/* zptrfs.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 doublecomplex c_b16 = {1.,0.};
/* Subroutine */ int zptrfs_(char *uplo, integer *n, integer *nrhs,
doublereal *d__, doublecomplex *e, doublereal *df, doublecomplex *ef,
doublecomplex *b, integer *ldb, doublecomplex *x, integer *ldx,
doublereal *ferr, doublereal *berr, doublecomplex *work, doublereal *
rwork, integer *info)
{
/* System generated locals */
integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5,
i__6;
doublereal d__1, d__2, d__3, d__4, d__5, d__6, d__7, d__8, d__9, d__10,
d__11, d__12;
doublecomplex z__1, z__2, z__3;
/* Builtin functions */
double d_imag(doublecomplex *);
void d_cnjg(doublecomplex *, doublecomplex *);
double z_abs(doublecomplex *);
/* Local variables */
integer i__, j;
doublereal s;
doublecomplex bi, cx, dx, ex;
integer ix, nz;
doublereal eps, safe1, safe2;
extern logical lsame_(char *, char *);
integer count;
logical upper;
extern /* Subroutine */ int zaxpy_(integer *, doublecomplex *,
doublecomplex *, integer *, doublecomplex *, integer *);
extern doublereal dlamch_(char *);
extern integer idamax_(integer *, doublereal *, integer *);
doublereal safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
doublereal lstres;
extern /* Subroutine */ int zpttrs_(char *, integer *, integer *,
doublereal *, doublecomplex *, doublecomplex *, integer *,
integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* ZPTRFS improves the computed solution to a system of linear */
/* equations when the coefficient matrix is Hermitian positive definite */
/* and tridiagonal, and provides error bounds and backward error */
/* estimates for the solution. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* Specifies whether the superdiagonal or the subdiagonal of the */
/* tridiagonal matrix A is stored and the form of the */
/* factorization: */
/* = 'U': E is the superdiagonal of A, and A = U**H*D*U; */
/* = 'L': E is the subdiagonal of A, and A = L*D*L**H. */
/* (The two forms are equivalent if A is real.) */
/* 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) DOUBLE PRECISION array, dimension (N) */
/* The n real diagonal elements of the tridiagonal matrix A. */
/* E (input) COMPLEX*16 array, dimension (N-1) */
/* The (n-1) off-diagonal elements of the tridiagonal matrix A */
/* (see UPLO). */
/* DF (input) DOUBLE PRECISION array, dimension (N) */
/* The n diagonal elements of the diagonal matrix D from */
/* the factorization computed by ZPTTRF. */
/* EF (input) COMPLEX*16 array, dimension (N-1) */
/* The (n-1) off-diagonal elements of the unit bidiagonal */
/* factor U or L from the factorization computed by ZPTTRF */
/* (see UPLO). */
/* B (input) COMPLEX*16 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) COMPLEX*16 array, dimension (LDX,NRHS) */
/* On entry, the solution matrix X, as computed by ZPTTRS. */
/* 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 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) 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) COMPLEX*16 array, dimension (N) */
/* RWORK (workspace) DOUBLE PRECISION 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 .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. 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;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*nrhs < 0) {
*info = -3;
} else if (*ldb < max(1,*n)) {
*info = -9;
} else if (*ldx < max(1,*n)) {
*info = -11;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZPTRFS", &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 */
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 - A * X. Also compute */
/* abs(A)*abs(x) + abs(b) for use in the backward error bound. */
if (upper) {
if (*n == 1) {
i__2 = j * b_dim1 + 1;
bi.r = b[i__2].r, bi.i = b[i__2].i;
i__2 = j * x_dim1 + 1;
z__1.r = d__[1] * x[i__2].r, z__1.i = d__[1] * x[i__2].i;
dx.r = z__1.r, dx.i = z__1.i;
z__1.r = bi.r - dx.r, z__1.i = bi.i - dx.i;
work[1].r = z__1.r, work[1].i = z__1.i;
rwork[1] = (d__1 = bi.r, abs(d__1)) + (d__2 = d_imag(&bi),
abs(d__2)) + ((d__3 = dx.r, abs(d__3)) + (d__4 =
d_imag(&dx), abs(d__4)));
} else {
i__2 = j * b_dim1 + 1;
bi.r = b[i__2].r, bi.i = b[i__2].i;
i__2 = j * x_dim1 + 1;
z__1.r = d__[1] * x[i__2].r, z__1.i = d__[1] * x[i__2].i;
dx.r = z__1.r, dx.i = z__1.i;
i__2 = j * x_dim1 + 2;
z__1.r = e[1].r * x[i__2].r - e[1].i * x[i__2].i, z__1.i = e[
1].r * x[i__2].i + e[1].i * x[i__2].r;
ex.r = z__1.r, ex.i = z__1.i;
z__2.r = bi.r - dx.r, z__2.i = bi.i - dx.i;
z__1.r = z__2.r - ex.r, z__1.i = z__2.i - ex.i;
work[1].r = z__1.r, work[1].i = z__1.i;
i__2 = j * x_dim1 + 2;
rwork[1] = (d__1 = bi.r, abs(d__1)) + (d__2 = d_imag(&bi),
abs(d__2)) + ((d__3 = dx.r, abs(d__3)) + (d__4 =
d_imag(&dx), abs(d__4))) + ((d__5 = e[1].r, abs(d__5))
+ (d__6 = d_imag(&e[1]), abs(d__6))) * ((d__7 = x[
i__2].r, abs(d__7)) + (d__8 = d_imag(&x[j * x_dim1 +
2]), abs(d__8)));
i__2 = *n - 1;
for (i__ = 2; i__ <= i__2; ++i__) {
i__3 = i__ + j * b_dim1;
bi.r = b[i__3].r, bi.i = b[i__3].i;
d_cnjg(&z__2, &e[i__ - 1]);
i__3 = i__ - 1 + j * x_dim1;
z__1.r = z__2.r * x[i__3].r - z__2.i * x[i__3].i, z__1.i =
z__2.r * x[i__3].i + z__2.i * x[i__3].r;
cx.r = z__1.r, cx.i = z__1.i;
i__3 = i__;
i__4 = i__ + j * x_dim1;
z__1.r = d__[i__3] * x[i__4].r, z__1.i = d__[i__3] * x[
i__4].i;
dx.r = z__1.r, dx.i = z__1.i;
i__3 = i__;
i__4 = i__ + 1 + j * x_dim1;
z__1.r = e[i__3].r * x[i__4].r - e[i__3].i * x[i__4].i,
z__1.i = e[i__3].r * x[i__4].i + e[i__3].i * x[
i__4].r;
ex.r = z__1.r, ex.i = z__1.i;
i__3 = i__;
z__3.r = bi.r - cx.r, z__3.i = bi.i - cx.i;
z__2.r = z__3.r - dx.r, z__2.i = z__3.i - dx.i;
z__1.r = z__2.r - ex.r, z__1.i = z__2.i - ex.i;
work[i__3].r = z__1.r, work[i__3].i = z__1.i;
i__3 = i__ - 1;
i__4 = i__ - 1 + j * x_dim1;
i__5 = i__;
i__6 = i__ + 1 + j * x_dim1;
rwork[i__] = (d__1 = bi.r, abs(d__1)) + (d__2 = d_imag(&
bi), abs(d__2)) + ((d__3 = e[i__3].r, abs(d__3))
+ (d__4 = d_imag(&e[i__ - 1]), abs(d__4))) * ((
d__5 = x[i__4].r, abs(d__5)) + (d__6 = d_imag(&x[
i__ - 1 + j * x_dim1]), abs(d__6))) + ((d__7 =
dx.r, abs(d__7)) + (d__8 = d_imag(&dx), abs(d__8))
) + ((d__9 = e[i__5].r, abs(d__9)) + (d__10 =
d_imag(&e[i__]), abs(d__10))) * ((d__11 = x[i__6]
.r, abs(d__11)) + (d__12 = d_imag(&x[i__ + 1 + j *
x_dim1]), abs(d__12)));
/* L30: */
}
i__2 = *n + j * b_dim1;
bi.r = b[i__2].r, bi.i = b[i__2].i;
d_cnjg(&z__2, &e[*n - 1]);
i__2 = *n - 1 + j * x_dim1;
z__1.r = z__2.r * x[i__2].r - z__2.i * x[i__2].i, z__1.i =
z__2.r * x[i__2].i + z__2.i * x[i__2].r;
cx.r = z__1.r, cx.i = z__1.i;
i__2 = *n;
i__3 = *n + j * x_dim1;
z__1.r = d__[i__2] * x[i__3].r, z__1.i = d__[i__2] * x[i__3]
.i;
dx.r = z__1.r, dx.i = z__1.i;
i__2 = *n;
z__2.r = bi.r - cx.r, z__2.i = bi.i - cx.i;
z__1.r = z__2.r - dx.r, z__1.i = z__2.i - dx.i;
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
i__2 = *n - 1;
i__3 = *n - 1 + j * x_dim1;
rwork[*n] = (d__1 = bi.r, abs(d__1)) + (d__2 = d_imag(&bi),
abs(d__2)) + ((d__3 = e[i__2].r, abs(d__3)) + (d__4 =
d_imag(&e[*n - 1]), abs(d__4))) * ((d__5 = x[i__3].r,
abs(d__5)) + (d__6 = d_imag(&x[*n - 1 + j * x_dim1]),
abs(d__6))) + ((d__7 = dx.r, abs(d__7)) + (d__8 =
d_imag(&dx), abs(d__8)));
}
} else {
if (*n == 1) {
i__2 = j * b_dim1 + 1;
bi.r = b[i__2].r, bi.i = b[i__2].i;
i__2 = j * x_dim1 + 1;
z__1.r = d__[1] * x[i__2].r, z__1.i = d__[1] * x[i__2].i;
dx.r = z__1.r, dx.i = z__1.i;
z__1.r = bi.r - dx.r, z__1.i = bi.i - dx.i;
work[1].r = z__1.r, work[1].i = z__1.i;
rwork[1] = (d__1 = bi.r, abs(d__1)) + (d__2 = d_imag(&bi),
abs(d__2)) + ((d__3 = dx.r, abs(d__3)) + (d__4 =
d_imag(&dx), abs(d__4)));
} else {
i__2 = j * b_dim1 + 1;
bi.r = b[i__2].r, bi.i = b[i__2].i;
i__2 = j * x_dim1 + 1;
z__1.r = d__[1] * x[i__2].r, z__1.i = d__[1] * x[i__2].i;
dx.r = z__1.r, dx.i = z__1.i;
d_cnjg(&z__2, &e[1]);
i__2 = j * x_dim1 + 2;
z__1.r = z__2.r * x[i__2].r - z__2.i * x[i__2].i, z__1.i =
z__2.r * x[i__2].i + z__2.i * x[i__2].r;
ex.r = z__1.r, ex.i = z__1.i;
z__2.r = bi.r - dx.r, z__2.i = bi.i - dx.i;
z__1.r = z__2.r - ex.r, z__1.i = z__2.i - ex.i;
work[1].r = z__1.r, work[1].i = z__1.i;
i__2 = j * x_dim1 + 2;
rwork[1] = (d__1 = bi.r, abs(d__1)) + (d__2 = d_imag(&bi),
abs(d__2)) + ((d__3 = dx.r, abs(d__3)) + (d__4 =
d_imag(&dx), abs(d__4))) + ((d__5 = e[1].r, abs(d__5))
+ (d__6 = d_imag(&e[1]), abs(d__6))) * ((d__7 = x[
i__2].r, abs(d__7)) + (d__8 = d_imag(&x[j * x_dim1 +
2]), abs(d__8)));
i__2 = *n - 1;
for (i__ = 2; i__ <= i__2; ++i__) {
i__3 = i__ + j * b_dim1;
bi.r = b[i__3].r, bi.i = b[i__3].i;
i__3 = i__ - 1;
i__4 = i__ - 1 + j * x_dim1;
z__1.r = e[i__3].r * x[i__4].r - e[i__3].i * x[i__4].i,
z__1.i = e[i__3].r * x[i__4].i + e[i__3].i * x[
i__4].r;
cx.r = z__1.r, cx.i = z__1.i;
i__3 = i__;
i__4 = i__ + j * x_dim1;
z__1.r = d__[i__3] * x[i__4].r, z__1.i = d__[i__3] * x[
i__4].i;
dx.r = z__1.r, dx.i = z__1.i;
d_cnjg(&z__2, &e[i__]);
i__3 = i__ + 1 + j * x_dim1;
z__1.r = z__2.r * x[i__3].r - z__2.i * x[i__3].i, z__1.i =
z__2.r * x[i__3].i + z__2.i * x[i__3].r;
ex.r = z__1.r, ex.i = z__1.i;
i__3 = i__;
z__3.r = bi.r - cx.r, z__3.i = bi.i - cx.i;
z__2.r = z__3.r - dx.r, z__2.i = z__3.i - dx.i;
z__1.r = z__2.r - ex.r, z__1.i = z__2.i - ex.i;
work[i__3].r = z__1.r, work[i__3].i = z__1.i;
i__3 = i__ - 1;
i__4 = i__ - 1 + j * x_dim1;
i__5 = i__;
i__6 = i__ + 1 + j * x_dim1;
rwork[i__] = (d__1 = bi.r, abs(d__1)) + (d__2 = d_imag(&
bi), abs(d__2)) + ((d__3 = e[i__3].r, abs(d__3))
+ (d__4 = d_imag(&e[i__ - 1]), abs(d__4))) * ((
d__5 = x[i__4].r, abs(d__5)) + (d__6 = d_imag(&x[
i__ - 1 + j * x_dim1]), abs(d__6))) + ((d__7 =
dx.r, abs(d__7)) + (d__8 = d_imag(&dx), abs(d__8))
) + ((d__9 = e[i__5].r, abs(d__9)) + (d__10 =
d_imag(&e[i__]), abs(d__10))) * ((d__11 = x[i__6]
.r, abs(d__11)) + (d__12 = d_imag(&x[i__ + 1 + j *
x_dim1]), abs(d__12)));
/* L40: */
}
i__2 = *n + j * b_dim1;
bi.r = b[i__2].r, bi.i = b[i__2].i;
i__2 = *n - 1;
i__3 = *n - 1 + j * x_dim1;
z__1.r = e[i__2].r * x[i__3].r - e[i__2].i * x[i__3].i,
z__1.i = e[i__2].r * x[i__3].i + e[i__2].i * x[i__3]
.r;
cx.r = z__1.r, cx.i = z__1.i;
i__2 = *n;
i__3 = *n + j * x_dim1;
z__1.r = d__[i__2] * x[i__3].r, z__1.i = d__[i__2] * x[i__3]
.i;
dx.r = z__1.r, dx.i = z__1.i;
i__2 = *n;
z__2.r = bi.r - cx.r, z__2.i = bi.i - cx.i;
z__1.r = z__2.r - dx.r, z__1.i = z__2.i - dx.i;
work[i__2].r = z__1.r, work[i__2].i = z__1.i;
i__2 = *n - 1;
i__3 = *n - 1 + j * x_dim1;
rwork[*n] = (d__1 = bi.r, abs(d__1)) + (d__2 = d_imag(&bi),
abs(d__2)) + ((d__3 = e[i__2].r, abs(d__3)) + (d__4 =
d_imag(&e[*n - 1]), abs(d__4))) * ((d__5 = x[i__3].r,
abs(d__5)) + (d__6 = d_imag(&x[*n - 1 + j * x_dim1]),
abs(d__6))) + ((d__7 = dx.r, abs(d__7)) + (d__8 =
d_imag(&dx), abs(d__8)));
}
}
/* 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.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
/* Computing MAX */
i__3 = i__;
d__3 = s, d__4 = ((d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2))) / rwork[i__];
s = max(d__3,d__4);
} else {
/* Computing MAX */
i__3 = i__;
d__3 = s, d__4 = ((d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + safe1) / (rwork[i__]
+ safe1);
s = max(d__3,d__4);
}
/* 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. */
zpttrs_(uplo, n, &c__1, &df[1], &ef[1], &work[1], n, info);
zaxpy_(n, &c_b16, &work[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 (rwork[i__] > safe2) {
i__3 = i__;
rwork[i__] = (d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + nz * eps * rwork[i__]
;
} else {
i__3 = i__;
rwork[i__] = (d__1 = work[i__3].r, abs(d__1)) + (d__2 =
d_imag(&work[i__]), abs(d__2)) + nz * eps * rwork[i__]
+ safe1;
}
/* L60: */
}
ix = idamax_(n, &rwork[1], &c__1);
ferr[j] = rwork[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. */
rwork[1] = 1.;
i__2 = *n;
for (i__ = 2; i__ <= i__2; ++i__) {
rwork[i__] = rwork[i__ - 1] * z_abs(&ef[i__ - 1]) + 1.;
/* L70: */
}
/* Solve D * M(L)' * x = b. */
rwork[*n] /= df[*n];
for (i__ = *n - 1; i__ >= 1; --i__) {
rwork[i__] = rwork[i__] / df[i__] + rwork[i__ + 1] * z_abs(&ef[
i__]);
/* L80: */
}
/* Compute norm(inv(A)) = max(x(i)), 1<=i<=n. */
ix = idamax_(n, &rwork[1], &c__1);
ferr[j] *= (d__1 = rwork[ix], abs(d__1));
/* Normalize error. */
lstres = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
d__1 = lstres, d__2 = z_abs(&x[i__ + j * x_dim1]);
lstres = max(d__1,d__2);
/* L90: */
}
if (lstres != 0.) {
ferr[j] /= lstres;
}
/* L100: */
}
return 0;
/* End of ZPTRFS */
} /* zptrfs_ */