/* cppsvx.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 cppsvx_(char *fact, char *uplo, integer *n, integer *
nrhs, complex *ap, complex *afp, char *equed, real *s, complex *b,
integer *ldb, complex *x, integer *ldx, real *rcond, real *ferr, real
*berr, complex *work, real *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;
real r__1, r__2;
complex q__1;
/* Local variables */
integer i__, j;
real amax, smin, smax;
extern logical lsame_(char *, char *);
real scond, anorm;
extern /* Subroutine */ int ccopy_(integer *, complex *, integer *,
complex *, integer *);
logical equil, rcequ;
extern doublereal clanhp_(char *, char *, integer *, complex *, real *), slamch_(char *);
extern /* Subroutine */ int claqhp_(char *, integer *, complex *, real *,
real *, real *, char *);
logical nofact;
extern /* Subroutine */ int clacpy_(char *, integer *, integer *, complex
*, integer *, complex *, integer *), xerbla_(char *,
integer *);
real bignum;
extern /* Subroutine */ int cppcon_(char *, integer *, complex *, real *,
real *, complex *, real *, integer *);
integer infequ;
extern /* Subroutine */ int cppequ_(char *, integer *, complex *, real *,
real *, real *, integer *), cpprfs_(char *, integer *,
integer *, complex *, complex *, complex *, integer *, complex *,
integer *, real *, real *, complex *, real *, integer *),
cpptrf_(char *, integer *, complex *, integer *);
real smlnum;
extern /* Subroutine */ int cpptrs_(char *, integer *, integer *, complex
*, complex *, integer *, integer *);
/* -- LAPACK driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CPPSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to */
/* compute the solution to a complex system of linear equations */
/* A * X = B, */
/* where A is an N-by-N Hermitian positive definite matrix stored in */
/* packed format and X and B are N-by-NRHS matrices. */
/* Error bounds on the solution and a condition estimate are also */
/* provided. */
/* Description */
/* =========== */
/* The following steps are performed: */
/* 1. If FACT = 'E', real scaling factors are computed to equilibrate */
/* the system: */
/* diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B */
/* Whether or not the system will be equilibrated depends on the */
/* scaling of the matrix A, but if equilibration is used, A is */
/* overwritten by diag(S)*A*diag(S) and B by diag(S)*B. */
/* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to */
/* factor the matrix A (after equilibration if FACT = 'E') as */
/* A = U'* U , if UPLO = 'U', or */
/* A = L * L', if UPLO = 'L', */
/* where U is an upper triangular matrix, L is a lower triangular */
/* matrix, and ' indicates conjugate transpose. */
/* 3. If the leading i-by-i principal minor is not positive definite, */
/* then the routine returns with INFO = i. Otherwise, the factored */
/* form of A is used to estimate the condition number of the matrix */
/* A. If the reciprocal of the condition number is less than machine */
/* precision, INFO = N+1 is returned as a warning, but the routine */
/* still goes on to solve for X and compute error bounds as */
/* described below. */
/* 4. The system of equations is solved for X using the factored form */
/* of A. */
/* 5. Iterative refinement is applied to improve the computed solution */
/* matrix and calculate error bounds and backward error estimates */
/* for it. */
/* 6. If equilibration was used, the matrix X is premultiplied by */
/* diag(S) so that it solves the original system before */
/* equilibration. */
/* Arguments */
/* ========= */
/* FACT (input) CHARACTER*1 */
/* Specifies whether or not the factored form of the matrix A is */
/* supplied on entry, and if not, whether the matrix A should be */
/* equilibrated before it is factored. */
/* = 'F': On entry, AFP contains the factored form of A. */
/* If EQUED = 'Y', the matrix A has been equilibrated */
/* with scaling factors given by S. AP and AFP will not */
/* be modified. */
/* = 'N': The matrix A will be copied to AFP and factored. */
/* = 'E': The matrix A will be equilibrated if necessary, then */
/* copied to AFP and factored. */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The number of linear equations, i.e., 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 matrices B and X. NRHS >= 0. */
/* AP (input/output) COMPLEX array, dimension (N*(N+1)/2) */
/* On entry, the upper or lower triangle of the Hermitian matrix */
/* A, packed columnwise in a linear array, except if FACT = 'F' */
/* and EQUED = 'Y', then A must contain the equilibrated matrix */
/* diag(S)*A*diag(S). The j-th column of A is stored in the */
/* array AP as follows: */
/* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; */
/* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. */
/* See below for further details. A is not modified if */
/* FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. */
/* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by */
/* diag(S)*A*diag(S). */
/* AFP (input or output) COMPLEX array, dimension (N*(N+1)/2) */
/* If FACT = 'F', then AFP is an input argument and on entry */
/* contains the triangular factor U or L from the Cholesky */
/* factorization A = U**H*U or A = L*L**H, in the same storage */
/* format as A. If EQUED .ne. 'N', then AFP is the factored */
/* form of the equilibrated matrix A. */
/* If FACT = 'N', then AFP is an output argument and on exit */
/* returns the triangular factor U or L from the Cholesky */
/* factorization A = U**H*U or A = L*L**H of the original */
/* matrix A. */
/* If FACT = 'E', then AFP is an output argument and on exit */
/* returns the triangular factor U or L from the Cholesky */
/* factorization A = U**H*U or A = L*L**H of the equilibrated */
/* matrix A (see the description of AP for the form of the */
/* equilibrated matrix). */
/* EQUED (input or output) CHARACTER*1 */
/* Specifies the form of equilibration that was done. */
/* = 'N': No equilibration (always true if FACT = 'N'). */
/* = 'Y': Equilibration was done, i.e., A has been replaced by */
/* diag(S) * A * diag(S). */
/* EQUED is an input argument if FACT = 'F'; otherwise, it is an */
/* output argument. */
/* S (input or output) REAL array, dimension (N) */
/* The scale factors for A; not accessed if EQUED = 'N'. S is */
/* an input argument if FACT = 'F'; otherwise, S is an output */
/* argument. If FACT = 'F' and EQUED = 'Y', each element of S */
/* must be positive. */
/* B (input/output) COMPLEX array, dimension (LDB,NRHS) */
/* On entry, the N-by-NRHS right hand side matrix B. */
/* On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', */
/* B is overwritten by diag(S) * B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* X (output) COMPLEX array, dimension (LDX,NRHS) */
/* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to */
/* the original system of equations. Note that if EQUED = 'Y', */
/* A and B are modified on exit, and the solution to the */
/* equilibrated system is inv(diag(S))*X. */
/* LDX (input) INTEGER */
/* The leading dimension of the array X. LDX >= max(1,N). */
/* RCOND (output) REAL */
/* The estimate of the reciprocal condition number of the matrix */
/* A after equilibration (if done). If RCOND is less than the */
/* machine precision (in particular, if RCOND = 0), the matrix */
/* is singular to working precision. This condition is */
/* indicated by a return code of INFO > 0. */
/* 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 */
/* > 0: if INFO = i, and i is */
/* <= N: the leading minor of order i of A is */
/* not positive definite, so the factorization */
/* could not be completed, and the solution has not */
/* been computed. RCOND = 0 is returned. */
/* = N+1: U is nonsingular, but RCOND is less than machine */
/* precision, meaning that the matrix is singular */
/* to working precision. Nevertheless, the */
/* solution and error bounds are computed because */
/* there are a number of situations where the */
/* computed solution can be more accurate than the */
/* value of RCOND would suggest. */
/* Further Details */
/* =============== */
/* The packed storage scheme is illustrated by the following example */
/* when N = 4, UPLO = 'U': */
/* Two-dimensional storage of the Hermitian matrix A: */
/* a11 a12 a13 a14 */
/* a22 a23 a24 */
/* a33 a34 (aij = conjg(aji)) */
/* a44 */
/* Packed storage of the upper triangle of A: */
/* AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
--ap;
--afp;
--s;
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;
nofact = lsame_(fact, "N");
equil = lsame_(fact, "E");
if (nofact || equil) {
*(unsigned char *)equed = 'N';
rcequ = FALSE_;
} else {
rcequ = lsame_(equed, "Y");
smlnum = slamch_("Safe minimum");
bignum = 1.f / smlnum;
}
/* Test the input parameters. */
if (! nofact && ! equil && ! lsame_(fact, "F")) {
*info = -1;
} else if (! lsame_(uplo, "U") && ! lsame_(uplo,
"L")) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*nrhs < 0) {
*info = -4;
} else if (lsame_(fact, "F") && ! (rcequ || lsame_(
equed, "N"))) {
*info = -7;
} else {
if (rcequ) {
smin = bignum;
smax = 0.f;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* Computing MIN */
r__1 = smin, r__2 = s[j];
smin = dmin(r__1,r__2);
/* Computing MAX */
r__1 = smax, r__2 = s[j];
smax = dmax(r__1,r__2);
/* L10: */
}
if (smin <= 0.f) {
*info = -8;
} else if (*n > 0) {
scond = dmax(smin,smlnum) / dmin(smax,bignum);
} else {
scond = 1.f;
}
}
if (*info == 0) {
if (*ldb < max(1,*n)) {
*info = -10;
} else if (*ldx < max(1,*n)) {
*info = -12;
}
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CPPSVX", &i__1);
return 0;
}
if (equil) {
/* Compute row and column scalings to equilibrate the matrix A. */
cppequ_(uplo, n, &ap[1], &s[1], &scond, &amax, &infequ);
if (infequ == 0) {
/* Equilibrate the matrix. */
claqhp_(uplo, n, &ap[1], &s[1], &scond, &amax, equed);
rcequ = lsame_(equed, "Y");
}
}
/* Scale the right-hand side. */
if (rcequ) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__ + j * b_dim1;
i__4 = i__;
i__5 = i__ + j * b_dim1;
q__1.r = s[i__4] * b[i__5].r, q__1.i = s[i__4] * b[i__5].i;
b[i__3].r = q__1.r, b[i__3].i = q__1.i;
/* L20: */
}
/* L30: */
}
}
if (nofact || equil) {
/* Compute the Cholesky factorization A = U'*U or A = L*L'. */
i__1 = *n * (*n + 1) / 2;
ccopy_(&i__1, &ap[1], &c__1, &afp[1], &c__1);
cpptrf_(uplo, n, &afp[1], info);
/* Return if INFO is non-zero. */
if (*info > 0) {
*rcond = 0.f;
return 0;
}
}
/* Compute the norm of the matrix A. */
anorm = clanhp_("I", uplo, n, &ap[1], &rwork[1]);
/* Compute the reciprocal of the condition number of A. */
cppcon_(uplo, n, &afp[1], &anorm, rcond, &work[1], &rwork[1], info);
/* Compute the solution matrix X. */
clacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx);
cpptrs_(uplo, n, nrhs, &afp[1], &x[x_offset], ldx, info);
/* Use iterative refinement to improve the computed solution and */
/* compute error bounds and backward error estimates for it. */
cpprfs_(uplo, n, nrhs, &ap[1], &afp[1], &b[b_offset], ldb, &x[x_offset],
ldx, &ferr[1], &berr[1], &work[1], &rwork[1], info);
/* Transform the solution matrix X to a solution of the original */
/* system. */
if (rcequ) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__ + j * x_dim1;
i__4 = i__;
i__5 = i__ + j * x_dim1;
q__1.r = s[i__4] * x[i__5].r, q__1.i = s[i__4] * x[i__5].i;
x[i__3].r = q__1.r, x[i__3].i = q__1.i;
/* L40: */
}
/* L50: */
}
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
ferr[j] /= scond;
/* L60: */
}
}
/* Set INFO = N+1 if the matrix is singular to working precision. */
if (*rcond < slamch_("Epsilon")) {
*info = *n + 1;
}
return 0;
/* End of CPPSVX */
} /* cppsvx_ */