/* Subroutine */ int zherfsx_(char *uplo, char *equed, integer *n, integer *
nrhs, doublecomplex *a, integer *lda, doublecomplex *af, integer *
ldaf, integer *ipiv, doublereal *s, doublecomplex *b, integer *ldb,
doublecomplex *x, integer *ldx, doublereal *rcond, 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;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
doublereal illrcond_thresh__, unstable_thresh__, err_lbnd__;
integer ref_type__;
integer j;
doublereal rcond_tmp__;
integer prec_type__;
doublereal cwise_wrong__;
extern /* Subroutine */ int zla_herfsx_extended__(integer *, char *,
integer *, integer *, doublecomplex *, integer *, doublecomplex *,
integer *, integer *, logical *, doublereal *, doublecomplex *,
integer *, doublecomplex *, integer *, doublereal *, integer *,
doublereal *, doublereal *, doublecomplex *, doublereal *,
doublecomplex *, doublecomplex *, doublereal *, integer *,
doublereal *, doublereal *, logical *, integer *, ftnlen);
char norm[1];
logical ignore_cwise__;
extern logical lsame_(char *, char *);
doublereal anorm;
logical rcequ;
extern doublereal zla_hercond_c__(char *, integer *, doublecomplex *,
integer *, doublecomplex *, integer *, integer *, doublereal *,
logical *, integer *, doublecomplex *, doublereal *, ftnlen),
zla_hercond_x__(char *, integer *, doublecomplex *, integer *,
doublecomplex *, integer *, integer *, doublecomplex *, integer *,
doublecomplex *, doublereal *, ftnlen), dlamch_(char *);
extern /* Subroutine */ int xerbla_(char *, integer *);
extern doublereal zlanhe_(char *, char *, integer *, doublecomplex *,
integer *, doublereal *);
extern /* Subroutine */ int zhecon_(char *, integer *, doublecomplex *,
integer *, integer *, doublereal *, doublereal *, doublecomplex *,
integer *);
extern integer ilaprec_(char *);
integer ithresh, n_norms__;
doublereal rthresh;
/* -- LAPACK 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. -- */
/* .. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* Purpose */
/* ======= */
/* ZHERFSX improves the computed solution to a system of linear */
/* equations when the coefficient matrix is Hermitian indefinite, and */
/* provides error bounds and backward error estimates for the */
/* solution. In addition to normwise error bound, the code provides */
/* maximum componentwise error bound if possible. See comments for */
/* ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds. */
/* The original system of linear equations may have been equilibrated */
/* before calling this routine, as described by arguments EQUED and S */
/* below. In this case, the solution and error bounds returned are */
/* for the original unequilibrated system. */
/* 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. */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* EQUED (input) CHARACTER*1 */
/* Specifies the form of equilibration that was done to A */
/* before calling this routine. This is needed to compute */
/* the solution and error bounds correctly. */
/* = 'N': No equilibration */
/* = 'Y': Both row and column equilibration, i.e., A has been */
/* replaced by diag(S) * A * diag(S). */
/* The right hand side B has been changed accordingly. */
/* N (input) INTEGER */
/* 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) COMPLEX*16 array, dimension (LDA,N) */
/* 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. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* AF (input) COMPLEX*16 array, dimension (LDAF,N) */
/* The factored form of the matrix A. AF contains the block */
/* diagonal matrix D and the multipliers used to obtain the */
/* factor U or L from the factorization A = U*D*U**T or A = */
/* L*D*L**T as computed by DSYTRF. */
/* LDAF (input) INTEGER */
/* The leading dimension of the array AF. LDAF >= max(1,N). */
/* IPIV (input) INTEGER array, dimension (N) */
/* Details of the interchanges and the block structure of D */
/* as determined by DSYTRF. */
/* S (input or output) DOUBLE PRECISION array, dimension (N) */
/* The 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) COMPLEX*16 array, dimension (LDB,NRHS) */
/* The right hand side matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* X (input/output) COMPLEX*16 array, dimension (LDX,NRHS) */
/* On entry, the solution matrix X, as computed by DGETRS. */
/* On exit, the improved solution matrix 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. */
/* BERR (output) DOUBLE PRECISION array, dimension (NRHS) */
/* Componentwise relative backward error. This is the */
/* componentwise relative backward error of each solution vector X(j) */
/* (i.e., the smallest relative change in any element of A or B that */
/* makes X(j) an exact solution). */
/* N_ERR_BNDS (input) INTEGER */
/* Number of error bounds to return for each right hand side */
/* and each type (normwise or componentwise). See ERR_BNDS_NORM and */
/* ERR_BNDS_COMP below. */
/* ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) */
/* For each right-hand side, this array contains information about */
/* various error bounds and condition numbers corresponding to the */
/* normwise relative error, which is defined as follows: */
/* Normwise relative error in the ith solution vector: */
/* max_j (abs(XTRUE(j,i) - X(j,i))) */
/* ------------------------------ */
/* max_j abs(X(j,i)) */
/* The array is indexed by the type of error information as described */
/* below. There currently are up to three pieces of information */
/* returned. */
/* The first index in ERR_BNDS_NORM(i,:) corresponds to the ith */
/* right-hand side. */
/* The second index in ERR_BNDS_NORM(:,err) contains the following */
/* three fields: */
/* err = 1 "Trust/don't trust" boolean. Trust the answer if the */
/* reciprocal condition number is less than the threshold */
/* sqrt(n) * dlamch('Epsilon'). */
/* err = 2 "Guaranteed" error bound: The estimated forward error, */
/* almost certainly within a factor of 10 of the true error */
/* so long as the next entry is greater than the threshold */
/* sqrt(n) * dlamch('Epsilon'). This error bound should only */
/* be trusted if the previous boolean is true. */
/* err = 3 Reciprocal condition number: Estimated normwise */
/* reciprocal condition number. Compared with the threshold */
/* sqrt(n) * dlamch('Epsilon') to determine if the error */
/* estimate is "guaranteed". These reciprocal condition */
/* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */
/* appropriately scaled matrix Z. */
/* Let Z = S*A, where S scales each row by a power of the */
/* radix so all absolute row sums of Z are approximately 1. */
/* See Lapack Working Note 165 for further details and extra */
/* cautions. */
/* ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) */
/* For each right-hand side, this array contains information about */
/* various error bounds and condition numbers corresponding to the */
/* componentwise relative error, which is defined as follows: */
/* Componentwise relative error in the ith solution vector: */
/* abs(XTRUE(j,i) - X(j,i)) */
/* max_j ---------------------- */
/* abs(X(j,i)) */
/* The array is indexed by the right-hand side i (on which the */
/* componentwise relative error depends), and the type of error */
/* information as described below. There currently are up to three */
/* pieces of information returned for each right-hand side. If */
/* componentwise accuracy is not requested (PARAMS(3) = 0.0), then */
/* ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most */
/* the first (:,N_ERR_BNDS) entries are returned. */
/* The first index in ERR_BNDS_COMP(i,:) corresponds to the ith */
/* right-hand side. */
/* The second index in ERR_BNDS_COMP(:,err) contains the following */
/* three fields: */
/* err = 1 "Trust/don't trust" boolean. Trust the answer if the */
/* reciprocal condition number is less than the threshold */
/* sqrt(n) * dlamch('Epsilon'). */
/* err = 2 "Guaranteed" error bound: The estimated forward error, */
/* almost certainly within a factor of 10 of the true error */
/* so long as the next entry is greater than the threshold */
/* sqrt(n) * dlamch('Epsilon'). This error bound should only */
/* be trusted if the previous boolean is true. */
/* err = 3 Reciprocal condition number: Estimated componentwise */
/* reciprocal condition number. Compared with the threshold */
/* sqrt(n) * dlamch('Epsilon') to determine if the error */
/* estimate is "guaranteed". These reciprocal condition */
/* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */
/* appropriately scaled matrix Z. */
/* Let Z = S*(A*diag(x)), where x is the solution for the */
/* current right-hand side and S scales each row of */
/* A*diag(x) by a power of the radix so all absolute row */
/* sums of Z are approximately 1. */
/* See Lapack Working Note 165 for further details and extra */
/* cautions. */
/* NPARAMS (input) INTEGER */
/* Specifies the number of parameters set in PARAMS. If .LE. 0, the */
/* PARAMS array is never referenced and default values are used. */
/* PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS */
/* Specifies algorithm parameters. If an entry is .LT. 0.0, then */
/* that entry will be filled with default value used for that */
/* parameter. Only positions up to NPARAMS are accessed; defaults */
/* are used for higher-numbered parameters. */
/* PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative */
/* refinement or not. */
/* Default: 1.0D+0 */
/* = 0.0 : No refinement is performed, and no error bounds are */
/* computed. */
/* = 1.0 : Use the double-precision refinement algorithm, */
/* possibly with doubled-single computations if the */
/* compilation environment does not support DOUBLE */
/* PRECISION. */
/* (other values are reserved for future use) */
/* PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual */
/* computations allowed for refinement. */
/* Default: 10 */
/* Aggressive: Set to 100 to permit convergence using approximate */
/* factorizations or factorizations other than LU. If */
/* the factorization uses a technique other than */
/* Gaussian elimination, the guarantees in */
/* err_bnds_norm and err_bnds_comp may no longer be */
/* trustworthy. */
/* PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code */
/* will attempt to find a solution with small componentwise */
/* relative error in the double-precision algorithm. Positive */
/* is true, 0.0 is false. */
/* Default: 1.0 (attempt componentwise convergence) */
/* WORK (workspace) COMPLEX*16 array, dimension (2*N) */
/* RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) */
/* INFO (output) INTEGER */
/* = 0: Successful exit. The solution to every right-hand side is */
/* guaranteed. */
/* < 0: If INFO = -i, the i-th argument had an illegal value */
/* > 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 Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Check the input parameters. */
/* 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;
--s;
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;
ref_type__ = 1;
if (*nparams >= 1) {
if (params[1] < 0.) {
params[1] = 1.;
} else {
ref_type__ = (integer) params[1];
}
}
/* Set default parameters. */
illrcond_thresh__ = (doublereal) (*n) * dlamch_("Epsilon");
ithresh = 10;
rthresh = .5;
unstable_thresh__ = .25;
ignore_cwise__ = FALSE_;
if (*nparams >= 2) {
if (params[2] < 0.) {
params[2] = (doublereal) ithresh;
} else {
ithresh = (integer) params[2];
}
}
if (*nparams >= 3) {
if (params[3] < 0.) {
if (ignore_cwise__) {
params[3] = 0.;
} else {
params[3] = 1.;
}
} else {
ignore_cwise__ = params[3] == 0.;
}
}
if (ref_type__ == 0 || *n_err_bnds__ == 0) {
n_norms__ = 0;
} else if (ignore_cwise__) {
n_norms__ = 1;
} else {
n_norms__ = 2;
}
rcequ = lsame_(equed, "Y");
/* Test input parameters. */
if (! lsame_(uplo, "U") && ! lsame_(uplo, "L")) {
*info = -1;
} else if (! rcequ && ! lsame_(equed, "N")) {
*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 (*ldb < max(1,*n)) {
*info = -11;
} else if (*ldx < max(1,*n)) {
*info = -13;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZHERFSX", &i__1);
return 0;
}
/* Quick return if possible. */
if (*n == 0 || *nrhs == 0) {
*rcond = 1.;
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
berr[j] = 0.;
if (*n_err_bnds__ >= 1) {
err_bnds_norm__[j + err_bnds_norm_dim1] = 1.;
err_bnds_comp__[j + err_bnds_comp_dim1] = 1.;
} else if (*n_err_bnds__ >= 2) {
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = 0.;
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = 0.;
} else if (*n_err_bnds__ >= 3) {
err_bnds_norm__[j + err_bnds_norm_dim1 * 3] = 1.;
err_bnds_comp__[j + err_bnds_comp_dim1 * 3] = 1.;
}
}
return 0;
}
/* Default to failure. */
*rcond = 0.;
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
berr[j] = 1.;
if (*n_err_bnds__ >= 1) {
err_bnds_norm__[j + err_bnds_norm_dim1] = 1.;
err_bnds_comp__[j + err_bnds_comp_dim1] = 1.;
} else if (*n_err_bnds__ >= 2) {
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = 1.;
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = 1.;
} else if (*n_err_bnds__ >= 3) {
err_bnds_norm__[j + err_bnds_norm_dim1 * 3] = 0.;
err_bnds_comp__[j + err_bnds_comp_dim1 * 3] = 0.;
}
}
/* Compute the norm of A and the reciprocal of the condition */
/* number of A. */
*(unsigned char *)norm = 'I';
anorm = zlanhe_(norm, uplo, n, &a[a_offset], lda, &rwork[1]);
zhecon_(uplo, n, &af[af_offset], ldaf, &ipiv[1], &anorm, rcond, &work[1],
info);
/* Perform refinement on each right-hand side */
if (ref_type__ != 0) {
prec_type__ = ilaprec_("E");
zla_herfsx_extended__(&prec_type__, uplo, n, nrhs, &a[a_offset], lda,
&af[af_offset], ldaf, &ipiv[1], &rcequ, &s[1], &b[b_offset],
ldb, &x[x_offset], ldx, &berr[1], &n_norms__, &
err_bnds_norm__[err_bnds_norm_offset], &err_bnds_comp__[
err_bnds_comp_offset], &work[1], &rwork[1], &work[*n + 1],
(doublecomplex *)(&rwork[1]), rcond, &ithresh, &rthresh, &unstable_thresh__, &
ignore_cwise__, info, (ftnlen)1);
}
/* Computing MAX */
d__1 = 10., d__2 = sqrt((doublereal) (*n));
err_lbnd__ = max(d__1,d__2) * dlamch_("Epsilon");
if (*n_err_bnds__ >= 1 && n_norms__ >= 1) {
/* Compute scaled normwise condition number cond(A*C). */
if (rcequ) {
rcond_tmp__ = zla_hercond_c__(uplo, n, &a[a_offset], lda, &af[
af_offset], ldaf, &ipiv[1], &s[1], &c_true, info, &work[1]
, &rwork[1], (ftnlen)1);
} else {
rcond_tmp__ = zla_hercond_c__(uplo, n, &a[a_offset], lda, &af[
af_offset], ldaf, &ipiv[1], &s[1], &c_false, info, &work[
1], &rwork[1], (ftnlen)1);
}
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
/* Cap the error at 1.0. */
if (*n_err_bnds__ >= 2 && err_bnds_norm__[j + (err_bnds_norm_dim1
<< 1)] > 1.) {
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = 1.;
}
/* Threshold the error (see LAWN). */
if (rcond_tmp__ < illrcond_thresh__) {
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = 1.;
err_bnds_norm__[j + err_bnds_norm_dim1] = 0.;
if (*info <= *n) {
*info = *n + j;
}
} else if (err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] <
err_lbnd__) {
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = err_lbnd__;
err_bnds_norm__[j + err_bnds_norm_dim1] = 1.;
}
/* Save the condition number. */
if (*n_err_bnds__ >= 3) {
err_bnds_norm__[j + err_bnds_norm_dim1 * 3] = rcond_tmp__;
}
}
}
if (*n_err_bnds__ >= 1 && n_norms__ >= 2) {
/* Compute componentwise condition number cond(A*diag(Y(:,J))) for */
/* each right-hand side using the current solution as an estimate of */
/* the true solution. If the componentwise error estimate is too */
/* large, then the solution is a lousy estimate of truth and the */
/* estimated RCOND may be too optimistic. To avoid misleading users, */
/* the inverse condition number is set to 0.0 when the estimated */
/* cwise error is at least CWISE_WRONG. */
cwise_wrong__ = sqrt(dlamch_("Epsilon"));
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
if (err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] <
cwise_wrong__) {
rcond_tmp__ = zla_hercond_x__(uplo, n, &a[a_offset], lda, &af[
af_offset], ldaf, &ipiv[1], &x[j * x_dim1 + 1], info,
&work[1], &rwork[1], (ftnlen)1);
} else {
rcond_tmp__ = 0.;
}
/* Cap the error at 1.0. */
if (*n_err_bnds__ >= 2 && err_bnds_comp__[j + (err_bnds_comp_dim1
<< 1)] > 1.) {
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = 1.;
}
/* Threshold the error (see LAWN). */
if (rcond_tmp__ < illrcond_thresh__) {
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = 1.;
err_bnds_comp__[j + err_bnds_comp_dim1] = 0.;
if (params[3] == 1. && *info < *n + j) {
*info = *n + j;
}
} else if (err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] <
err_lbnd__) {
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = err_lbnd__;
err_bnds_comp__[j + err_bnds_comp_dim1] = 1.;
}
/* Save the condition number. */
if (*n_err_bnds__ >= 3) {
err_bnds_comp__[j + err_bnds_comp_dim1 * 3] = rcond_tmp__;
}
}
}
return 0;
/* End of ZHERFSX */
} /* zherfsx_ */
/* Subroutine */ int dgbrfsx_(char *trans, char *equed, integer *n, integer *
kl, integer *ku, integer *nrhs, doublereal *ab, integer *ldab,
doublereal *afb, integer *ldafb, integer *ipiv, doublereal *r__,
doublereal *c__, doublereal *b, integer *ldb, doublereal *x, integer *
ldx, doublereal *rcond, doublereal *berr, integer *n_err_bnds__,
doublereal *err_bnds_norm__, doublereal *err_bnds_comp__, integer *
nparams, doublereal *params, doublereal *work, integer *iwork,
integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, afb_dim1, afb_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 */
doublereal illrcond_thresh__, unstable_thresh__, err_lbnd__;
integer ref_type__;
integer j;
doublereal rcond_tmp__;
integer prec_type__, trans_type__;
doublereal cwise_wrong__;
char norm[1];
logical ignore_cwise__;
doublereal anorm;
logical colequ, notran, rowequ;
integer ithresh, n_norms__;
doublereal rthresh;
/* -- LAPACK 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. -- */
/* Purpose */
/* ======= */
/* DGBRFSX improves the computed solution to a system of linear */
/* equations and provides error bounds and backward error estimates */
/* for the solution. In addition to normwise error bound, the code */
/* provides maximum componentwise error bound if possible. See */
/* comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the */
/* error bounds. */
/* The original system of linear equations may have been equilibrated */
/* before calling this routine, as described by arguments EQUED, R */
/* and C below. In this case, the solution and error bounds returned */
/* are for the original unequilibrated system. */
/* 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. */
/* 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) */
/* EQUED (input) CHARACTER*1 */
/* Specifies the form of equilibration that was done to A */
/* before calling this routine. This is needed to compute */
/* the solution and error bounds correctly. */
/* = 'N': No equilibration */
/* = '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). */
/* The right hand side B has been changed accordingly. */
/* N (input) INTEGER */
/* 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) DOUBLE PRECISION array, dimension (LDAB,N) */
/* The original band matrix A, stored 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). */
/* LDAB (input) INTEGER */
/* The leading dimension of the array AB. LDAB >= KL+KU+1. */
/* AFB (input) DOUBLE PRECISION array, dimension (LDAFB,N) */
/* Details of the LU factorization of the band matrix A, as */
/* computed by DGBTRF. 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. */
/* LDAFB (input) INTEGER */
/* The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. */
/* IPIV (input) INTEGER array, dimension (N) */
/* The pivot indices from DGETRF; for 1<=i<=N, row i of the */
/* matrix was interchanged with row IPIV(i). */
/* 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) DOUBLE PRECISION array, dimension (LDB,NRHS) */
/* The right hand side matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) */
/* On entry, the solution matrix X, as computed by DGETRS. */
/* On exit, the improved solution matrix 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. */
/* BERR (output) DOUBLE PRECISION array, dimension (NRHS) */
/* Componentwise relative backward error. This is the */
/* componentwise relative backward error of each solution vector X(j) */
/* (i.e., the smallest relative change in any element of A or B that */
/* makes X(j) an exact solution). */
/* N_ERR_BNDS (input) INTEGER */
/* Number of error bounds to return for each right hand side */
/* and each type (normwise or componentwise). See ERR_BNDS_NORM and */
/* ERR_BNDS_COMP below. */
/* ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) */
/* For each right-hand side, this array contains information about */
/* various error bounds and condition numbers corresponding to the */
/* normwise relative error, which is defined as follows: */
/* Normwise relative error in the ith solution vector: */
/* max_j (abs(XTRUE(j,i) - X(j,i))) */
/* ------------------------------ */
/* max_j abs(X(j,i)) */
/* The array is indexed by the type of error information as described */
/* below. There currently are up to three pieces of information */
/* returned. */
/* The first index in ERR_BNDS_NORM(i,:) corresponds to the ith */
/* right-hand side. */
/* The second index in ERR_BNDS_NORM(:,err) contains the following */
/* three fields: */
/* err = 1 "Trust/don't trust" boolean. Trust the answer if the */
/* reciprocal condition number is less than the threshold */
/* sqrt(n) * dlamch('Epsilon'). */
/* err = 2 "Guaranteed" error bound: The estimated forward error, */
/* almost certainly within a factor of 10 of the true error */
/* so long as the next entry is greater than the threshold */
/* sqrt(n) * dlamch('Epsilon'). This error bound should only */
/* be trusted if the previous boolean is true. */
/* err = 3 Reciprocal condition number: Estimated normwise */
/* reciprocal condition number. Compared with the threshold */
/* sqrt(n) * dlamch('Epsilon') to determine if the error */
/* estimate is "guaranteed". These reciprocal condition */
/* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */
/* appropriately scaled matrix Z. */
/* Let Z = S*A, where S scales each row by a power of the */
/* radix so all absolute row sums of Z are approximately 1. */
/* See Lapack Working Note 165 for further details and extra */
/* cautions. */
/* ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) */
/* For each right-hand side, this array contains information about */
/* various error bounds and condition numbers corresponding to the */
/* componentwise relative error, which is defined as follows: */
/* Componentwise relative error in the ith solution vector: */
/* abs(XTRUE(j,i) - X(j,i)) */
/* max_j ---------------------- */
/* abs(X(j,i)) */
/* The array is indexed by the right-hand side i (on which the */
/* componentwise relative error depends), and the type of error */
/* information as described below. There currently are up to three */
/* pieces of information returned for each right-hand side. If */
/* componentwise accuracy is not requested (PARAMS(3) = 0.0), then */
/* ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most */
/* the first (:,N_ERR_BNDS) entries are returned. */
/* The first index in ERR_BNDS_COMP(i,:) corresponds to the ith */
/* right-hand side. */
/* The second index in ERR_BNDS_COMP(:,err) contains the following */
/* three fields: */
/* err = 1 "Trust/don't trust" boolean. Trust the answer if the */
/* reciprocal condition number is less than the threshold */
/* sqrt(n) * dlamch('Epsilon'). */
/* err = 2 "Guaranteed" error bound: The estimated forward error, */
/* almost certainly within a factor of 10 of the true error */
/* so long as the next entry is greater than the threshold */
/* sqrt(n) * dlamch('Epsilon'). This error bound should only */
/* be trusted if the previous boolean is true. */
/* err = 3 Reciprocal condition number: Estimated componentwise */
/* reciprocal condition number. Compared with the threshold */
/* sqrt(n) * dlamch('Epsilon') to determine if the error */
/* estimate is "guaranteed". These reciprocal condition */
/* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */
/* appropriately scaled matrix Z. */
/* Let Z = S*(A*diag(x)), where x is the solution for the */
/* current right-hand side and S scales each row of */
/* A*diag(x) by a power of the radix so all absolute row */
/* sums of Z are approximately 1. */
/* See Lapack Working Note 165 for further details and extra */
/* cautions. */
/* NPARAMS (input) INTEGER */
/* Specifies the number of parameters set in PARAMS. If .LE. 0, the */
/* PARAMS array is never referenced and default values are used. */
/* PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS */
/* Specifies algorithm parameters. If an entry is .LT. 0.0, then */
/* that entry will be filled with default value used for that */
/* parameter. Only positions up to NPARAMS are accessed; defaults */
/* are used for higher-numbered parameters. */
/* PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative */
/* refinement or not. */
/* Default: 1.0D+0 */
/* = 0.0 : No refinement is performed, and no error bounds are */
/* computed. */
/* = 1.0 : Use the double-precision refinement algorithm, */
/* possibly with doubled-single computations if the */
/* compilation environment does not support DOUBLE */
/* PRECISION. */
/* (other values are reserved for future use) */
/* PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual */
/* computations allowed for refinement. */
/* Default: 10 */
/* Aggressive: Set to 100 to permit convergence using approximate */
/* factorizations or factorizations other than LU. If */
/* the factorization uses a technique other than */
/* Gaussian elimination, the guarantees in */
/* err_bnds_norm and err_bnds_comp may no longer be */
/* trustworthy. */
/* PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code */
/* will attempt to find a solution with small componentwise */
/* relative error in the double-precision algorithm. Positive */
/* is true, 0.0 is false. */
/* Default: 1.0 (attempt componentwise convergence) */
/* WORK (workspace) DOUBLE PRECISION array, dimension (4*N) */
/* IWORK (workspace) INTEGER array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: Successful exit. The solution to every right-hand side is */
/* guaranteed. */
/* < 0: If INFO = -i, the i-th argument had an illegal value */
/* > 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. */
/* ================================================================== */
/* Check the input parameters. */
/* 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;
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
afb_dim1 = *ldafb;
afb_offset = 1 + afb_dim1;
afb -= afb_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;
--iwork;
/* Function Body */
*info = 0;
trans_type__ = ilatrans_(trans);
ref_type__ = 1;
if (*nparams >= 1) {
if (params[1] < 0.) {
params[1] = 1.;
} else {
ref_type__ = (integer) params[1];
}
}
/* Set default parameters. */
illrcond_thresh__ = (doublereal) (*n) * dlamch_("Epsilon");
ithresh = 10;
rthresh = .5;
unstable_thresh__ = .25;
ignore_cwise__ = FALSE_;
if (*nparams >= 2) {
if (params[2] < 0.) {
params[2] = (doublereal) ithresh;
} else {
ithresh = (integer) params[2];
}
}
if (*nparams >= 3) {
if (params[3] < 0.) {
if (ignore_cwise__) {
params[3] = 0.;
} else {
params[3] = 1.;
}
} else {
ignore_cwise__ = params[3] == 0.;
}
}
if (ref_type__ == 0 || *n_err_bnds__ == 0) {
n_norms__ = 0;
} else if (ignore_cwise__) {
n_norms__ = 1;
} else {
n_norms__ = 2;
}
notran = lsame_(trans, "N");
rowequ = lsame_(equed, "R") || lsame_(equed, "B");
colequ = lsame_(equed, "C") || lsame_(equed, "B");
/* Test input parameters. */
if (trans_type__ == -1) {
*info = -1;
} else if (! rowequ && ! colequ && ! lsame_(equed, "N")) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*kl < 0) {
*info = -4;
} else if (*ku < 0) {
*info = -5;
} else if (*nrhs < 0) {
*info = -6;
} else if (*ldab < *kl + *ku + 1) {
*info = -8;
} else if (*ldafb < (*kl << 1) + *ku + 1) {
*info = -10;
} else if (*ldb < max(1,*n)) {
*info = -13;
} else if (*ldx < max(1,*n)) {
*info = -15;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DGBRFSX", &i__1);
return 0;
}
/* Quick return if possible. */
if (*n == 0 || *nrhs == 0) {
*rcond = 1.;
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
berr[j] = 0.;
if (*n_err_bnds__ >= 1) {
err_bnds_norm__[j + err_bnds_norm_dim1] = 1.;
err_bnds_comp__[j + err_bnds_comp_dim1] = 1.;
} else if (*n_err_bnds__ >= 2) {
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = 0.;
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = 0.;
} else if (*n_err_bnds__ >= 3) {
err_bnds_norm__[j + err_bnds_norm_dim1 * 3] = 1.;
err_bnds_comp__[j + err_bnds_comp_dim1 * 3] = 1.;
}
}
return 0;
}
/* Default to failure. */
*rcond = 0.;
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
berr[j] = 1.;
if (*n_err_bnds__ >= 1) {
err_bnds_norm__[j + err_bnds_norm_dim1] = 1.;
err_bnds_comp__[j + err_bnds_comp_dim1] = 1.;
} else if (*n_err_bnds__ >= 2) {
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = 1.;
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = 1.;
} else if (*n_err_bnds__ >= 3) {
err_bnds_norm__[j + err_bnds_norm_dim1 * 3] = 0.;
err_bnds_comp__[j + err_bnds_comp_dim1 * 3] = 0.;
}
}
/* Compute the norm of A and the reciprocal of the condition */
/* number of A. */
if (notran) {
*(unsigned char *)norm = 'I';
} else {
*(unsigned char *)norm = '1';
}
anorm = dlangb_(norm, n, kl, ku, &ab[ab_offset], ldab, &work[1]);
dgbcon_(norm, n, kl, ku, &afb[afb_offset], ldafb, &ipiv[1], &anorm, rcond,
&work[1], &iwork[1], info);
/* Perform refinement on each right-hand side */
if (ref_type__ != 0) {
prec_type__ = ilaprec_("E");
if (notran) {
dla_gbrfsx_extended__(&prec_type__, &trans_type__, n, kl, ku,
nrhs, &ab[ab_offset], ldab, &afb[afb_offset], ldafb, &
ipiv[1], &colequ, &c__[1], &b[b_offset], ldb, &x[x_offset]
, ldx, &berr[1], &n_norms__, &err_bnds_norm__[
err_bnds_norm_offset], &err_bnds_comp__[
err_bnds_comp_offset], &work[*n + 1], &work[1], &work[(*n
<< 1) + 1], &work[1], rcond, &ithresh, &rthresh, &
unstable_thresh__, &ignore_cwise__, info);
} else {
dla_gbrfsx_extended__(&prec_type__, &trans_type__, n, kl, ku,
nrhs, &ab[ab_offset], ldab, &afb[afb_offset], ldafb, &
ipiv[1], &rowequ, &r__[1], &b[b_offset], ldb, &x[x_offset]
, ldx, &berr[1], &n_norms__, &err_bnds_norm__[
err_bnds_norm_offset], &err_bnds_comp__[
err_bnds_comp_offset], &work[*n + 1], &work[1], &work[(*n
<< 1) + 1], &work[1], rcond, &ithresh, &rthresh, &
unstable_thresh__, &ignore_cwise__, info);
}
}
/* Computing MAX */
d__1 = 10., d__2 = sqrt((doublereal) (*n));
err_lbnd__ = max(d__1,d__2) * dlamch_("Epsilon");
if (*n_err_bnds__ >= 1 && n_norms__ >= 1) {
/* Compute scaled normwise condition number cond(A*C). */
if (colequ && notran) {
rcond_tmp__ = dla_gbrcond__(trans, n, kl, ku, &ab[ab_offset],
ldab, &afb[afb_offset], ldafb, &ipiv[1], &c_n1, &c__[1],
info, &work[1], &iwork[1], (ftnlen)1);
} else if (rowequ && ! notran) {
rcond_tmp__ = dla_gbrcond__(trans, n, kl, ku, &ab[ab_offset],
ldab, &afb[afb_offset], ldafb, &ipiv[1], &c_n1, &r__[1],
info, &work[1], &iwork[1], (ftnlen)1);
} else {
rcond_tmp__ = dla_gbrcond__(trans, n, kl, ku, &ab[ab_offset],
ldab, &afb[afb_offset], ldafb, &ipiv[1], &c__0, &r__[1],
info, &work[1], &iwork[1], (ftnlen)1);
}
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
/* Cap the error at 1.0. */
if (*n_err_bnds__ >= 2 && err_bnds_norm__[j + (err_bnds_norm_dim1
<< 1)] > 1.) {
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = 1.;
}
/* Threshold the error (see LAWN). */
if (rcond_tmp__ < illrcond_thresh__) {
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = 1.;
err_bnds_norm__[j + err_bnds_norm_dim1] = 0.;
if (*info <= *n) {
*info = *n + j;
}
} else if (err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] <
err_lbnd__) {
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = err_lbnd__;
err_bnds_norm__[j + err_bnds_norm_dim1] = 1.;
}
/* Save the condition number. */
if (*n_err_bnds__ >= 3) {
err_bnds_norm__[j + err_bnds_norm_dim1 * 3] = rcond_tmp__;
}
}
}
if (*n_err_bnds__ >= 1 && n_norms__ >= 2) {
/* Compute componentwise condition number cond(A*diag(Y(:,J))) for */
/* each right-hand side using the current solution as an estimate of */
/* the true solution. If the componentwise error estimate is too */
/* large, then the solution is a lousy estimate of truth and the */
/* estimated RCOND may be too optimistic. To avoid misleading users, */
/* the inverse condition number is set to 0.0 when the estimated */
/* cwise error is at least CWISE_WRONG. */
cwise_wrong__ = sqrt(dlamch_("Epsilon"));
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
if (err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] <
cwise_wrong__) {
rcond_tmp__ = dla_gbrcond__(trans, n, kl, ku, &ab[ab_offset],
ldab, &afb[afb_offset], ldafb, &ipiv[1], &c__1, &x[j *
x_dim1 + 1], info, &work[1], &iwork[1], (ftnlen)1);
} else {
rcond_tmp__ = 0.;
}
/* Cap the error at 1.0. */
if (*n_err_bnds__ >= 2 && err_bnds_comp__[j + (err_bnds_comp_dim1
<< 1)] > 1.) {
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = 1.;
}
/* Threshold the error (see LAWN). */
if (rcond_tmp__ < illrcond_thresh__) {
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = 1.;
err_bnds_comp__[j + err_bnds_comp_dim1] = 0.;
if (params[3] == 1. && *info < *n + j) {
*info = *n + j;
}
} else if (err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] <
err_lbnd__) {
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = err_lbnd__;
err_bnds_comp__[j + err_bnds_comp_dim1] = 1.;
}
/* Save the condition number. */
if (*n_err_bnds__ >= 3) {
err_bnds_comp__[j + err_bnds_comp_dim1 * 3] = rcond_tmp__;
}
}
}
return 0;
/* End of DGBRFSX */
} /* dgbrfsx_ */
/* Subroutine */
int zgbrfsx_(char *trans, char *equed, integer *n, integer * kl, integer *ku, integer *nrhs, doublecomplex *ab, integer *ldab, doublecomplex *afb, integer *ldafb, integer *ipiv, doublereal *r__, doublereal *c__, doublecomplex *b, integer *ldb, doublecomplex *x, integer *ldx, doublereal *rcond, 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 ab_dim1, ab_offset, afb_dim1, afb_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;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
doublereal illrcond_thresh__, unstable_thresh__, err_lbnd__;
integer ref_type__;
extern integer ilatrans_(char *);
integer j;
doublereal rcond_tmp__;
integer prec_type__, trans_type__;
doublereal cwise_wrong__;
extern /* Subroutine */
int zla_gbrfsx_extended_(integer *, integer *, integer *, integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, integer *, logical *, doublereal *, doublecomplex *, integer *, doublecomplex *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublecomplex *, doublereal *, doublecomplex *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, logical *, integer *);
char norm[1];
logical ignore_cwise__;
extern logical lsame_(char *, char *);
doublereal anorm;
extern doublereal zla_gbrcond_c_(char *, integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, integer * , doublereal *, logical *, integer *, doublecomplex *, doublereal *), zla_gbrcond_x_(char *, integer *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublereal *), dlamch_(char *);
extern /* Subroutine */
int xerbla_(char *, integer *);
extern doublereal zlangb_(char *, integer *, integer *, integer *, doublecomplex *, integer *, doublereal *);
extern /* Subroutine */
int zgbcon_(char *, integer *, integer *, integer *, doublecomplex *, integer *, integer *, doublereal *, doublereal *, doublecomplex *, doublereal *, integer *);
logical colequ, notran, rowequ;
extern integer ilaprec_(char *);
integer ithresh, n_norms__;
doublereal rthresh;
/* -- LAPACK computational routine (version 3.4.1) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* April 2012 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Check the input parameters. */
/* 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;
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
afb_dim1 = *ldafb;
afb_offset = 1 + afb_dim1;
afb -= afb_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;
trans_type__ = ilatrans_(trans);
ref_type__ = 1;
if (*nparams >= 1)
{
if (params[1] < 0.)
{
params[1] = 1.;
}
else
{
ref_type__ = (integer) params[1];
}
}
/* Set default parameters. */
illrcond_thresh__ = (doublereal) (*n) * dlamch_("Epsilon");
ithresh = 10;
rthresh = .5;
unstable_thresh__ = .25;
ignore_cwise__ = FALSE_;
if (*nparams >= 2)
{
if (params[2] < 0.)
{
params[2] = (doublereal) ithresh;
}
else
{
ithresh = (integer) params[2];
}
}
if (*nparams >= 3)
{
if (params[3] < 0.)
{
if (ignore_cwise__)
{
params[3] = 0.;
}
else
{
params[3] = 1.;
}
}
else
{
ignore_cwise__ = params[3] == 0.;
}
}
if (ref_type__ == 0 || *n_err_bnds__ == 0)
{
n_norms__ = 0;
}
else if (ignore_cwise__)
{
n_norms__ = 1;
}
else
{
n_norms__ = 2;
}
notran = lsame_(trans, "N");
rowequ = lsame_(equed, "R") || lsame_(equed, "B");
colequ = lsame_(equed, "C") || lsame_(equed, "B");
/* Test input parameters. */
if (trans_type__ == -1)
{
*info = -1;
}
else if (! rowequ && ! colequ && ! lsame_(equed, "N"))
{
*info = -2;
}
else if (*n < 0)
{
*info = -3;
}
else if (*kl < 0)
{
*info = -4;
}
else if (*ku < 0)
{
*info = -5;
}
else if (*nrhs < 0)
{
*info = -6;
}
else if (*ldab < *kl + *ku + 1)
{
*info = -8;
}
else if (*ldafb < (*kl << 1) + *ku + 1)
{
*info = -10;
}
else if (*ldb < max(1,*n))
{
*info = -13;
}
else if (*ldx < max(1,*n))
{
*info = -15;
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("ZGBRFSX", &i__1);
return 0;
}
/* Quick return if possible. */
if (*n == 0 || *nrhs == 0)
{
*rcond = 1.;
i__1 = *nrhs;
for (j = 1;
j <= i__1;
++j)
{
berr[j] = 0.;
if (*n_err_bnds__ >= 1)
{
err_bnds_norm__[j + err_bnds_norm_dim1] = 1.;
err_bnds_comp__[j + err_bnds_comp_dim1] = 1.;
}
if (*n_err_bnds__ >= 2)
{
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = 0.;
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = 0.;
}
if (*n_err_bnds__ >= 3)
{
err_bnds_norm__[j + err_bnds_norm_dim1 * 3] = 1.;
err_bnds_comp__[j + err_bnds_comp_dim1 * 3] = 1.;
}
}
return 0;
}
/* Default to failure. */
*rcond = 0.;
i__1 = *nrhs;
for (j = 1;
j <= i__1;
++j)
{
berr[j] = 1.;
if (*n_err_bnds__ >= 1)
{
err_bnds_norm__[j + err_bnds_norm_dim1] = 1.;
err_bnds_comp__[j + err_bnds_comp_dim1] = 1.;
}
if (*n_err_bnds__ >= 2)
{
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = 1.;
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = 1.;
}
if (*n_err_bnds__ >= 3)
{
err_bnds_norm__[j + err_bnds_norm_dim1 * 3] = 0.;
err_bnds_comp__[j + err_bnds_comp_dim1 * 3] = 0.;
}
}
/* Compute the norm of A and the reciprocal of the condition */
/* number of A. */
if (notran)
{
*(unsigned char *)norm = 'I';
}
else
{
*(unsigned char *)norm = '1';
}
anorm = zlangb_(norm, n, kl, ku, &ab[ab_offset], ldab, &rwork[1]);
zgbcon_(norm, n, kl, ku, &afb[afb_offset], ldafb, &ipiv[1], &anorm, rcond, &work[1], &rwork[1], info);
/* Perform refinement on each right-hand side */
if (ref_type__ != 0)
{
prec_type__ = ilaprec_("E");
if (notran)
{
zla_gbrfsx_extended_(&prec_type__, &trans_type__, n, kl, ku, nrhs, &ab[ab_offset], ldab, &afb[afb_offset], ldafb, & ipiv[1], &colequ, &c__[1], &b[b_offset], ldb, &x[x_offset] , ldx, &berr[1], &n_norms__, &err_bnds_norm__[ err_bnds_norm_offset], &err_bnds_comp__[ err_bnds_comp_offset], &work[1], &rwork[1], &work[*n + 1], &rwork[1], rcond, &ithresh, &rthresh, &unstable_thresh__, &ignore_cwise__, info);
}
else
{
zla_gbrfsx_extended_(&prec_type__, &trans_type__, n, kl, ku, nrhs, &ab[ab_offset], ldab, &afb[afb_offset], ldafb, & ipiv[1], &rowequ, &r__[1], &b[b_offset], ldb, &x[x_offset] , ldx, &berr[1], &n_norms__, &err_bnds_norm__[ err_bnds_norm_offset], &err_bnds_comp__[ err_bnds_comp_offset], &work[1], &rwork[1], &work[*n + 1], &rwork[1], rcond, &ithresh, &rthresh, &unstable_thresh__, &ignore_cwise__, info);
}
}
/* Computing MAX */
d__1 = 10.;
d__2 = sqrt((doublereal) (*n)); // , expr subst
err_lbnd__ = max(d__1,d__2) * dlamch_("Epsilon");
if (*n_err_bnds__ >= 1 && n_norms__ >= 1)
{
/* Compute scaled normwise condition number cond(A*C). */
if (colequ && notran)
{
rcond_tmp__ = zla_gbrcond_c_(trans, n, kl, ku, &ab[ab_offset], ldab, &afb[afb_offset], ldafb, &ipiv[1], &c__[1], &c_true, info, &work[1], &rwork[1]);
}
else if (rowequ && ! notran)
{
rcond_tmp__ = zla_gbrcond_c_(trans, n, kl, ku, &ab[ab_offset], ldab, &afb[afb_offset], ldafb, &ipiv[1], &r__[1], &c_true, info, &work[1], &rwork[1]);
}
else
{
rcond_tmp__ = zla_gbrcond_c_(trans, n, kl, ku, &ab[ab_offset], ldab, &afb[afb_offset], ldafb, &ipiv[1], &c__[1], & c_false, info, &work[1], &rwork[1]);
}
i__1 = *nrhs;
for (j = 1;
j <= i__1;
++j)
{
/* Cap the error at 1.0. */
if (*n_err_bnds__ >= 2 && err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] > 1.)
{
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = 1.;
}
/* Threshold the error (see LAWN). */
if (rcond_tmp__ < illrcond_thresh__)
{
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = 1.;
err_bnds_norm__[j + err_bnds_norm_dim1] = 0.;
if (*info <= *n)
{
*info = *n + j;
}
}
else if (err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] < err_lbnd__)
{
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = err_lbnd__;
err_bnds_norm__[j + err_bnds_norm_dim1] = 1.;
}
/* Save the condition number. */
if (*n_err_bnds__ >= 3)
{
err_bnds_norm__[j + err_bnds_norm_dim1 * 3] = rcond_tmp__;
}
}
}
if (*n_err_bnds__ >= 1 && n_norms__ >= 2)
{
/* Compute componentwise condition number cond(A*diag(Y(:,J))) for */
/* each right-hand side using the current solution as an estimate of */
/* the true solution. If the componentwise error estimate is too */
/* large, then the solution is a lousy estimate of truth and the */
/* estimated RCOND may be too optimistic. To avoid misleading users, */
/* the inverse condition number is set to 0.0 when the estimated */
/* cwise error is at least CWISE_WRONG. */
cwise_wrong__ = sqrt(dlamch_("Epsilon"));
i__1 = *nrhs;
for (j = 1;
j <= i__1;
++j)
{
if (err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] < cwise_wrong__)
{
rcond_tmp__ = zla_gbrcond_x_(trans, n, kl, ku, &ab[ab_offset] , ldab, &afb[afb_offset], ldafb, &ipiv[1], &x[j * x_dim1 + 1], info, &work[1], &rwork[1]);
}
else
{
rcond_tmp__ = 0.;
}
/* Cap the error at 1.0. */
if (*n_err_bnds__ >= 2 && err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] > 1.)
{
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = 1.;
}
/* Threshold the error (see LAWN). */
if (rcond_tmp__ < illrcond_thresh__)
{
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = 1.;
err_bnds_comp__[j + err_bnds_comp_dim1] = 0.;
if (params[3] == 1. && *info < *n + j)
{
*info = *n + j;
}
}
else if (err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] < err_lbnd__)
{
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = err_lbnd__;
err_bnds_comp__[j + err_bnds_comp_dim1] = 1.;
}
/* Save the condition number. */
if (*n_err_bnds__ >= 3)
{
err_bnds_comp__[j + err_bnds_comp_dim1 * 3] = rcond_tmp__;
}
}
}
return 0;
/* End of ZGBRFSX */
}