/* Subroutine */ int ctimsp_(char *line, integer *nn, integer *nval, integer *
nns, integer *nsval, integer *la, real *timmin, complex *a, complex *
b, complex *work, integer *iwork, real *reslts, integer *ldr1,
integer *ldr2, integer *ldr3, integer *nout, ftnlen line_len)
{
/* Initialized data */
static char uplos[1*2] = "U" "L";
static char subnam[6*3] = "CSPTRF" "CSPTRS" "CSPTRI";
/* Format strings */
static char fmt_9999[] = "(1x,a6,\002 timing run not attempted\002,/)";
static char fmt_9998[] = "(/\002 *** Speed of \002,a6,\002 in megaflops "
"***\002,/)";
static char fmt_9997[] = "(5x,a6,\002 with UPLO = '\002,a1,\002'\002,/)";
/* System generated locals */
integer reslts_dim1, reslts_dim2, reslts_dim3, reslts_offset, i__1, i__2;
/* Builtin functions
Subroutine */ int s_copy(char *, char *, ftnlen, ftnlen);
integer s_wsfe(cilist *), do_fio(integer *, char *, ftnlen), e_wsfe(void);
/* Local variables */
static integer info;
static char path[3];
static real time;
static integer isub, nrhs;
static char uplo[1];
static integer i__, n;
static char cname[6];
static integer laval[1];
extern logical lsame_(char *, char *);
extern /* Subroutine */ int ccopy_(integer *, complex *, integer *,
complex *, integer *);
extern doublereal sopla_(char *, integer *, integer *, integer *, integer
*, integer *);
static integer iuplo;
static real s1, s2;
static integer ic, in;
extern /* Subroutine */ int atimck_(integer *, char *, integer *, integer
*, integer *, integer *, integer *, integer *, ftnlen);
extern doublereal second_(void);
extern /* Subroutine */ int ctimmg_(integer *, integer *, integer *,
complex *, integer *, integer *, integer *), atimin_(char *, char
*, integer *, char *, logical *, integer *, integer *, ftnlen,
ftnlen, ftnlen);
extern doublereal smflop_(real *, real *, integer *);
static real untime;
extern /* Subroutine */ int csptrf_(char *, integer *, complex *, integer
*, integer *);
static logical timsub[3];
extern /* Subroutine */ int csptri_(char *, integer *, complex *, integer
*, complex *, integer *), sprtbl_(char *, char *, integer
*, integer *, integer *, integer *, integer *, real *, integer *,
integer *, integer *, ftnlen, ftnlen), csptrs_(char *, integer *,
integer *, complex *, integer *, complex *, integer *, integer *);
static integer lda, ldb, icl, mat;
static real ops;
/* Fortran I/O blocks */
static cilist io___8 = { 0, 0, 0, fmt_9999, 0 };
static cilist io___26 = { 0, 0, 0, fmt_9998, 0 };
static cilist io___27 = { 0, 0, 0, fmt_9997, 0 };
#define subnam_ref(a_0,a_1) &subnam[(a_1)*6 + a_0 - 6]
#define reslts_ref(a_1,a_2,a_3,a_4) reslts[(((a_4)*reslts_dim3 + (a_3))*\
reslts_dim2 + (a_2))*reslts_dim1 + a_1]
/* -- LAPACK timing routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
March 31, 1993
Purpose
=======
CTIMSP times CSPTRF, -TRS, and -TRI.
Arguments
=========
LINE (input) CHARACTER*80
The input line that requested this routine. The first six
characters contain either the name of a subroutine or a
generic path name. The remaining characters may be used to
specify the individual routines to be timed. See ATIMIN for
a full description of the format of the input line.
NN (input) INTEGER
The number of values of N contained in the vector NVAL.
NVAL (input) INTEGER array, dimension (NN)
The values of the matrix size N.
NNS (input) INTEGER
The number of values of NRHS contained in the vector NSVAL.
NSVAL (input) INTEGER array, dimension (NNS)
The values of the number of right hand sides NRHS.
LA (input) INTEGER
The size of the arrays A, B, and C.
TIMMIN (input) REAL
The minimum time a subroutine will be timed.
A (workspace) COMPLEX array, dimension (LA)
B (workspace) COMPLEX array, dimension (LA)
WORK (workspace) COMPLEX array, dimension (NMAX)
IWORK (workspace) INTEGER array, dimension (NMAX)
where NMAX is the maximum value of N permitted.
RESLTS (output) REAL array, dimension
(LDR1,LDR2,LDR3,NSUBS)
The timing results for each subroutine over the relevant
values of N.
LDR1 (input) INTEGER
The first dimension of RESLTS. LDR1 >= max(4,NNB).
LDR2 (input) INTEGER
The second dimension of RESLTS. LDR2 >= max(1,NN).
LDR3 (input) INTEGER
The third dimension of RESLTS. LDR3 >= 2.
NOUT (input) INTEGER
The unit number for output.
=====================================================================
Parameter adjustments */
--nval;
--nsval;
--a;
--b;
--work;
--iwork;
reslts_dim1 = *ldr1;
reslts_dim2 = *ldr2;
reslts_dim3 = *ldr3;
reslts_offset = 1 + reslts_dim1 * (1 + reslts_dim2 * (1 + reslts_dim3 * 1)
);
reslts -= reslts_offset;
/* Function Body
Extract the timing request from the input line. */
s_copy(path, "Complex precision", (ftnlen)1, (ftnlen)17);
s_copy(path + 1, "SP", (ftnlen)2, (ftnlen)2);
atimin_(path, line, &c__3, subnam, timsub, nout, &info, (ftnlen)3, (
ftnlen)80, (ftnlen)6);
if (info != 0) {
goto L120;
}
/* Check that N*(N+1)/2 <= LA for the input values. */
s_copy(cname, line, (ftnlen)6, (ftnlen)6);
laval[0] = *la;
atimck_(&c__4, cname, nn, &nval[1], &c__1, laval, nout, &info, (ftnlen)6);
if (info > 0) {
io___8.ciunit = *nout;
s_wsfe(&io___8);
do_fio(&c__1, cname, (ftnlen)6);
e_wsfe();
goto L120;
}
/* Do first for UPLO = 'U', then for UPLO = 'L' */
for (iuplo = 1; iuplo <= 2; ++iuplo) {
*(unsigned char *)uplo = *(unsigned char *)&uplos[iuplo - 1];
if (lsame_(uplo, "U")) {
mat = 9;
} else {
mat = -9;
}
/* Do for each value of N in NVAL. */
i__1 = *nn;
for (in = 1; in <= i__1; ++in) {
n = nval[in];
lda = n * (n + 1) / 2;
/* Time CSPTRF */
if (timsub[0]) {
ctimmg_(&mat, &n, &n, &a[1], &lda, &c__0, &c__0);
ic = 0;
s1 = second_();
L10:
csptrf_(uplo, &n, &a[1], &iwork[1], &info);
s2 = second_();
time = s2 - s1;
++ic;
if (time < *timmin) {
ctimmg_(&mat, &n, &n, &a[1], &lda, &c__0, &c__0);
goto L10;
}
/* Subtract the time used in CTIMMG. */
icl = 1;
s1 = second_();
L20:
s2 = second_();
untime = s2 - s1;
++icl;
if (icl <= ic) {
ctimmg_(&mat, &n, &n, &a[1], &lda, &c__0, &c__0);
goto L20;
}
time = (time - untime) / (real) ic;
ops = sopla_("CSPTRF", &n, &n, &c__0, &c__0, &c__0)
;
reslts_ref(1, in, iuplo, 1) = smflop_(&ops, &time, &info);
} else {
ic = 0;
ctimmg_(&mat, &n, &n, &a[1], &lda, &c__0, &c__0);
}
/* Generate another matrix and factor it using CSPTRF so
that the factored form can be used in timing the other
routines. */
if (ic != 1) {
csptrf_(uplo, &n, &a[1], &iwork[1], &info);
}
/* Time CSPTRI */
if (timsub[2]) {
ccopy_(&lda, &a[1], &c__1, &b[1], &c__1);
ic = 0;
s1 = second_();
L30:
csptri_(uplo, &n, &b[1], &iwork[1], &work[1], &info);
s2 = second_();
time = s2 - s1;
++ic;
if (time < *timmin) {
ccopy_(&lda, &a[1], &c__1, &b[1], &c__1);
goto L30;
}
/* Subtract the time used in CCOPY. */
icl = 1;
s1 = second_();
L40:
s2 = second_();
untime = s2 - s1;
++icl;
if (icl <= ic) {
ccopy_(&lda, &a[1], &c__1, &b[1], &c__1);
goto L40;
}
time = (time - untime) / (real) ic;
ops = sopla_("CSPTRI", &n, &n, &c__0, &c__0, &c__0)
;
reslts_ref(1, in, iuplo, 3) = smflop_(&ops, &time, &info);
}
/* Time CSPTRS */
if (timsub[1]) {
i__2 = *nns;
for (i__ = 1; i__ <= i__2; ++i__) {
nrhs = nsval[i__];
ldb = n;
if (ldb % 2 == 0) {
++ldb;
}
ctimmg_(&c__0, &n, &nrhs, &b[1], &ldb, &c__0, &c__0);
ic = 0;
s1 = second_();
L50:
csptrs_(uplo, &n, &nrhs, &a[1], &iwork[1], &b[1], &ldb, &
info);
s2 = second_();
time = s2 - s1;
++ic;
if (time < *timmin) {
ctimmg_(&c__0, &n, &nrhs, &b[1], &ldb, &c__0, &c__0);
goto L50;
}
/* Subtract the time used in CTIMMG. */
icl = 1;
s1 = second_();
L60:
s2 = second_();
untime = s2 - s1;
++icl;
if (icl <= ic) {
ctimmg_(&c__0, &n, &nrhs, &b[1], &ldb, &c__0, &c__0);
goto L60;
}
time = (time - untime) / (real) ic;
ops = sopla_("CSPTRS", &n, &nrhs, &c__0, &c__0, &c__0);
reslts_ref(i__, in, iuplo, 2) = smflop_(&ops, &time, &
info);
/* L70: */
}
}
/* L80: */
}
/* L90: */
}
/* Print tables of results for each timed routine. */
for (isub = 1; isub <= 3; ++isub) {
if (! timsub[isub - 1]) {
goto L110;
}
io___26.ciunit = *nout;
s_wsfe(&io___26);
do_fio(&c__1, subnam_ref(0, isub), (ftnlen)6);
e_wsfe();
for (iuplo = 1; iuplo <= 2; ++iuplo) {
io___27.ciunit = *nout;
s_wsfe(&io___27);
do_fio(&c__1, subnam_ref(0, isub), (ftnlen)6);
do_fio(&c__1, uplos + (iuplo - 1), (ftnlen)1);
e_wsfe();
if (isub == 1) {
sprtbl_(" ", "N", &c__1, laval, nn, &nval[1], &c__1, &
reslts_ref(1, 1, iuplo, 1), ldr1, ldr2, nout, (ftnlen)
1, (ftnlen)1);
} else if (isub == 2) {
sprtbl_("NRHS", "N", nns, &nsval[1], nn, &nval[1], &c__1, &
reslts_ref(1, 1, iuplo, 2), ldr1, ldr2, nout, (ftnlen)
4, (ftnlen)1);
} else if (isub == 3) {
sprtbl_(" ", "N", &c__1, laval, nn, &nval[1], &c__1, &
reslts_ref(1, 1, iuplo, 3), ldr1, ldr2, nout, (ftnlen)
1, (ftnlen)1);
}
/* L100: */
}
L110:
;
}
L120:
return 0;
/* End of CTIMSP */
} /* ctimsp_ */
/* Subroutine */ int cspsvx_(char *fact, char *uplo, integer *n, integer *
nrhs, complex *ap, complex *afp, integer *ipiv, 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;
/* Local variables */
extern logical lsame_(char *, char *);
real anorm;
extern /* Subroutine */ int ccopy_(integer *, complex *, integer *,
complex *, integer *);
extern doublereal slamch_(char *);
logical nofact;
extern /* Subroutine */ int clacpy_(char *, integer *, integer *, complex
*, integer *, complex *, integer *), xerbla_(char *,
integer *);
extern doublereal clansp_(char *, char *, integer *, complex *, real *);
extern /* Subroutine */ int cspcon_(char *, integer *, complex *, integer
*, real *, real *, complex *, integer *), csprfs_(char *,
integer *, integer *, complex *, complex *, integer *, complex *,
integer *, complex *, integer *, real *, real *, complex *, real *
, integer *), csptrf_(char *, integer *, complex *,
integer *, integer *), csptrs_(char *, integer *, integer
*, complex *, integer *, 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 */
/* ======= */
/* CSPSVX uses the diagonal pivoting factorization A = U*D*U**T or */
/* A = L*D*L**T to compute the solution to a complex system of linear */
/* equations A * X = B, where A is an N-by-N symmetric 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 = 'N', the diagonal pivoting method is used to factor A as */
/* A = U * D * U**T, if UPLO = 'U', or */
/* A = L * D * L**T, if UPLO = 'L', */
/* where U (or L) is a product of permutation and unit upper (lower) */
/* triangular matrices and D is symmetric and block diagonal with */
/* 1-by-1 and 2-by-2 diagonal blocks. */
/* 2. If some D(i,i)=0, so that D is exactly singular, 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. */
/* 3. The system of equations is solved for X using the factored form */
/* of A. */
/* 4. Iterative refinement is applied to improve the computed solution */
/* matrix and calculate error bounds and backward error estimates */
/* for it. */
/* Arguments */
/* ========= */
/* FACT (input) CHARACTER*1 */
/* Specifies whether or not the factored form of A has been */
/* supplied on entry. */
/* = 'F': On entry, AFP and IPIV contain the factored form */
/* of A. AP, AFP and IPIV will not be modified. */
/* = 'N': The matrix A will be 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) COMPLEX array, dimension (N*(N+1)/2) */
/* The upper or lower triangle of the symmetric matrix A, packed */
/* columnwise in a linear array. 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)*(2*n-j)/2) = A(i,j) for j<=i<=n. */
/* See below for further details. */
/* 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 block diagonal matrix D and the multipliers used */
/* to obtain the factor U or L from the factorization */
/* A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, stored as */
/* a packed triangular matrix in the same storage format as A. */
/* If FACT = 'N', then AFP is an output argument and on exit */
/* contains the block diagonal matrix D and the multipliers used */
/* to obtain the factor U or L from the factorization */
/* A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, stored as */
/* a packed triangular matrix in the same storage format as A. */
/* IPIV (input or output) INTEGER array, dimension (N) */
/* If FACT = 'F', then IPIV is an input argument and on entry */
/* contains details of the interchanges and the block structure */
/* of D, as determined by CSPTRF. */
/* If IPIV(k) > 0, then rows and columns k and IPIV(k) were */
/* interchanged and D(k,k) is a 1-by-1 diagonal block. */
/* If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and */
/* columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) */
/* is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = */
/* IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were */
/* interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. */
/* If FACT = 'N', then IPIV is an output argument and on exit */
/* contains details of the interchanges and the block structure */
/* of D, as determined by CSPTRF. */
/* B (input) COMPLEX array, dimension (LDB,NRHS) */
/* The N-by-NRHS right hand side matrix 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. */
/* 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. 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: D(i,i) is exactly zero. The factorization */
/* has been completed but the factor D is exactly */
/* singular, so the solution and error bounds could */
/* not be computed. RCOND = 0 is returned. */
/* = N+1: D 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 symmetric matrix A: */
/* a11 a12 a13 a14 */
/* a22 a23 a24 */
/* a33 a34 (aij = 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 .. */
/* Test the input parameters. */
/* Parameter adjustments */
--ap;
--afp;
--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;
--rwork;
/* Function Body */
*info = 0;
nofact = lsame_(fact, "N");
if (! nofact && ! 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 (*ldb < max(1,*n)) {
*info = -9;
} else if (*ldx < max(1,*n)) {
*info = -11;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CSPSVX", &i__1);
return 0;
}
if (nofact) {
/* Compute the factorization A = U*D*U' or A = L*D*L'. */
i__1 = *n * (*n + 1) / 2;
ccopy_(&i__1, &ap[1], &c__1, &afp[1], &c__1);
csptrf_(uplo, n, &afp[1], &ipiv[1], info);
/* Return if INFO is non-zero. */
if (*info > 0) {
*rcond = 0.f;
return 0;
}
}
/* Compute the norm of the matrix A. */
anorm = clansp_("I", uplo, n, &ap[1], &rwork[1]);
/* Compute the reciprocal of the condition number of A. */
cspcon_(uplo, n, &afp[1], &ipiv[1], &anorm, rcond, &work[1], info);
/* Compute the solution vectors X. */
clacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx);
csptrs_(uplo, n, nrhs, &afp[1], &ipiv[1], &x[x_offset], ldx, info);
/* Use iterative refinement to improve the computed solutions and */
/* compute error bounds and backward error estimates for them. */
csprfs_(uplo, n, nrhs, &ap[1], &afp[1], &ipiv[1], &b[b_offset], ldb, &x[
x_offset], ldx, &ferr[1], &berr[1], &work[1], &rwork[1], info);
/* Set INFO = N+1 if the matrix is singular to working precision. */
if (*rcond < slamch_("Epsilon")) {
*info = *n + 1;
}
return 0;
/* End of CSPSVX */
} /* cspsvx_ */
/* Subroutine */ int csprfs_(char *uplo, integer *n, integer *nrhs, complex *
ap, complex *afp, integer *ipiv, complex *b, integer *ldb, complex *x,
integer *ldx, real *ferr, real *berr, complex *work, real *rwork,
integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
Purpose
=======
CSPRFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric indefinite
and packed, and provides error bounds and backward error estimates
for the solution.
Arguments
=========
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
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 matrices B and X. NRHS >= 0.
AP (input) COMPLEX array, dimension (N*(N+1)/2)
The upper or lower triangle of the symmetric matrix A, packed
columnwise in a linear array. 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
AFP (input) COMPLEX array, dimension (N*(N+1)/2)
The factored form of the matrix A. AFP contains the block
diagonal matrix D and the multipliers used to obtain the
factor U or L from the factorization A = U*D*U**T or
A = L*D*L**T as computed by CSPTRF, stored as a packed
triangular matrix.
IPIV (input) INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSPTRF.
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/output) COMPLEX array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by CSPTRS.
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 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
Internal Parameters
===================
ITMAX is the maximum number of steps of iterative refinement.
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static complex c_b1 = {1.f,0.f};
static integer c__1 = 1;
/* 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, r__3, r__4;
complex q__1;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
static integer kase;
static real safe1, safe2;
static integer i__, j, k;
static real s;
extern logical lsame_(char *, char *);
extern /* Subroutine */ int ccopy_(integer *, complex *, integer *,
complex *, integer *), caxpy_(integer *, complex *, complex *,
integer *, complex *, integer *);
static integer count;
extern /* Subroutine */ int cspmv_(char *, integer *, complex *, complex *
, complex *, integer *, complex *, complex *, integer *);
static logical upper;
static integer ik, kk;
extern /* Subroutine */ int clacon_(integer *, complex *, complex *, real
*, integer *);
static real xk;
extern doublereal slamch_(char *);
static integer nz;
static real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
static real lstres;
extern /* Subroutine */ int csptrs_(char *, integer *, integer *, complex
*, integer *, complex *, integer *, integer *);
static real eps;
#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]
#define x_subscr(a_1,a_2) (a_2)*x_dim1 + a_1
#define x_ref(a_1,a_2) x[x_subscr(a_1,a_2)]
--ap;
--afp;
--ipiv;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1 * 1;
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 = -8;
} else if (*ldx < max(1,*n)) {
*info = -10;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CSPRFS", &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 = *n + 1;
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 */
ccopy_(n, &b_ref(1, j), &c__1, &work[1], &c__1);
q__1.r = -1.f, q__1.i = 0.f;
cspmv_(uplo, n, &q__1, &ap[1], &x_ref(1, j), &c__1, &c_b1, &work[1], &
c__1);
/* 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. */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = b_subscr(i__, j);
rwork[i__] = (r__1 = b[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
b_ref(i__, j)), dabs(r__2));
/* L30: */
}
/* Compute abs(A)*abs(X) + abs(B). */
kk = 1;
if (upper) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.f;
i__3 = x_subscr(k, j);
xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&x_ref(k,
j)), dabs(r__2));
ik = kk;
i__3 = k - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = ik;
rwork[i__] += ((r__1 = ap[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&ap[ik]), dabs(r__2))) * xk;
i__4 = ik;
i__5 = x_subscr(i__, j);
s += ((r__1 = ap[i__4].r, dabs(r__1)) + (r__2 = r_imag(&
ap[ik]), dabs(r__2))) * ((r__3 = x[i__5].r, dabs(
r__3)) + (r__4 = r_imag(&x_ref(i__, j)), dabs(
r__4)));
++ik;
/* L40: */
}
i__3 = kk + k - 1;
rwork[k] = rwork[k] + ((r__1 = ap[i__3].r, dabs(r__1)) + (
r__2 = r_imag(&ap[kk + k - 1]), dabs(r__2))) * xk + s;
kk += k;
/* L50: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.f;
i__3 = x_subscr(k, j);
xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&x_ref(k,
j)), dabs(r__2));
i__3 = kk;
rwork[k] += ((r__1 = ap[i__3].r, dabs(r__1)) + (r__2 = r_imag(
&ap[kk]), dabs(r__2))) * xk;
ik = kk + 1;
i__3 = *n;
for (i__ = k + 1; i__ <= i__3; ++i__) {
i__4 = ik;
rwork[i__] += ((r__1 = ap[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&ap[ik]), dabs(r__2))) * xk;
i__4 = ik;
i__5 = x_subscr(i__, j);
s += ((r__1 = ap[i__4].r, dabs(r__1)) + (r__2 = r_imag(&
ap[ik]), dabs(r__2))) * ((r__3 = x[i__5].r, dabs(
r__3)) + (r__4 = r_imag(&x_ref(i__, j)), dabs(
r__4)));
++ik;
/* L60: */
}
rwork[k] += s;
kk += *n - k + 1;
/* L70: */
}
}
s = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
/* Computing MAX */
i__3 = i__;
r__3 = s, r__4 = ((r__1 = work[i__3].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__3 = i__;
r__3 = s, r__4 = ((r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + safe1) / (rwork[i__]
+ safe1);
s = dmax(r__3,r__4);
}
/* L80: */
}
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. */
csptrs_(uplo, n, &c__1, &afp[1], &ipiv[1], &work[1], n, info);
caxpy_(n, &c_b1, &work[1], &c__1, &x_ref(1, j), &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.
Use CLACON to estimate the infinity-norm of the matrix
inv(A) * diag(W),
where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
i__3 = i__;
rwork[i__] = (r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[
i__];
} else {
i__3 = i__;
rwork[i__] = (r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[
i__] + safe1;
}
/* L90: */
}
kase = 0;
L100:
clacon_(n, &work[*n + 1], &work[1], &ferr[j], &kase);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(A'). */
csptrs_(uplo, n, &c__1, &afp[1], &ipiv[1], &work[1], n, info);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
q__1.r = rwork[i__4] * work[i__5].r, q__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
/* L110: */
}
} else if (kase == 2) {
/* Multiply by inv(A)*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
q__1.r = rwork[i__4] * work[i__5].r, q__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
/* L120: */
}
csptrs_(uplo, n, &c__1, &afp[1], &ipiv[1], &work[1], n, info);
}
goto L100;
}
/* Normalize error. */
lstres = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
i__3 = x_subscr(i__, j);
r__3 = lstres, r__4 = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&x_ref(i__, j)), dabs(r__2));
lstres = dmax(r__3,r__4);
/* L130: */
}
if (lstres != 0.f) {
ferr[j] /= lstres;
}
/* L140: */
}
return 0;
/* End of CSPRFS */
} /* csprfs_ */
/* Subroutine */ int cspsv_(char *uplo, integer *n, integer *nrhs, complex *
ap, integer *ipiv, complex *b, integer *ldb, integer *info)
{
/* -- LAPACK driver routine (version 2.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
March 31, 1993
Purpose
=======
CSPSV computes the solution to a complex system of linear equations
A * X = B,
where A is an N-by-N symmetric matrix stored in packed format and X
and B are N-by-NRHS matrices.
The diagonal pivoting method is used to factor A as
A = U * D * U**T, if UPLO = 'U', or
A = L * D * L**T, if UPLO = 'L',
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, D is symmetric and block diagonal with 1-by-1
and 2-by-2 diagonal blocks. The factored form of A is then used to
solve the system of equations A * X = B.
Arguments
=========
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 matrix B. NRHS >= 0.
AP (input/output) COMPLEX array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
A, packed columnwise in a linear array. 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.
On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, stored as
a packed triangular matrix in the same storage format as A.
IPIV (output) INTEGER array, dimension (N)
Details of the interchanges and the block structure of D, as
determined by CSPTRF. If IPIV(k) > 0, then rows and columns
k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
then rows and columns k-1 and -IPIV(k) were interchanged and
D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and
IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
-IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
diagonal block.
B (input/output) COMPLEX array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit, if INFO = 0, the N-by-NRHS solution matrix X.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular, so the solution could not be
computed.
Further Details
===============
The packed storage scheme is illustrated by the following example
when N = 4, UPLO = 'U':
Two-dimensional storage of the symmetric matrix A:
a11 a12 a13 a14
a22 a23 a24
a33 a34 (aij = aji)
a44
Packed storage of the upper triangle of A:
AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
=====================================================================
Test the input parameters.
Parameter adjustments
Function Body */
/* System generated locals */
integer b_dim1, b_offset, i__1;
/* Local variables */
extern logical lsame_(char *, char *);
extern /* Subroutine */ int xerbla_(char *, integer *), csptrf_(
char *, integer *, complex *, integer *, integer *),
csptrs_(char *, integer *, integer *, complex *, integer *,
complex *, integer *, integer *);
#define AP(I) ap[(I)-1]
#define IPIV(I) ipiv[(I)-1]
#define B(I,J) b[(I)-1 + ((J)-1)* ( *ldb)]
*info = 0;
if (! lsame_(uplo, "U") && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*nrhs < 0) {
*info = -3;
} else if (*ldb < max(1,*n)) {
*info = -7;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CSPSV ", &i__1);
return 0;
}
/* Compute the factorization A = U*D*U' or A = L*D*L'. */
csptrf_(uplo, n, &AP(1), &IPIV(1), info);
if (*info == 0) {
/* Solve the system A*X = B, overwriting B with X. */
csptrs_(uplo, n, nrhs, &AP(1), &IPIV(1), &B(1,1), ldb, info);
}
return 0;
/* End of CSPSV */
} /* cspsv_ */
