer *, doublecomplex *, integer *, doublereal *, doublereal *, doublereal *, doublereal * , doublereal *, char *); integer infequ; logical colequ; doublereal rowcnd; logical notran; extern /* Subroutine */ int zgetrf_(integer *, integer *, doublecomplex *, integer *, integer *, integer *), zlacpy_(char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *); doublereal smlnum; extern /* Subroutine */ int zgetrs_(char *, integer *, integer *, doublecomplex *, integer *, integer *, doublecomplex *, integer *, integer *); logical rowequ; extern /* Subroutine */ int zlascl2_(integer *, integer *, doublereal *, doublecomplex *, integer *), zgeequb_(integer *, integer *, doublecomplex *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *), zgerfsx_( char *, char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, integer *, doublereal *, doublereal *, doublecomplex *, integer *, doublecomplex *, integer *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, integer *, doublereal *, doublecomplex *, doublereal *, integer * ); /* -- LAPACK driver routine (version 3.2) -- */ /* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- */ /* -- Jason Riedy of Univ. of California Berkeley. -- */ /* -- November 2008 -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* -- Univ. of California Berkeley and NAG Ltd. -- */ /* .. */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* ZGESVXX 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. ZGESVXX 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. */ /* ZGESVXX 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 */ /* ZGESVXX 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 ZGESVXX 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) */ /* 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 N-by-N matrix A. If FACT = 'F' and EQUED is */ /* not 'N', then A must have been equilibrated by the scaling */ /* factors in R and/or C. A 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). */ /* 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 factors L and U from the factorization */ /* A = P*L*U as computed by ZGETRF. If EQUED .ne. 'N', then */ /* AF 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). */ /* 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 the pivot indices from the factorization A = P*L*U */ /* as computed by ZGETRF; 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 .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Executable Statements .. */ /* Parameter adjustments */ err_bnds_comp_dim1 = *nrhs; err_bnds_comp_offset = 1 + err_bnds_comp_dim1; err_bnds_comp__ -= err_bnds_comp_offset; err_bnds_norm_dim1 = *nrhs; err_bnds_norm_offset = 1 + err_bnds_norm_dim1; err_bnds_norm__ -= err_bnds_norm_offset; a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; af_dim1 = *ldaf; af_offset = 1 + af_dim1; af -= af_offset; --ipiv; --r__; --c__; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; x_dim1 = *ldx; x_offset = 1 + x_dim1; x -= x_offset; --berr; --params; --work; --rwork; /* Function Body */ *info = 0; nofact = lsame_(fact, "N"); equil = lsame_(fact, "E"); notran = lsame_(trans, "N"); smlnum = dlamch_("Safe minimum"); bignum = 1. / smlnum; if (nofact || equil) { *(unsigned char *)equed = 'N'; rowequ = FALSE_; colequ = FALSE_; } else { rowequ = lsame_(equed, "R") || lsame_(equed, "B"); colequ = lsame_(equed, "C") || lsame_(equed, "B"); } /* 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 ZGERFSX. */ *rpvgrw = 0.; /* Test the input parameters. PARAMS is not tested until ZGERFSX. */ if (! nofact && ! equil && ! lsame_(fact, "F")) { *info = -1; } else if (! notran && ! lsame_(trans, "T") && ! lsame_(trans, "C")) { *info = -2; } else if (*n < 0) { *info = -3; } else if (*nrhs < 0) { *info = -4; } else if (*lda < max(1,*n)) { *info = -6; } else if (*ldaf < max(1,*n)) { *info = -8; } else if (lsame_(fact, "F") && ! (rowequ || colequ || lsame_(equed, "N"))) { *info = -10; } else { if (rowequ) { rcmin = bignum; rcmax = 0.; i__1 = *n; for (j = 1; j <= i__1; ++j) { /* Computing MIN */ d__1 = rcmin, d__2 = r__[j]; rcmin = min(d__1,d__2); /* Computing MAX */ d__1 = rcmax, d__2 = r__[j]; rcmax = max(d__1,d__2); /* L10: */ } if (rcmin <= 0.) { *info = -11; } else if (*n > 0) { rowcnd = max(rcmin,smlnum) / min(rcmax,bignum); } else { rowcnd = 1.; } } if (colequ && *info == 0) { rcmin = bignum; rcmax = 0.; i__1 = *n; for (j = 1; j <= i__1; ++j) { /* Computing MIN */ d__1 = rcmin, d__2 = c__[j]; rcmin = min(d__1,d__2); /* Computing MAX */ d__1 = rcmax, d__2 = c__[j]; rcmax = max(d__1,d__2); /* L20: */ } if (rcmin <= 0.) { *info = -12; } else if (*n > 0) { colcnd = max(rcmin,smlnum) / min(rcmax,bignum); } else { colcnd = 1.; } } if (*info == 0) { if (*ldb < max(1,*n)) { *info = -14; } else if (*ldx < max(1,*n)) { *info = -16; } } } if (*info != 0) { i__1 = -(*info); xerbla_("ZGESVXX", &i__1); return 0; } if (equil) { /* Compute row and column scalings to equilibrate the matrix A. */ zgeequb_(n, n, &a[a_offset], lda, &r__[1], &c__[1], &rowcnd, &colcnd, &amax, &infequ); if (infequ == 0) { /* Equilibrate the matrix. */ zlaqge_(n, n, &a[a_offset], lda, &r__[1], &c__[1], &rowcnd, & colcnd, &amax, equed); rowequ = lsame_(equed, "R") || lsame_(equed, "B"); colequ = lsame_(equed, "C") || lsame_(equed, "B"); } /* If the scaling factors are not applied, set them to 1.0. */ if (! rowequ) { i__1 = *n; for (j = 1; j <= i__1; ++j) { r__[j] = 1.; } } if (! colequ) { i__1 = *n; for (j = 1; j <= i__1; ++j) { c__[j] = 1.; } } } /* Scale the right-hand side. */ if (notran) { if (rowequ) { zlascl2_(n, nrhs, &r__[1], &b[b_offset], ldb); } } else { if (colequ) { zlascl2_(n, nrhs, &c__[1], &b[b_offset], ldb); } } if (nofact || equil) { /* Compute the LU factorization of A. */ zlacpy_("Full", n, n, &a[a_offset], lda, &af[af_offset], ldaf); zgetrf_(n, n, &af[af_offset], ldaf, &ipiv[1], info); /* Return if INFO is non-zero. */ if (*info > 0) { /* Pivot in column INFO is exactly 0 */ /* Compute the reciprocal pivot growth factor of the */ /* leading rank-deficient INFO columns of A. */ *rpvgrw = zla_rpvgrw__(n, info, &a[a_offset], lda, &af[af_offset], ldaf); return 0; } } /* Compute the reciprocal pivot growth factor RPVGRW. */ *rpvgrw = zla_rpvgrw__(n, n, &a[a_offset], lda, &af[af_offset], ldaf); /* Compute the solution matrix X. */ zlacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx); zgetrs_(trans, n, nrhs, &af[af_offset], ldaf, &ipiv[1], &x[x_offset], ldx, info); /* Use iterative refinement to improve the computed solution and */ /* compute error bounds and backward error estimates for it. */ zgerfsx_(trans, equed, n, nrhs, &a[a_offset], lda, &af[af_offset], ldaf, & ipiv[1], &r__[1], &c__[1], &b[b_offset], ldb, &x[x_offset], ldx, rcond, &berr[1], n_err_bnds__, &err_bnds_norm__[ err_bnds_norm_offset], &err_bnds_comp__[err_bnds_comp_offset], nparams, ¶ms[1], &work[1], &rwork[1], info); /* Scale solutions. */ if (colequ && notran) { zlascl2_(n, nrhs, &c__[1], &x[x_offset], ldx); } else if (rowequ && ! notran) { zlascl2_(n, nrhs, &r__[1], &x[x_offset], ldx); } return 0; /* End of ZGESVXX */ } /* zgesvxx_ */