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/* sptrfs.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 real c_b11 = 1.f;
/* Subroutine */ int sptrfs_(integer *n, integer *nrhs, real *d__, real *e,
real *df, real *ef, real *b, integer *ldb, real *x, integer *ldx,
real *ferr, real *berr, real *work, integer *info)
{
/* System generated locals */
integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2;
real r__1, r__2, r__3;
/* Local variables */
integer i__, j;
real s, bi, cx, dx, ex;
integer ix, nz;
real eps, safe1, safe2;
integer count;
extern /* Subroutine */ int saxpy_(integer *, real *, real *, integer *,
real *, integer *);
extern doublereal slamch_(char *);
real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
extern integer isamax_(integer *, real *, integer *);
real lstres;
extern /* Subroutine */ int spttrs_(integer *, integer *, real *, real *,
real *, integer *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SPTRFS improves the computed solution to a system of linear */
/* equations when the coefficient matrix is symmetric positive definite */
/* and tridiagonal, and provides error bounds and backward error */
/* estimates for the solution. */
/* Arguments */
/* ========= */
/* 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. */
/* D (input) REAL array, dimension (N) */
/* The n diagonal elements of the tridiagonal matrix A. */
/* E (input) REAL array, dimension (N-1) */
/* The (n-1) subdiagonal elements of the tridiagonal matrix A. */
/* DF (input) REAL array, dimension (N) */
/* The n diagonal elements of the diagonal matrix D from the */
/* factorization computed by SPTTRF. */
/* EF (input) REAL array, dimension (N-1) */
/* The (n-1) subdiagonal elements of the unit bidiagonal factor */
/* L from the factorization computed by SPTTRF. */
/* 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 SPTTRS. */
/* 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 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). */
/* 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 (2*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 .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
--e;
--df;
--ef;
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;
/* Function Body */
*info = 0;
if (*n < 0) {
*info = -1;
} else if (*nrhs < 0) {
*info = -2;
} else if (*ldb < max(1,*n)) {
*info = -8;
} else if (*ldx < max(1,*n)) {
*info = -10;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SPTRFS", &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;
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = 4;
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 - A * X. Also compute */
/* abs(A)*abs(x) + abs(b) for use in the backward error bound. */
if (*n == 1) {
bi = b[j * b_dim1 + 1];
dx = d__[1] * x[j * x_dim1 + 1];
work[*n + 1] = bi - dx;
work[1] = dabs(bi) + dabs(dx);
} else {
bi = b[j * b_dim1 + 1];
dx = d__[1] * x[j * x_dim1 + 1];
ex = e[1] * x[j * x_dim1 + 2];
work[*n + 1] = bi - dx - ex;
work[1] = dabs(bi) + dabs(dx) + dabs(ex);
i__2 = *n - 1;
for (i__ = 2; i__ <= i__2; ++i__) {
bi = b[i__ + j * b_dim1];
cx = e[i__ - 1] * x[i__ - 1 + j * x_dim1];
dx = d__[i__] * x[i__ + j * x_dim1];
ex = e[i__] * x[i__ + 1 + j * x_dim1];
work[*n + i__] = bi - cx - dx - ex;
work[i__] = dabs(bi) + dabs(cx) + dabs(dx) + dabs(ex);
/* L30: */
}
bi = b[*n + j * b_dim1];
cx = e[*n - 1] * x[*n - 1 + j * x_dim1];
dx = d__[*n] * x[*n + j * x_dim1];
work[*n + *n] = bi - cx - dx;
work[*n] = dabs(bi) + dabs(cx) + dabs(dx);
}
/* 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. */
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);
}
/* L40: */
}
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. */
spttrs_(n, &c__1, &df[1], &ef[1], &work[*n + 1], n, info);
saxpy_(n, &c_b11, &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. */
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;
}
/* L50: */
}
ix = isamax_(n, &work[1], &c__1);
ferr[j] = work[ix];
/* Estimate the norm of inv(A). */
/* Solve M(A) * x = e, where M(A) = (m(i,j)) is given by */
/* m(i,j) = abs(A(i,j)), i = j, */
/* m(i,j) = -abs(A(i,j)), i .ne. j, */
/* and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)'. */
/* Solve M(L) * x = e. */
work[1] = 1.f;
i__2 = *n;
for (i__ = 2; i__ <= i__2; ++i__) {
work[i__] = work[i__ - 1] * (r__1 = ef[i__ - 1], dabs(r__1)) +
1.f;
/* L60: */
}
/* Solve D * M(L)' * x = b. */
work[*n] /= df[*n];
for (i__ = *n - 1; i__ >= 1; --i__) {
work[i__] = work[i__] / df[i__] + work[i__ + 1] * (r__1 = ef[i__],
dabs(r__1));
/* L70: */
}
/* Compute norm(inv(A)) = max(x(i)), 1<=i<=n. */
ix = isamax_(n, &work[1], &c__1);
ferr[j] *= (r__1 = work[ix], dabs(r__1));
/* 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[i__ + j * x_dim1], dabs(r__1));
lstres = dmax(r__2,r__3);
/* L80: */
}
if (lstres != 0.f) {
ferr[j] /= lstres;
}
/* L90: */
}
return 0;
/* End of SPTRFS */
} /* sptrfs_ */
|
