/* ctbrfs.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;
/* Subroutine */ int ctbrfs_(char *uplo, char *trans, char *diag, integer *n,
integer *kd, integer *nrhs, complex *ab, integer *ldab, complex *b,
integer *ldb, complex *x, integer *ldx, real *ferr, real *berr,
complex *work, real *rwork, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, b_dim1, b_offset, x_dim1, x_offset, i__1,
i__2, i__3, i__4, i__5;
real r__1, r__2, r__3, r__4;
complex q__1;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
integer i__, j, k;
real s, xk;
integer nz;
real eps;
integer kase;
real safe1, safe2;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int ctbmv_(char *, char *, char *, integer *,
integer *, complex *, integer *, complex *, integer *), ccopy_(integer *, complex *, integer *, complex *
, integer *), ctbsv_(char *, char *, char *, integer *, integer *,
complex *, integer *, complex *, integer *), caxpy_(integer *, complex *, complex *, integer *,
complex *, integer *);
logical upper;
extern /* Subroutine */ int clacn2_(integer *, complex *, complex *, real
*, integer *, integer *);
extern doublereal slamch_(char *);
real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
logical notran;
char transn[1], transt[1];
logical nounit;
real lstres;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CTBRFS provides error bounds and backward error estimates for the */
/* solution to a system of linear equations with a triangular band */
/* coefficient matrix. */
/* The solution matrix X must be computed by CTBTRS or some other */
/* means before entering this routine. CTBRFS does not do iterative */
/* refinement because doing so cannot improve the backward error. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': A is upper triangular; */
/* = 'L': A is lower triangular. */
/* 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) */
/* DIAG (input) CHARACTER*1 */
/* = 'N': A is non-unit triangular; */
/* = 'U': A is unit triangular. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* KD (input) INTEGER */
/* The number of superdiagonals or subdiagonals of the */
/* triangular band matrix A. 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) COMPLEX array, dimension (LDAB,N) */
/* The upper or lower triangular 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). */
/* If DIAG = 'U', the diagonal elements of A are not referenced */
/* and are assumed to be 1. */
/* LDAB (input) INTEGER */
/* The leading dimension of the array AB. LDAB >= KD+1. */
/* B (input) COMPLEX 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) COMPLEX array, dimension (LDX,NRHS) */
/* The 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) COMPLEX array, dimension (2*N) */
/* RWORK (workspace) REAL array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_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;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
notran = lsame_(trans, "N");
nounit = lsame_(diag, "N");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (! notran && ! lsame_(trans, "T") && !
lsame_(trans, "C")) {
*info = -2;
} else if (! nounit && ! lsame_(diag, "U")) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*kd < 0) {
*info = -5;
} else if (*nrhs < 0) {
*info = -6;
} else if (*ldab < *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_("CTBRFS", &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 *)transn = 'N';
*(unsigned char *)transt = 'C';
} else {
*(unsigned char *)transn = 'C';
*(unsigned char *)transt = 'N';
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = *kd + 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) {
/* Compute residual R = B - op(A) * X, */
/* where op(A) = A, A**T, or A**H, depending on TRANS. */
ccopy_(n, &x[j * x_dim1 + 1], &c__1, &work[1], &c__1);
ctbmv_(uplo, trans, diag, n, kd, &ab[ab_offset], ldab, &work[1], &
c__1);
q__1.r = -1.f, q__1.i = -0.f;
caxpy_(n, &q__1, &b[j * b_dim1 + 1], &c__1, &work[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__) {
i__3 = i__ + j * b_dim1;
rwork[i__] = (r__1 = b[i__3].r, dabs(r__1)) + (r__2 = r_imag(&b[
i__ + j * b_dim1]), dabs(r__2));
/* L20: */
}
if (notran) {
/* Compute abs(A)*abs(X) + abs(B). */
if (upper) {
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = k + j * x_dim1;
xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
x[k + j * x_dim1]), dabs(r__2));
/* Computing MAX */
i__3 = 1, i__4 = k - *kd;
i__5 = k;
for (i__ = max(i__3,i__4); i__ <= i__5; ++i__) {
i__3 = *kd + 1 + i__ - k + k * ab_dim1;
rwork[i__] += ((r__1 = ab[i__3].r, dabs(r__1)) + (
r__2 = r_imag(&ab[*kd + 1 + i__ - k + k *
ab_dim1]), dabs(r__2))) * xk;
/* L30: */
}
/* L40: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__5 = k + j * x_dim1;
xk = (r__1 = x[i__5].r, dabs(r__1)) + (r__2 = r_imag(&
x[k + j * x_dim1]), dabs(r__2));
/* Computing MAX */
i__5 = 1, i__3 = k - *kd;
i__4 = k - 1;
for (i__ = max(i__5,i__3); i__ <= i__4; ++i__) {
i__5 = *kd + 1 + i__ - k + k * ab_dim1;
rwork[i__] += ((r__1 = ab[i__5].r, dabs(r__1)) + (
r__2 = r_imag(&ab[*kd + 1 + i__ - k + k *
ab_dim1]), dabs(r__2))) * xk;
/* L50: */
}
rwork[k] += xk;
/* L60: */
}
}
} else {
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__4 = k + j * x_dim1;
xk = (r__1 = x[i__4].r, dabs(r__1)) + (r__2 = r_imag(&
x[k + j * x_dim1]), dabs(r__2));
/* Computing MIN */
i__5 = *n, i__3 = k + *kd;
i__4 = min(i__5,i__3);
for (i__ = k; i__ <= i__4; ++i__) {
i__5 = i__ + 1 - k + k * ab_dim1;
rwork[i__] += ((r__1 = ab[i__5].r, dabs(r__1)) + (
r__2 = r_imag(&ab[i__ + 1 - k + k *
ab_dim1]), dabs(r__2))) * xk;
/* L70: */
}
/* L80: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__4 = k + j * x_dim1;
xk = (r__1 = x[i__4].r, dabs(r__1)) + (r__2 = r_imag(&
x[k + j * x_dim1]), dabs(r__2));
/* Computing MIN */
i__5 = *n, i__3 = k + *kd;
i__4 = min(i__5,i__3);
for (i__ = k + 1; i__ <= i__4; ++i__) {
i__5 = i__ + 1 - k + k * ab_dim1;
rwork[i__] += ((r__1 = ab[i__5].r, dabs(r__1)) + (
r__2 = r_imag(&ab[i__ + 1 - k + k *
ab_dim1]), dabs(r__2))) * xk;
/* L90: */
}
rwork[k] += xk;
/* L100: */
}
}
}
} else {
/* Compute abs(A**H)*abs(X) + abs(B). */
if (upper) {
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.f;
/* Computing MAX */
i__4 = 1, i__5 = k - *kd;
i__3 = k;
for (i__ = max(i__4,i__5); i__ <= i__3; ++i__) {
i__4 = *kd + 1 + i__ - k + k * ab_dim1;
i__5 = i__ + j * x_dim1;
s += ((r__1 = ab[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&ab[*kd + 1 + i__ - k + k *
ab_dim1]), dabs(r__2))) * ((r__3 = x[i__5]
.r, dabs(r__3)) + (r__4 = r_imag(&x[i__ +
j * x_dim1]), dabs(r__4)));
/* L110: */
}
rwork[k] += s;
/* L120: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = k + j * x_dim1;
s = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
x[k + j * x_dim1]), dabs(r__2));
/* Computing MAX */
i__3 = 1, i__4 = k - *kd;
i__5 = k - 1;
for (i__ = max(i__3,i__4); i__ <= i__5; ++i__) {
i__3 = *kd + 1 + i__ - k + k * ab_dim1;
i__4 = i__ + j * x_dim1;
s += ((r__1 = ab[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&ab[*kd + 1 + i__ - k + k *
ab_dim1]), dabs(r__2))) * ((r__3 = x[i__4]
.r, dabs(r__3)) + (r__4 = r_imag(&x[i__ +
j * x_dim1]), dabs(r__4)));
/* L130: */
}
rwork[k] += s;
/* L140: */
}
}
} else {
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.f;
/* Computing MIN */
i__3 = *n, i__4 = k + *kd;
i__5 = min(i__3,i__4);
for (i__ = k; i__ <= i__5; ++i__) {
i__3 = i__ + 1 - k + k * ab_dim1;
i__4 = i__ + j * x_dim1;
s += ((r__1 = ab[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&ab[i__ + 1 - k + k * ab_dim1]),
dabs(r__2))) * ((r__3 = x[i__4].r, dabs(
r__3)) + (r__4 = r_imag(&x[i__ + j *
x_dim1]), dabs(r__4)));
/* L150: */
}
rwork[k] += s;
/* L160: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__5 = k + j * x_dim1;
s = (r__1 = x[i__5].r, dabs(r__1)) + (r__2 = r_imag(&
x[k + j * x_dim1]), dabs(r__2));
/* Computing MIN */
i__3 = *n, i__4 = k + *kd;
i__5 = min(i__3,i__4);
for (i__ = k + 1; i__ <= i__5; ++i__) {
i__3 = i__ + 1 - k + k * ab_dim1;
i__4 = i__ + j * x_dim1;
s += ((r__1 = ab[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&ab[i__ + 1 - k + k * ab_dim1]),
dabs(r__2))) * ((r__3 = x[i__4].r, dabs(
r__3)) + (r__4 = r_imag(&x[i__ + j *
x_dim1]), dabs(r__4)));
/* L170: */
}
rwork[k] += s;
/* L180: */
}
}
}
}
s = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
/* Computing MAX */
i__5 = i__;
r__3 = s, r__4 = ((r__1 = work[i__5].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2))) / rwork[i__];
s = dmax(r__3,r__4);
} else {
/* Computing MAX */
i__5 = i__;
r__3 = s, r__4 = ((r__1 = work[i__5].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + safe1) / (rwork[i__]
+ safe1);
s = dmax(r__3,r__4);
}
/* L190: */
}
berr[j] = s;
/* 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 CLACN2 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 (rwork[i__] > safe2) {
i__5 = i__;
rwork[i__] = (r__1 = work[i__5].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[
i__];
} else {
i__5 = i__;
rwork[i__] = (r__1 = work[i__5].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[
i__] + safe1;
}
/* L200: */
}
kase = 0;
L210:
clacn2_(n, &work[*n + 1], &work[1], &ferr[j], &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(op(A)**H). */
ctbsv_(uplo, transt, diag, n, kd, &ab[ab_offset], ldab, &work[
1], &c__1);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__5 = i__;
i__3 = i__;
i__4 = i__;
q__1.r = rwork[i__3] * work[i__4].r, q__1.i = rwork[i__3]
* work[i__4].i;
work[i__5].r = q__1.r, work[i__5].i = q__1.i;
/* L220: */
}
} else {
/* Multiply by inv(op(A))*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__5 = i__;
i__3 = i__;
i__4 = i__;
q__1.r = rwork[i__3] * work[i__4].r, q__1.i = rwork[i__3]
* work[i__4].i;
work[i__5].r = q__1.r, work[i__5].i = q__1.i;
/* L230: */
}
ctbsv_(uplo, transn, diag, n, kd, &ab[ab_offset], ldab, &work[
1], &c__1);
}
goto L210;
}
/* Normalize error. */
lstres = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
i__5 = i__ + j * x_dim1;
r__3 = lstres, r__4 = (r__1 = x[i__5].r, dabs(r__1)) + (r__2 =
r_imag(&x[i__ + j * x_dim1]), dabs(r__2));
lstres = dmax(r__3,r__4);
/* L240: */
}
if (lstres != 0.f) {
ferr[j] /= lstres;
}
/* L250: */
}
return 0;
/* End of CTBRFS */
} /* ctbrfs_ */