/* dgtrfs.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_b18 = -1.;
static doublereal c_b19 = 1.;
/* Subroutine */ int dgtrfs_(char *trans, integer *n, integer *nrhs,
doublereal *dl, doublereal *d__, doublereal *du, doublereal *dlf,
doublereal *df, doublereal *duf, doublereal *du2, integer *ipiv,
doublereal *b, integer *ldb, doublereal *x, integer *ldx, doublereal *
ferr, doublereal *berr, doublereal *work, integer *iwork, integer *
info)
{
/* System generated locals */
integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2;
doublereal d__1, d__2, d__3, d__4;
/* Local variables */
integer i__, j;
doublereal s;
integer nz;
doublereal eps;
integer kase;
doublereal safe1, safe2;
extern logical lsame_(char *, char *);
integer isave[3];
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *), daxpy_(integer *, doublereal *,
doublereal *, integer *, doublereal *, integer *);
integer count;
extern /* Subroutine */ int dlacn2_(integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *);
extern doublereal dlamch_(char *);
extern /* Subroutine */ int dlagtm_(char *, integer *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, integer *, doublereal *, doublereal *, integer *);
doublereal safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
logical notran;
char transn[1];
extern /* Subroutine */ int dgttrs_(char *, integer *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *, integer *,
doublereal *, integer *, integer *);
char transt[1];
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 */
/* ======= */
/* DGTRFS improves the computed solution to a system of linear */
/* equations when the coefficient matrix is tridiagonal, 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. */
/* NRHS (input) INTEGER */
/* The number of right hand sides, i.e., the number of columns */
/* of the matrix B. NRHS >= 0. */
/* DL (input) DOUBLE PRECISION array, dimension (N-1) */
/* The (n-1) subdiagonal elements of A. */
/* D (input) DOUBLE PRECISION array, dimension (N) */
/* The diagonal elements of A. */
/* DU (input) DOUBLE PRECISION array, dimension (N-1) */
/* The (n-1) superdiagonal elements of A. */
/* DLF (input) DOUBLE PRECISION array, dimension (N-1) */
/* The (n-1) multipliers that define the matrix L from the */
/* LU factorization of A as computed by DGTTRF. */
/* DF (input) DOUBLE PRECISION array, dimension (N) */
/* The n diagonal elements of the upper triangular matrix U from */
/* the LU factorization of A. */
/* DUF (input) DOUBLE PRECISION array, dimension (N-1) */
/* The (n-1) elements of the first superdiagonal of U. */
/* DU2 (input) DOUBLE PRECISION array, dimension (N-2) */
/* The (n-2) elements of the second superdiagonal of U. */
/* IPIV (input) INTEGER array, dimension (N) */
/* The pivot indices; for 1 <= i <= n, row i of the matrix was */
/* interchanged with row IPIV(i). IPIV(i) will always be either */
/* i or i+1; IPIV(i) = i indicates a row interchange was not */
/* required. */
/* 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 DGTTRS. */
/* 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 */
--dl;
--d__;
--du;
--dlf;
--df;
--duf;
--du2;
--ipiv;
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;
notran = lsame_(trans, "N");
if (! notran && ! lsame_(trans, "T") && ! lsame_(
trans, "C")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*nrhs < 0) {
*info = -3;
} else if (*ldb < max(1,*n)) {
*info = -13;
} else if (*ldx < max(1,*n)) {
*info = -15;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DGTRFS", &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;
}
if (notran) {
*(unsigned char *)transn = 'N';
*(unsigned char *)transt = 'T';
} else {
*(unsigned char *)transn = 'T';
*(unsigned char *)transt = 'N';
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = 4;
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 - op(A) * X, */
/* where op(A) = A, A**T, or A**H, depending on TRANS. */
dcopy_(n, &b[j * b_dim1 + 1], &c__1, &work[*n + 1], &c__1);
dlagtm_(trans, n, &c__1, &c_b18, &dl[1], &d__[1], &du[1], &x[j *
x_dim1 + 1], ldx, &c_b19, &work[*n + 1], n);
/* Compute abs(op(A))*abs(x) + abs(b) for use in the backward */
/* error bound. */
if (notran) {
if (*n == 1) {
work[1] = (d__1 = b[j * b_dim1 + 1], abs(d__1)) + (d__2 = d__[
1] * x[j * x_dim1 + 1], abs(d__2));
} else {
work[1] = (d__1 = b[j * b_dim1 + 1], abs(d__1)) + (d__2 = d__[
1] * x[j * x_dim1 + 1], abs(d__2)) + (d__3 = du[1] *
x[j * x_dim1 + 2], abs(d__3));
i__2 = *n - 1;
for (i__ = 2; i__ <= i__2; ++i__) {
work[i__] = (d__1 = b[i__ + j * b_dim1], abs(d__1)) + (
d__2 = dl[i__ - 1] * x[i__ - 1 + j * x_dim1], abs(
d__2)) + (d__3 = d__[i__] * x[i__ + j * x_dim1],
abs(d__3)) + (d__4 = du[i__] * x[i__ + 1 + j *
x_dim1], abs(d__4));
/* L30: */
}
work[*n] = (d__1 = b[*n + j * b_dim1], abs(d__1)) + (d__2 =
dl[*n - 1] * x[*n - 1 + j * x_dim1], abs(d__2)) + (
d__3 = d__[*n] * x[*n + j * x_dim1], abs(d__3));
}
} else {
if (*n == 1) {
work[1] = (d__1 = b[j * b_dim1 + 1], abs(d__1)) + (d__2 = d__[
1] * x[j * x_dim1 + 1], abs(d__2));
} else {
work[1] = (d__1 = b[j * b_dim1 + 1], abs(d__1)) + (d__2 = d__[
1] * x[j * x_dim1 + 1], abs(d__2)) + (d__3 = dl[1] *
x[j * x_dim1 + 2], abs(d__3));
i__2 = *n - 1;
for (i__ = 2; i__ <= i__2; ++i__) {
work[i__] = (d__1 = b[i__ + j * b_dim1], abs(d__1)) + (
d__2 = du[i__ - 1] * x[i__ - 1 + j * x_dim1], abs(
d__2)) + (d__3 = d__[i__] * x[i__ + j * x_dim1],
abs(d__3)) + (d__4 = dl[i__] * x[i__ + 1 + j *
x_dim1], abs(d__4));
/* L40: */
}
work[*n] = (d__1 = b[*n + j * b_dim1], abs(d__1)) + (d__2 =
du[*n - 1] * x[*n - 1 + j * x_dim1], abs(d__2)) + (
d__3 = d__[*n] * x[*n + j * x_dim1], abs(d__3));
}
}
/* 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. */
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);
}
/* L50: */
}
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. */
dgttrs_(trans, n, &c__1, &dlf[1], &df[1], &duf[1], &du2[1], &ipiv[
1], &work[*n + 1], n, info);
daxpy_(n, &c_b19, &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(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 DLACN2 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__] = (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;
}
/* L60: */
}
kase = 0;
L70:
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(op(A)**T). */
dgttrs_(transt, n, &c__1, &dlf[1], &df[1], &duf[1], &du2[1], &
ipiv[1], &work[*n + 1], n, info);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[*n + i__] = work[i__] * work[*n + i__];
/* L80: */
}
} else {
/* Multiply by inv(op(A))*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
work[*n + i__] = work[i__] * work[*n + i__];
/* L90: */
}
dgttrs_(transn, n, &c__1, &dlf[1], &df[1], &duf[1], &du2[1], &
ipiv[1], &work[*n + 1], n, info);
}
goto L70;
}
/* 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);
/* L100: */
}
if (lstres != 0.) {
ferr[j] /= lstres;
}
/* L110: */
}
return 0;
/* End of DGTRFS */
} /* dgtrfs_ */
