int
f2c_sgbmv(char *trans, integer *M, integer *N, integer *KL, integer *KU,
real *alpha,
real *A, integer *lda,
real *X, integer *incX,
real *beta,
real *Y, integer *incY)
{
sgbmv_(trans, M, N, KL, KU,
alpha, A, lda, X, incX, beta, Y, incY);
return 0;
}
/* Subroutine */ int sgbt02_(char *trans, integer *m, integer *n, integer *kl,
integer *ku, integer *nrhs, real *a, integer *lda, real *x, integer *
ldx, real *b, integer *ldb, real *resid)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, x_dim1, x_offset, i__1, i__2,
i__3;
real r__1, r__2;
/* Local variables */
integer j, i1, i2, n1, kd;
real eps;
real anorm, bnorm;
real xnorm;
/* -- LAPACK test routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SGBT02 computes the residual for a solution of a banded system of */
/* equations A*x = b or A'*x = b: */
/* RESID = norm( B - A*X ) / ( norm(A) * norm(X) * EPS). */
/* where EPS is the machine precision. */
/* Arguments */
/* ========= */
/* TRANS (input) CHARACTER*1 */
/* Specifies the form of the system of equations: */
/* = 'N': A *x = b */
/* = 'T': A'*x = b, where A' is the transpose of A */
/* = 'C': A'*x = b, where A' is the transpose of A */
/* M (input) INTEGER */
/* The number of rows of the matrix A. M >= 0. */
/* N (input) INTEGER */
/* The number of columns 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 columns of B. NRHS >= 0. */
/* A (input) REAL array, dimension (LDA,N) */
/* The original matrix A in band storage, stored in rows 1 to */
/* KL+KU+1. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,KL+KU+1). */
/* X (input) REAL array, dimension (LDX,NRHS) */
/* The computed solution vectors for the system of linear */
/* equations. */
/* LDX (input) INTEGER */
/* The leading dimension of the array X. If TRANS = 'N', */
/* LDX >= max(1,N); if TRANS = 'T' or 'C', LDX >= max(1,M). */
/* B (input/output) REAL array, dimension (LDB,NRHS) */
/* On entry, the right hand side vectors for the system of */
/* linear equations. */
/* On exit, B is overwritten with the difference B - A*X. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. IF TRANS = 'N', */
/* LDB >= max(1,M); if TRANS = 'T' or 'C', LDB >= max(1,N). */
/* RESID (output) REAL */
/* The maximum over the number of right hand sides of */
/* norm(B - A*X) / ( norm(A) * norm(X) * EPS ). */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Quick return if N = 0 pr NRHS = 0 */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
/* Function Body */
if (*m <= 0 || *n <= 0 || *nrhs <= 0) {
*resid = 0.f;
return 0;
}
/* Exit with RESID = 1/EPS if ANORM = 0. */
eps = slamch_("Epsilon");
kd = *ku + 1;
anorm = 0.f;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* Computing MAX */
i__2 = kd + 1 - j;
i1 = max(i__2,1);
/* Computing MIN */
i__2 = kd + *m - j, i__3 = *kl + kd;
i2 = min(i__2,i__3);
/* Computing MAX */
i__2 = i2 - i1 + 1;
r__1 = anorm, r__2 = sasum_(&i__2, &a[i1 + j * a_dim1], &c__1);
anorm = dmax(r__1,r__2);
/* L10: */
}
if (anorm <= 0.f) {
*resid = 1.f / eps;
return 0;
}
if (lsame_(trans, "T") || lsame_(trans, "C")) {
n1 = *n;
} else {
n1 = *m;
}
/* Compute B - A*X (or B - A'*X ) */
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
sgbmv_(trans, m, n, kl, ku, &c_b8, &a[a_offset], lda, &x[j * x_dim1 +
1], &c__1, &c_b10, &b[j * b_dim1 + 1], &c__1);
/* L20: */
}
/* Compute the maximum over the number of right hand sides of */
/* norm(B - A*X) / ( norm(A) * norm(X) * EPS ). */
*resid = 0.f;
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
bnorm = sasum_(&n1, &b[j * b_dim1 + 1], &c__1);
xnorm = sasum_(&n1, &x[j * x_dim1 + 1], &c__1);
if (xnorm <= 0.f) {
*resid = 1.f / eps;
} else {
/* Computing MAX */
r__1 = *resid, r__2 = bnorm / anorm / xnorm / eps;
*resid = dmax(r__1,r__2);
}
/* L30: */
}
return 0;
/* End of SGBT02 */
} /* sgbt02_ */
void
sgbmv(char trans, int m, int n, int kl, int ku, float alpha, float *a, int inca, float *x, int incx, float beta, float *y, int incy )
{
sgbmv_( &trans, &m, &n, &kl, &ku, &alpha, a, &inca, x, &incx, &beta, y, &incy );
}
/* Subroutine */ int sla_gbrfsx_extended__(integer *prec_type__, integer *
trans_type__, integer *n, integer *kl, integer *ku, integer *nrhs,
real *ab, integer *ldab, real *afb, integer *ldafb, integer *ipiv,
logical *colequ, real *c__, real *b, integer *ldb, real *y, integer *
ldy, real *berr_out__, integer *n_norms__, real *err_bnds_norm__,
real *err_bnds_comp__, real *res, real *ayb, real *dy, real *y_tail__,
real *rcond, integer *ithresh, real *rthresh, real *dz_ub__, logical
*ignore_cwise__, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, afb_dim1, afb_offset, b_dim1, b_offset,
y_dim1, y_offset, err_bnds_norm_dim1, err_bnds_norm_offset,
err_bnds_comp_dim1, err_bnds_comp_offset, i__1, i__2, i__3;
real r__1, r__2;
char ch__1[1];
/* Local variables */
real dxratmax, dzratmax;
integer i__, j, m;
extern /* Subroutine */ int sla_gbamv__(integer *, integer *, integer *,
integer *, integer *, real *, real *, integer *, real *, integer *
, real *, real *, integer *);
logical incr_prec__;
real prev_dz_z__, yk, final_dx_x__, final_dz_z__;
extern /* Subroutine */ int sla_wwaddw__(integer *, real *, real *, real *
);
real prevnormdx;
integer cnt;
real dyk, eps, incr_thresh__, dx_x__, dz_z__, ymin;
extern /* Subroutine */ int sla_lin_berr__(integer *, integer *, integer *
, real *, real *, real *), blas_sgbmv_x__(integer *, integer *,
integer *, integer *, integer *, real *, real *, integer *, real *
, integer *, real *, real *, integer *, integer *);
integer y_prec_state__;
extern /* Subroutine */ int blas_sgbmv2_x__(integer *, integer *, integer
*, integer *, integer *, real *, real *, integer *, real *, real *
, integer *, real *, real *, integer *, integer *), sgbmv_(char *,
integer *, integer *, integer *, integer *, real *, real *,
integer *, real *, integer *, real *, real *, integer *);
real dxrat, dzrat;
char trans[1];
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *);
real normx, normy;
extern /* Subroutine */ int saxpy_(integer *, real *, real *, integer *,
real *, integer *);
extern doublereal slamch_(char *);
extern /* Subroutine */ int sgbtrs_(char *, integer *, integer *, integer
*, integer *, real *, integer *, integer *, real *, integer *,
integer *);
real normdx;
extern /* Character */ VOID chla_transtype__(char *, ftnlen, integer *);
real hugeval;
integer x_state__, z_state__;
/* -- 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 */
/* ======= */
/* SLA_GBRFSX_EXTENDED improves the computed solution to a system of */
/* linear equations by performing extra-precise iterative refinement */
/* and provides error bounds and backward error estimates for the solution. */
/* This subroutine is called by SGBRFSX to perform iterative refinement. */
/* 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. Note that this */
/* subroutine is only resonsible for setting the second fields of */
/* ERR_BNDS_NORM and ERR_BNDS_COMP. */
/* Arguments */
/* ========= */
/* PREC_TYPE (input) INTEGER */
/* Specifies the intermediate precision to be used in refinement. */
/* The value is defined by ILAPREC(P) where P is a CHARACTER and */
/* P = 'S': Single */
/* = 'D': Double */
/* = 'I': Indigenous */
/* = 'X', 'E': Extra */
/* TRANS_TYPE (input) INTEGER */
/* Specifies the transposition operation on A. */
/* The value is defined by ILATRANS(T) where T is a CHARACTER and */
/* T = 'N': No transpose */
/* = 'T': Transpose */
/* = 'C': Conjugate 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 */
/* matrix B. */
/* A (input) REAL array, dimension (LDA,N) */
/* On entry, the N-by-N matrix A. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* AF (input) REAL array, dimension (LDAF,N) */
/* The factors L and U from the factorization */
/* A = P*L*U as computed by SGBTRF. */
/* LDAF (input) INTEGER */
/* The leading dimension of the array AF. LDAF >= max(1,N). */
/* IPIV (input) INTEGER array, dimension (N) */
/* The pivot indices from the factorization A = P*L*U */
/* as computed by SGBTRF; row i of the matrix was interchanged */
/* with row IPIV(i). */
/* COLEQU (input) LOGICAL */
/* If .TRUE. then column equilibration was done to A before calling */
/* this routine. This is needed to compute the solution and error */
/* bounds correctly. */
/* C (input) REAL array, dimension (N) */
/* The column scale factors for A. If COLEQU = .FALSE., C */
/* is not accessed. 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) REAL array, dimension (LDB,NRHS) */
/* The right-hand-side matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* Y (input/output) REAL array, dimension (LDY,NRHS) */
/* On entry, the solution matrix X, as computed by SGBTRS. */
/* On exit, the improved solution matrix Y. */
/* LDY (input) INTEGER */
/* The leading dimension of the array Y. LDY >= max(1,N). */
/* BERR_OUT (output) REAL array, dimension (NRHS) */
/* On exit, BERR_OUT(j) contains the componentwise relative backward */
/* error for right-hand-side j from the formula */
/* max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) */
/* where abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. This is computed by SLA_LIN_BERR. */
/* N_NORMS (input) INTEGER */
/* Determines which error bounds to return (see ERR_BNDS_NORM */
/* and ERR_BNDS_COMP). */
/* If N_NORMS >= 1 return normwise error bounds. */
/* If N_NORMS >= 2 return componentwise error bounds. */
/* ERR_BNDS_NORM (input/output) REAL 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) * slamch('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) * slamch('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) * slamch('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. */
/* This subroutine is only responsible for setting the second field */
/* above. */
/* See Lapack Working Note 165 for further details and extra */
/* cautions. */
/* ERR_BNDS_COMP (input/output) REAL 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) * slamch('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) * slamch('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) * slamch('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. */
/* This subroutine is only responsible for setting the second field */
/* above. */
/* See Lapack Working Note 165 for further details and extra */
/* cautions. */
/* RES (input) REAL array, dimension (N) */
/* Workspace to hold the intermediate residual. */
/* AYB (input) REAL array, dimension (N) */
/* Workspace. This can be the same workspace passed for Y_TAIL. */
/* DY (input) REAL array, dimension (N) */
/* Workspace to hold the intermediate solution. */
/* Y_TAIL (input) REAL array, dimension (N) */
/* Workspace to hold the trailing bits of the intermediate solution. */
/* RCOND (input) REAL */
/* Reciprocal scaled condition number. This is an estimate of the */
/* reciprocal Skeel condition number of the matrix A after */
/* equilibration (if done). If this is less than the machine */
/* precision (in particular, if it is zero), the matrix is singular */
/* to working precision. Note that the error may still be small even */
/* if this number is very small and the matrix appears ill- */
/* conditioned. */
/* ITHRESH (input) INTEGER */
/* The maximum number of residual computations allowed for */
/* refinement. The default is 10. For '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. */
/* RTHRESH (input) REAL */
/* Determines when to stop refinement if the error estimate stops */
/* decreasing. Refinement will stop when the next solution no longer */
/* satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is */
/* the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The */
/* default value is 0.5. For 'aggressive' set to 0.9 to permit */
/* convergence on extremely ill-conditioned matrices. See LAWN 165 */
/* for more details. */
/* DZ_UB (input) REAL */
/* Determines when to start considering componentwise convergence. */
/* Componentwise convergence is only considered after each component */
/* of the solution Y is stable, which we definte as the relative */
/* change in each component being less than DZ_UB. The default value */
/* is 0.25, requiring the first bit to be stable. See LAWN 165 for */
/* more details. */
/* IGNORE_CWISE (input) LOGICAL */
/* If .TRUE. then ignore componentwise convergence. Default value */
/* is .FALSE.. */
/* INFO (output) INTEGER */
/* = 0: Successful exit. */
/* < 0: if INFO = -i, the ith argument to SGBTRS had an illegal */
/* value */
/* ===================================================================== */
/* .. Local Scalars .. */
/* .. */
/* .. Parameters .. */
/* .. */
/* .. 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;
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
afb_dim1 = *ldafb;
afb_offset = 1 + afb_dim1;
afb -= afb_offset;
--ipiv;
--c__;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
y_dim1 = *ldy;
y_offset = 1 + y_dim1;
y -= y_offset;
--berr_out__;
--res;
--ayb;
--dy;
--y_tail__;
/* Function Body */
if (*info != 0) {
return 0;
}
chla_transtype__(ch__1, (ftnlen)1, trans_type__);
*(unsigned char *)trans = *(unsigned char *)&ch__1[0];
eps = slamch_("Epsilon");
hugeval = slamch_("Overflow");
/* Force HUGEVAL to Inf */
hugeval *= hugeval;
/* Using HUGEVAL may lead to spurious underflows. */
incr_thresh__ = (real) (*n) * eps;
m = *kl + *ku + 1;
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
y_prec_state__ = 1;
if (y_prec_state__ == 2) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
y_tail__[i__] = 0.f;
}
}
dxrat = 0.f;
dxratmax = 0.f;
dzrat = 0.f;
dzratmax = 0.f;
final_dx_x__ = hugeval;
final_dz_z__ = hugeval;
prevnormdx = hugeval;
prev_dz_z__ = hugeval;
dz_z__ = hugeval;
dx_x__ = hugeval;
x_state__ = 1;
z_state__ = 0;
incr_prec__ = FALSE_;
i__2 = *ithresh;
for (cnt = 1; cnt <= i__2; ++cnt) {
/* Compute residual RES = B_s - op(A_s) * Y, */
/* op(A) = A, A**T, or A**H depending on TRANS (and type). */
scopy_(n, &b[j * b_dim1 + 1], &c__1, &res[1], &c__1);
if (y_prec_state__ == 0) {
sgbmv_(trans, &m, n, kl, ku, &c_b6, &ab[ab_offset], ldab, &y[
j * y_dim1 + 1], &c__1, &c_b8, &res[1], &c__1);
} else if (y_prec_state__ == 1) {
blas_sgbmv_x__(trans_type__, n, n, kl, ku, &c_b6, &ab[
ab_offset], ldab, &y[j * y_dim1 + 1], &c__1, &c_b8, &
res[1], &c__1, prec_type__);
} else {
blas_sgbmv2_x__(trans_type__, n, n, kl, ku, &c_b6, &ab[
ab_offset], ldab, &y[j * y_dim1 + 1], &y_tail__[1], &
c__1, &c_b8, &res[1], &c__1, prec_type__);
}
/* XXX: RES is no longer needed. */
scopy_(n, &res[1], &c__1, &dy[1], &c__1);
sgbtrs_(trans, n, kl, ku, &c__1, &afb[afb_offset], ldafb, &ipiv[1]
, &dy[1], n, info);
/* Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT. */
normx = 0.f;
normy = 0.f;
normdx = 0.f;
dz_z__ = 0.f;
ymin = hugeval;
i__3 = *n;
for (i__ = 1; i__ <= i__3; ++i__) {
yk = (r__1 = y[i__ + j * y_dim1], dabs(r__1));
dyk = (r__1 = dy[i__], dabs(r__1));
if (yk != 0.f) {
/* Computing MAX */
r__1 = dz_z__, r__2 = dyk / yk;
dz_z__ = dmax(r__1,r__2);
} else if (dyk != 0.f) {
dz_z__ = hugeval;
}
ymin = dmin(ymin,yk);
normy = dmax(normy,yk);
if (*colequ) {
/* Computing MAX */
r__1 = normx, r__2 = yk * c__[i__];
normx = dmax(r__1,r__2);
/* Computing MAX */
r__1 = normdx, r__2 = dyk * c__[i__];
normdx = dmax(r__1,r__2);
} else {
normx = normy;
normdx = dmax(normdx,dyk);
}
}
if (normx != 0.f) {
dx_x__ = normdx / normx;
} else if (normdx == 0.f) {
dx_x__ = 0.f;
} else {
dx_x__ = hugeval;
}
dxrat = normdx / prevnormdx;
dzrat = dz_z__ / prev_dz_z__;
/* Check termination criteria. */
if (! (*ignore_cwise__) && ymin * *rcond < incr_thresh__ * normy
&& y_prec_state__ < 2) {
incr_prec__ = TRUE_;
}
if (x_state__ == 3 && dxrat <= *rthresh) {
x_state__ = 1;
}
if (x_state__ == 1) {
if (dx_x__ <= eps) {
x_state__ = 2;
} else if (dxrat > *rthresh) {
if (y_prec_state__ != 2) {
incr_prec__ = TRUE_;
} else {
x_state__ = 3;
}
} else {
if (dxrat > dxratmax) {
dxratmax = dxrat;
}
}
if (x_state__ > 1) {
final_dx_x__ = dx_x__;
}
}
if (z_state__ == 0 && dz_z__ <= *dz_ub__) {
z_state__ = 1;
}
if (z_state__ == 3 && dzrat <= *rthresh) {
z_state__ = 1;
}
if (z_state__ == 1) {
if (dz_z__ <= eps) {
z_state__ = 2;
} else if (dz_z__ > *dz_ub__) {
z_state__ = 0;
dzratmax = 0.f;
final_dz_z__ = hugeval;
} else if (dzrat > *rthresh) {
if (y_prec_state__ != 2) {
incr_prec__ = TRUE_;
} else {
z_state__ = 3;
}
} else {
if (dzrat > dzratmax) {
dzratmax = dzrat;
}
}
if (z_state__ > 1) {
final_dz_z__ = dz_z__;
}
}
/* Exit if both normwise and componentwise stopped working, */
/* but if componentwise is unstable, let it go at least two */
/* iterations. */
if (x_state__ != 1) {
if (*ignore_cwise__) {
goto L666;
}
if (z_state__ == 3 || z_state__ == 2) {
goto L666;
}
if (z_state__ == 0 && cnt > 1) {
goto L666;
}
}
if (incr_prec__) {
incr_prec__ = FALSE_;
++y_prec_state__;
i__3 = *n;
for (i__ = 1; i__ <= i__3; ++i__) {
y_tail__[i__] = 0.f;
}
}
prevnormdx = normdx;
prev_dz_z__ = dz_z__;
/* Update soluton. */
if (y_prec_state__ < 2) {
saxpy_(n, &c_b8, &dy[1], &c__1, &y[j * y_dim1 + 1], &c__1);
} else {
sla_wwaddw__(n, &y[j * y_dim1 + 1], &y_tail__[1], &dy[1]);
}
}
/* Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT. */
L666:
/* Set final_* when cnt hits ithresh. */
if (x_state__ == 1) {
final_dx_x__ = dx_x__;
}
if (z_state__ == 1) {
final_dz_z__ = dz_z__;
}
/* Compute error bounds. */
if (*n_norms__ >= 1) {
err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = final_dx_x__ / (
1 - dxratmax);
}
if (*n_norms__ >= 2) {
err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = final_dz_z__ / (
1 - dzratmax);
}
/* Compute componentwise relative backward error from formula */
/* max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) */
/* where abs(Z) is the componentwise absolute value of the matrix */
/* or vector Z. */
/* Compute residual RES = B_s - op(A_s) * Y, */
/* op(A) = A, A**T, or A**H depending on TRANS (and type). */
scopy_(n, &b[j * b_dim1 + 1], &c__1, &res[1], &c__1);
sgbmv_(trans, n, n, kl, ku, &c_b6, &ab[ab_offset], ldab, &y[j *
y_dim1 + 1], &c__1, &c_b8, &res[1], &c__1);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
ayb[i__] = (r__1 = b[i__ + j * b_dim1], dabs(r__1));
}
/* Compute abs(op(A_s))*abs(Y) + abs(B_s). */
sla_gbamv__(trans_type__, n, n, kl, ku, &c_b8, &ab[ab_offset], ldab, &
y[j * y_dim1 + 1], &c__1, &c_b8, &ayb[1], &c__1);
sla_lin_berr__(n, n, &c__1, &res[1], &ayb[1], &berr_out__[j]);
/* End of loop for each RHS */
}
return 0;
} /* sla_gbrfsx_extended__ */
/* Subroutine */ int sgbrfs_(char *trans, integer *n, integer *kl, integer *
ku, integer *nrhs, real *ab, integer *ldab, real *afb, integer *ldafb,
integer *ipiv, real *b, integer *ldb, real *x, integer *ldx, real *
ferr, real *berr, real *work, integer *iwork, integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
Purpose
=======
SGBRFS improves the computed solution to a system of linear
equations when the coefficient matrix is banded, and provides
error bounds and backward error estimates for the solution.
Arguments
=========
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 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) REAL 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) REAL array, dimension (LDAFB,N)
Details of the LU factorization of the band matrix A, as
computed by SGBTRF. 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 SGBTRF; for 1<=i<=N, row i of the
matrix was interchanged with row IPIV(i).
B (input) REAL 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) REAL array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by SGBTRS.
On exit, the improved solution matrix X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
FERR (output) REAL array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
BERR (output) REAL array, dimension (NRHS)
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).
WORK (workspace) REAL array, dimension (3*N)
IWORK (workspace) INTEGER array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Internal Parameters
===================
ITMAX is the maximum number of steps of iterative refinement.
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static real c_b15 = -1.f;
static real c_b17 = 1.f;
/* System generated locals */
integer ab_dim1, ab_offset, afb_dim1, afb_offset, b_dim1, b_offset,
x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5, i__6, i__7;
real r__1, r__2, r__3;
/* Local variables */
static integer kase;
static real safe1, safe2;
static integer i__, j, k;
static real s;
extern logical lsame_(char *, char *);
extern /* Subroutine */ int sgbmv_(char *, integer *, integer *, integer *
, integer *, real *, real *, integer *, real *, integer *, real *,
real *, integer *);
static integer count;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), saxpy_(integer *, real *, real *, integer *, real *,
integer *);
static integer kk;
static real xk;
extern doublereal slamch_(char *);
static integer nz;
static real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *), slacon_(
integer *, real *, real *, integer *, real *, integer *);
static logical notran;
extern /* Subroutine */ int sgbtrs_(char *, integer *, integer *, integer
*, integer *, real *, integer *, integer *, real *, integer *,
integer *);
static char transt[1];
static real lstres, eps;
#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
#define x_ref(a_1,a_2) x[(a_2)*x_dim1 + a_1]
#define ab_ref(a_1,a_2) ab[(a_2)*ab_dim1 + a_1]
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1 * 1;
ab -= ab_offset;
afb_dim1 = *ldafb;
afb_offset = 1 + afb_dim1 * 1;
afb -= afb_offset;
--ipiv;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1 * 1;
x -= x_offset;
--ferr;
--berr;
--work;
--iwork;
/* Function Body */
*info = 0;
notran = lsame_(trans, "N");
if (! notran && ! lsame_(trans, "T") && ! lsame_(
trans, "C")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*kl < 0) {
*info = -3;
} else if (*ku < 0) {
*info = -4;
} else if (*nrhs < 0) {
*info = -5;
} else if (*ldab < *kl + *ku + 1) {
*info = -7;
} else if (*ldafb < (*kl << 1) + *ku + 1) {
*info = -9;
} else if (*ldb < max(1,*n)) {
*info = -12;
} else if (*ldx < max(1,*n)) {
*info = -14;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGBRFS", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0 || *nrhs == 0) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
ferr[j] = 0.f;
berr[j] = 0.f;
/* L10: */
}
return 0;
}
if (notran) {
*(unsigned char *)transt = 'T';
} else {
*(unsigned char *)transt = 'N';
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1
Computing MIN */
i__1 = *kl + *ku + 2, i__2 = *n + 1;
nz = min(i__1,i__2);
eps = slamch_("Epsilon");
safmin = slamch_("Safe minimum");
safe1 = nz * safmin;
safe2 = safe1 / eps;
/* Do for each right hand side */
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
count = 1;
lstres = 3.f;
L20:
/* Loop until stopping criterion is satisfied.
Compute residual R = B - op(A) * X,
where op(A) = A, A**T, or A**H, depending on TRANS. */
scopy_(n, &b_ref(1, j), &c__1, &work[*n + 1], &c__1);
sgbmv_(trans, n, n, kl, ku, &c_b15, &ab[ab_offset], ldab, &x_ref(1, j)
, &c__1, &c_b17, &work[*n + 1], &c__1);
/* Compute componentwise relative backward error from formula
max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z. If the i-th component of the denominator is less
than SAFE2, then SAFE1 is added to the i-th components of the
numerator and denominator before dividing. */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[i__] = (r__1 = b_ref(i__, j), dabs(r__1));
/* L30: */
}
/* Compute abs(op(A))*abs(X) + abs(B). */
if (notran) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
kk = *ku + 1 - k;
xk = (r__1 = x_ref(k, j), dabs(r__1));
/* Computing MAX */
i__3 = 1, i__4 = k - *ku;
/* Computing MIN */
i__6 = *n, i__7 = k + *kl;
i__5 = min(i__6,i__7);
for (i__ = max(i__3,i__4); i__ <= i__5; ++i__) {
work[i__] += (r__1 = ab_ref(kk + i__, k), dabs(r__1)) *
xk;
/* L40: */
}
/* L50: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.f;
kk = *ku + 1 - k;
/* Computing MAX */
i__5 = 1, i__3 = k - *ku;
/* Computing MIN */
i__6 = *n, i__7 = k + *kl;
i__4 = min(i__6,i__7);
for (i__ = max(i__5,i__3); i__ <= i__4; ++i__) {
s += (r__1 = ab_ref(kk + i__, k), dabs(r__1)) * (r__2 =
x_ref(i__, j), dabs(r__2));
/* L60: */
}
work[k] += s;
/* L70: */
}
}
s = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (work[i__] > safe2) {
/* Computing MAX */
r__2 = s, r__3 = (r__1 = work[*n + i__], dabs(r__1)) / work[
i__];
s = dmax(r__2,r__3);
} else {
/* Computing MAX */
r__2 = s, r__3 = ((r__1 = work[*n + i__], dabs(r__1)) + safe1)
/ (work[i__] + safe1);
s = dmax(r__2,r__3);
}
/* L80: */
}
berr[j] = s;
/* Test stopping criterion. Continue iterating if
1) The residual BERR(J) is larger than machine epsilon, and
2) BERR(J) decreased by at least a factor of 2 during the
last iteration, and
3) At most ITMAX iterations tried. */
if (berr[j] > eps && berr[j] * 2.f <= lstres && count <= 5) {
/* Update solution and try again. */
sgbtrs_(trans, n, kl, ku, &c__1, &afb[afb_offset], ldafb, &ipiv[1]
, &work[*n + 1], n, info);
saxpy_(n, &c_b17, &work[*n + 1], &c__1, &x_ref(1, j), &c__1);
lstres = berr[j];
++count;
goto L20;
}
/* Bound error from formula
norm(X - XTRUE) / norm(X) .le. FERR =
norm( abs(inv(op(A)))*
( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
where
norm(Z) is the magnitude of the largest component of Z
inv(op(A)) is the inverse of op(A)
abs(Z) is the componentwise absolute value of the matrix or
vector Z
NZ is the maximum number of nonzeros in any row of A, plus 1
EPS is machine epsilon
The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
is incremented by SAFE1 if the i-th component of
abs(op(A))*abs(X) + abs(B) is less than SAFE2.
Use SLACON to estimate the infinity-norm of the matrix
inv(op(A)) * diag(W),
where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (work[i__] > safe2) {
work[i__] = (r__1 = work[*n + i__], dabs(r__1)) + nz * eps *
work[i__];
} else {
work[i__] = (r__1 = work[*n + i__], dabs(r__1)) + nz * eps *
work[i__] + safe1;
}
/* L90: */
}
kase = 0;
L100:
slacon_(n, &work[(*n << 1) + 1], &work[*n + 1], &iwork[1], &ferr[j], &
kase);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(op(A)**T). */
sgbtrs_(transt, n, kl, ku, &c__1, &afb[afb_offset], ldafb, &
ipiv[1], &work[*n + 1], n, info);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[*n + i__] *= work[i__];
/* L110: */
}
} else {
/* Multiply by inv(op(A))*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[*n + i__] *= work[i__];
/* L120: */
}
sgbtrs_(trans, n, kl, ku, &c__1, &afb[afb_offset], ldafb, &
ipiv[1], &work[*n + 1], n, info);
}
goto L100;
}
/* Normalize error. */
lstres = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
r__2 = lstres, r__3 = (r__1 = x_ref(i__, j), dabs(r__1));
lstres = dmax(r__2,r__3);
/* L130: */
}
if (lstres != 0.f) {
ferr[j] /= lstres;
}
/* L140: */
}
return 0;
/* End of SGBRFS */
} /* sgbrfs_ */
