/* dpbrfs.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"
/* Table of constant values */
static integer c__1 = 1;
static doublereal c_b12 = -1.;
static doublereal c_b14 = 1.;
/* Subroutine */ int dpbrfs_(char *uplo, integer *n, integer *kd, integer *
nrhs, doublereal *ab, integer *ldab, doublereal *afb, integer *ldafb,
doublereal *b, integer *ldb, doublereal *x, integer *ldx, doublereal *
ferr, doublereal *berr, 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, i__1, i__2, i__3, i__4, i__5;
doublereal d__1, d__2, d__3;
/* Local variables */
integer i__, j, k, l;
doublereal s, xk;
integer nz;
doublereal eps;
integer kase;
doublereal safe1, safe2;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int dsbmv_(char *, integer *, integer *,
doublereal *, doublereal *, integer *, doublereal *, integer *,
doublereal *, doublereal *, integer *), dcopy_(integer *,
doublereal *, integer *, doublereal *, integer *), daxpy_(integer
*, doublereal *, doublereal *, integer *, doublereal *, integer *)
;
integer count;
logical upper;
extern /* Subroutine */ int dlacn2_(integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
doublereal safmin;
extern /* Subroutine */ int xerbla_(char *, integer *), dpbtrs_(
char *, integer *, integer *, integer *, doublereal *, integer *,
doublereal *, integer *, integer *);
doublereal lstres;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DPBRFS improves the computed solution to a system of linear */
/* equations when the coefficient matrix is symmetric positive definite */
/* and banded, and provides error bounds and backward error estimates */
/* for the solution. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* KD (input) INTEGER */
/* The number of superdiagonals of the matrix A if UPLO = 'U', */
/* or the number of subdiagonals if UPLO = 'L'. KD >= 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 upper or lower triangle of the symmetric band matrix A, */
/* stored in the first KD+1 rows of the array. The j-th column */
/* of A is stored in the j-th column of the array AB as follows: */
/* if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; */
/* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). */
/* LDAB (input) INTEGER */
/* The leading dimension of the array AB. LDAB >= KD+1. */
/* AFB (input) DOUBLE PRECISION array, dimension (LDAFB,N) */
/* The triangular factor U or L from the Cholesky factorization */
/* A = U**T*U or A = L*L**T of the band matrix A as computed by */
/* DPBTRF, in the same storage format as A (see AB). */
/* LDAFB (input) INTEGER */
/* The leading dimension of the array AFB. LDAFB >= KD+1. */
/* 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 DPBTRS. */
/* On exit, the improved solution matrix X. */
/* LDX (input) INTEGER */
/* The leading dimension of the array X. LDX >= max(1,N). */
/* FERR (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
afb_dim1 = *ldafb;
afb_offset = 1 + afb_dim1;
afb -= afb_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
--ferr;
--berr;
--work;
--iwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*kd < 0) {
*info = -3;
} else if (*nrhs < 0) {
*info = -4;
} else if (*ldab < *kd + 1) {
*info = -6;
} else if (*ldafb < *kd + 1) {
*info = -8;
} else if (*ldb < max(1,*n)) {
*info = -10;
} else if (*ldx < max(1,*n)) {
*info = -12;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DPBRFS", &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.;
berr[j] = 0.;
/* L10: */
}
return 0;
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
/* Computing MIN */
i__1 = *n + 1, i__2 = (*kd << 1) + 2;
nz = min(i__1,i__2);
eps = dlamch_("Epsilon");
safmin = dlamch_("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.;
L20:
/* Loop until stopping criterion is satisfied. */
/* Compute residual R = B - A * X */
dcopy_(n, &b[j * b_dim1 + 1], &c__1, &work[*n + 1], &c__1);
dsbmv_(uplo, n, kd, &c_b12, &ab[ab_offset], ldab, &x[j * x_dim1 + 1],
&c__1, &c_b14, &work[*n + 1], &c__1);
/* Compute componentwise relative backward error from formula */
/* max(i) ( abs(R(i)) / ( abs(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__] = (d__1 = b[i__ + j * b_dim1], abs(d__1));
/* L30: */
}
/* Compute abs(A)*abs(X) + abs(B). */
if (upper) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.;
xk = (d__1 = x[k + j * x_dim1], abs(d__1));
l = *kd + 1 - k;
/* Computing MAX */
i__3 = 1, i__4 = k - *kd;
i__5 = k - 1;
for (i__ = max(i__3,i__4); i__ <= i__5; ++i__) {
work[i__] += (d__1 = ab[l + i__ + k * ab_dim1], abs(d__1))
* xk;
s += (d__1 = ab[l + i__ + k * ab_dim1], abs(d__1)) * (
d__2 = x[i__ + j * x_dim1], abs(d__2));
/* L40: */
}
work[k] = work[k] + (d__1 = ab[*kd + 1 + k * ab_dim1], abs(
d__1)) * xk + s;
/* L50: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.;
xk = (d__1 = x[k + j * x_dim1], abs(d__1));
work[k] += (d__1 = ab[k * ab_dim1 + 1], abs(d__1)) * xk;
l = 1 - k;
/* Computing MIN */
i__3 = *n, i__4 = k + *kd;
i__5 = min(i__3,i__4);
for (i__ = k + 1; i__ <= i__5; ++i__) {
work[i__] += (d__1 = ab[l + i__ + k * ab_dim1], abs(d__1))
* xk;
s += (d__1 = ab[l + i__ + k * ab_dim1], abs(d__1)) * (
d__2 = x[i__ + j * x_dim1], abs(d__2));
/* L60: */
}
work[k] += s;
/* L70: */
}
}
s = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (work[i__] > safe2) {
/* Computing MAX */
d__2 = s, d__3 = (d__1 = work[*n + i__], abs(d__1)) / work[
i__];
s = max(d__2,d__3);
} else {
/* Computing MAX */
d__2 = s, d__3 = ((d__1 = work[*n + i__], abs(d__1)) + safe1)
/ (work[i__] + safe1);
s = max(d__2,d__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. <= lstres && count <= 5) {
/* Update solution and try again. */
dpbtrs_(uplo, n, kd, &c__1, &afb[afb_offset], ldafb, &work[*n + 1]
, n, info);
daxpy_(n, &c_b14, &work[*n + 1], &c__1, &x[j * x_dim1 + 1], &c__1)
;
lstres = berr[j];
++count;
goto L20;
}
/* Bound error from formula */
/* norm(X - XTRUE) / norm(X) .le. FERR = */
/* norm( abs(inv(A))* */
/* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) */
/* where */
/* norm(Z) is the magnitude of the largest component of Z */
/* inv(A) is the inverse of 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(A)*abs(X)+abs(B)) */
/* is incremented by SAFE1 if the i-th component of */
/* abs(A)*abs(X) + abs(B) is less than SAFE2. */
/* Use DLACN2 to estimate the infinity-norm of the matrix */
/* inv(A) * diag(W), */
/* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (work[i__] > safe2) {
work[i__] = (d__1 = work[*n + i__], abs(d__1)) + nz * eps *
work[i__];
} else {
work[i__] = (d__1 = work[*n + i__], abs(d__1)) + nz * eps *
work[i__] + safe1;
}
/* L90: */
}
kase = 0;
L100:
dlacn2_(n, &work[(*n << 1) + 1], &work[*n + 1], &iwork[1], &ferr[j], &
kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(A'). */
dpbtrs_(uplo, n, kd, &c__1, &afb[afb_offset], ldafb, &work[*n
+ 1], n, info);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[*n + i__] *= work[i__];
/* L110: */
}
} else if (kase == 2) {
/* Multiply by inv(A)*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[*n + i__] *= work[i__];
/* L120: */
}
dpbtrs_(uplo, n, kd, &c__1, &afb[afb_offset], ldafb, &work[*n
+ 1], n, info);
}
goto L100;
}
/* Normalize error. */
lstres = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
d__2 = lstres, d__3 = (d__1 = x[i__ + j * x_dim1], abs(d__1));
lstres = max(d__2,d__3);
/* L130: */
}
if (lstres != 0.) {
ferr[j] /= lstres;
}
/* L140: */
}
return 0;
/* End of DPBRFS */
} /* dpbrfs_ */
