/* -- translated by f2c (version 20100827).
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"
/* > \brief \b DLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
=========== DOCUMENTATION ===========
Online html documentation available at
http://www.netlib.org/lapack/explore-html/
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f">
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f">
> [TXT]</a>
> \endhtmlonly
Definition:
===========
SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,
LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )
LOGICAL LTRANS
INTEGER INFO, LDA, LDB, LDX, NA, NW
DOUBLE PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), X( LDX, * )
> \par Purpose:
=============
>
> \verbatim
>
> DLALN2 solves a system of the form (ca A - w D ) X = s B
> or (ca A**T - w D) X = s B with possible scaling ("s") and
> perturbation of A. (A**T means A-transpose.)
>
> A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA
> real diagonal matrix, w is a real or complex value, and X and B are
> NA x 1 matrices -- real if w is real, complex if w is complex. NA
> may be 1 or 2.
>
> If w is complex, X and B are represented as NA x 2 matrices,
> the first column of each being the real part and the second
> being the imaginary part.
>
> "s" is a scaling factor (.LE. 1), computed by DLALN2, which is
> so chosen that X can be computed without overflow. X is further
> scaled if necessary to assure that norm(ca A - w D)*norm(X) is less
> than overflow.
>
> If both singular values of (ca A - w D) are less than SMIN,
> SMIN*identity will be used instead of (ca A - w D). If only one
> singular value is less than SMIN, one element of (ca A - w D) will be
> perturbed enough to make the smallest singular value roughly SMIN.
> If both singular values are at least SMIN, (ca A - w D) will not be
> perturbed. In any case, the perturbation will be at most some small
> multiple of max( SMIN, ulp*norm(ca A - w D) ). The singular values
> are computed by infinity-norm approximations, and thus will only be
> correct to a factor of 2 or so.
>
> Note: all input quantities are assumed to be smaller than overflow
> by a reasonable factor. (See BIGNUM.)
> \endverbatim
Arguments:
==========
> \param[in] LTRANS
> \verbatim
> LTRANS is LOGICAL
> =.TRUE.: A-transpose will be used.
> =.FALSE.: A will be used (not transposed.)
> \endverbatim
>
> \param[in] NA
> \verbatim
> NA is INTEGER
> The size of the matrix A. It may (only) be 1 or 2.
> \endverbatim
>
> \param[in] NW
> \verbatim
> NW is INTEGER
> 1 if "w" is real, 2 if "w" is complex. It may only be 1
> or 2.
> \endverbatim
>
> \param[in] SMIN
> \verbatim
> SMIN is DOUBLE PRECISION
> The desired lower bound on the singular values of A. This
> should be a safe distance away from underflow or overflow,
> say, between (underflow/machine precision) and (machine
> precision * overflow ). (See BIGNUM and ULP.)
> \endverbatim
>
> \param[in] CA
> \verbatim
> CA is DOUBLE PRECISION
> The coefficient c, which A is multiplied by.
> \endverbatim
>
> \param[in] A
> \verbatim
> A is DOUBLE PRECISION array, dimension (LDA,NA)
> The NA x NA matrix A.
> \endverbatim
>
> \param[in] LDA
> \verbatim
> LDA is INTEGER
> The leading dimension of A. It must be at least NA.
> \endverbatim
>
> \param[in] D1
> \verbatim
> D1 is DOUBLE PRECISION
> The 1,1 element in the diagonal matrix D.
> \endverbatim
>
> \param[in] D2
> \verbatim
> D2 is DOUBLE PRECISION
> The 2,2 element in the diagonal matrix D. Not used if NW=1.
> \endverbatim
>
> \param[in] B
> \verbatim
> B is DOUBLE PRECISION array, dimension (LDB,NW)
> The NA x NW matrix B (right-hand side). If NW=2 ("w" is
> complex), column 1 contains the real part of B and column 2
> contains the imaginary part.
> \endverbatim
>
> \param[in] LDB
> \verbatim
> LDB is INTEGER
> The leading dimension of B. It must be at least NA.
> \endverbatim
>
> \param[in] WR
> \verbatim
> WR is DOUBLE PRECISION
> The real part of the scalar "w".
> \endverbatim
>
> \param[in] WI
> \verbatim
> WI is DOUBLE PRECISION
> The imaginary part of the scalar "w". Not used if NW=1.
> \endverbatim
>
> \param[out] X
> \verbatim
> X is DOUBLE PRECISION array, dimension (LDX,NW)
> The NA x NW matrix X (unknowns), as computed by DLALN2.
> If NW=2 ("w" is complex), on exit, column 1 will contain
> the real part of X and column 2 will contain the imaginary
> part.
> \endverbatim
>
> \param[in] LDX
> \verbatim
> LDX is INTEGER
> The leading dimension of X. It must be at least NA.
> \endverbatim
>
> \param[out] SCALE
> \verbatim
> SCALE is DOUBLE PRECISION
> The scale factor that B must be multiplied by to insure
> that overflow does not occur when computing X. Thus,
> (ca A - w D) X will be SCALE*B, not B (ignoring
> perturbations of A.) It will be at most 1.
> \endverbatim
>
> \param[out] XNORM
> \verbatim
> XNORM is DOUBLE PRECISION
> The infinity-norm of X, when X is regarded as an NA x NW
> real matrix.
> \endverbatim
>
> \param[out] INFO
> \verbatim
> INFO is INTEGER
> An error flag. It will be set to zero if no error occurs,
> a negative number if an argument is in error, or a positive
> number if ca A - w D had to be perturbed.
> The possible values are:
> = 0: No error occurred, and (ca A - w D) did not have to be
> perturbed.
> = 1: (ca A - w D) had to be perturbed to make its smallest
> (or only) singular value greater than SMIN.
> NOTE: In the interests of speed, this routine does not
> check the inputs for errors.
> \endverbatim
Authors:
========
> \author Univ. of Tennessee
> \author Univ. of California Berkeley
> \author Univ. of Colorado Denver
> \author NAG Ltd.
> \date September 2012
> \ingroup doubleOTHERauxiliary
=====================================================================
Subroutine */ int igraphdlaln2_(logical *ltrans, integer *na, integer *nw,
doublereal *smin, doublereal *ca, doublereal *a, integer *lda,
doublereal *d1, doublereal *d2, doublereal *b, integer *ldb,
doublereal *wr, doublereal *wi, doublereal *x, integer *ldx,
doublereal *scale, doublereal *xnorm, integer *info)
{
/* Initialized data */
static logical zswap[4] = { FALSE_,FALSE_,TRUE_,TRUE_ };
static logical rswap[4] = { FALSE_,TRUE_,FALSE_,TRUE_ };
static integer ipivot[16] /* was [4][4] */ = { 1,2,3,4,2,1,4,3,3,4,1,2,
4,3,2,1 };
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, x_dim1, x_offset;
doublereal d__1, d__2, d__3, d__4, d__5, d__6;
IGRAPH_F77_SAVE doublereal equiv_0[4], equiv_1[4];
/* Local variables */
integer j;
#define ci (equiv_0)
#define cr (equiv_1)
doublereal bi1, bi2, br1, br2, xi1, xi2, xr1, xr2, ci21, ci22, cr21, cr22,
li21, csi, ui11, lr21, ui12, ui22;
#define civ (equiv_0)
doublereal csr, ur11, ur12, ur22;
#define crv (equiv_1)
doublereal bbnd, cmax, ui11r, ui12s, temp, ur11r, ur12s, u22abs;
integer icmax;
doublereal bnorm, cnorm, smini;
extern doublereal igraphdlamch_(char *);
extern /* Subroutine */ int igraphdladiv_(doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *);
doublereal bignum, smlnum;
/* -- LAPACK auxiliary routine (version 3.4.2) --
-- LAPACK is a software package provided by Univ. of Tennessee, --
-- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
September 2012
=====================================================================
Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
/* Function Body
Compute BIGNUM */
smlnum = 2. * igraphdlamch_("Safe minimum");
bignum = 1. / smlnum;
smini = max(*smin,smlnum);
/* Don't check for input errors */
*info = 0;
/* Standard Initializations */
*scale = 1.;
if (*na == 1) {
/* 1 x 1 (i.e., scalar) system C X = B */
if (*nw == 1) {
/* Real 1x1 system.
C = ca A - w D */
csr = *ca * a[a_dim1 + 1] - *wr * *d1;
cnorm = abs(csr);
/* If | C | < SMINI, use C = SMINI */
if (cnorm < smini) {
csr = smini;
cnorm = smini;
*info = 1;
}
/* Check scaling for X = B / C */
bnorm = (d__1 = b[b_dim1 + 1], abs(d__1));
if (cnorm < 1. && bnorm > 1.) {
if (bnorm > bignum * cnorm) {
*scale = 1. / bnorm;
}
}
/* Compute X */
x[x_dim1 + 1] = b[b_dim1 + 1] * *scale / csr;
*xnorm = (d__1 = x[x_dim1 + 1], abs(d__1));
} else {
/* Complex 1x1 system (w is complex)
C = ca A - w D */
csr = *ca * a[a_dim1 + 1] - *wr * *d1;
csi = -(*wi) * *d1;
cnorm = abs(csr) + abs(csi);
/* If | C | < SMINI, use C = SMINI */
if (cnorm < smini) {
csr = smini;
csi = 0.;
cnorm = smini;
*info = 1;
}
/* Check scaling for X = B / C */
bnorm = (d__1 = b[b_dim1 + 1], abs(d__1)) + (d__2 = b[(b_dim1 <<
1) + 1], abs(d__2));
if (cnorm < 1. && bnorm > 1.) {
if (bnorm > bignum * cnorm) {
*scale = 1. / bnorm;
}
}
/* Compute X */
d__1 = *scale * b[b_dim1 + 1];
d__2 = *scale * b[(b_dim1 << 1) + 1];
igraphdladiv_(&d__1, &d__2, &csr, &csi, &x[x_dim1 + 1], &x[(x_dim1 << 1)
+ 1]);
*xnorm = (d__1 = x[x_dim1 + 1], abs(d__1)) + (d__2 = x[(x_dim1 <<
1) + 1], abs(d__2));
}
} else {
/* 2x2 System
Compute the real part of C = ca A - w D (or ca A**T - w D ) */
cr[0] = *ca * a[a_dim1 + 1] - *wr * *d1;
cr[3] = *ca * a[(a_dim1 << 1) + 2] - *wr * *d2;
if (*ltrans) {
cr[2] = *ca * a[a_dim1 + 2];
cr[1] = *ca * a[(a_dim1 << 1) + 1];
} else {
cr[1] = *ca * a[a_dim1 + 2];
cr[2] = *ca * a[(a_dim1 << 1) + 1];
}
if (*nw == 1) {
/* Real 2x2 system (w is real)
Find the largest element in C */
cmax = 0.;
icmax = 0;
for (j = 1; j <= 4; ++j) {
if ((d__1 = crv[j - 1], abs(d__1)) > cmax) {
cmax = (d__1 = crv[j - 1], abs(d__1));
icmax = j;
}
/* L10: */
}
/* If norm(C) < SMINI, use SMINI*identity. */
if (cmax < smini) {
/* Computing MAX */
d__3 = (d__1 = b[b_dim1 + 1], abs(d__1)), d__4 = (d__2 = b[
b_dim1 + 2], abs(d__2));
bnorm = max(d__3,d__4);
if (smini < 1. && bnorm > 1.) {
if (bnorm > bignum * smini) {
*scale = 1. / bnorm;
}
}
temp = *scale / smini;
x[x_dim1 + 1] = temp * b[b_dim1 + 1];
x[x_dim1 + 2] = temp * b[b_dim1 + 2];
*xnorm = temp * bnorm;
*info = 1;
return 0;
}
/* Gaussian elimination with complete pivoting. */
ur11 = crv[icmax - 1];
cr21 = crv[ipivot[(icmax << 2) - 3] - 1];
ur12 = crv[ipivot[(icmax << 2) - 2] - 1];
cr22 = crv[ipivot[(icmax << 2) - 1] - 1];
ur11r = 1. / ur11;
lr21 = ur11r * cr21;
ur22 = cr22 - ur12 * lr21;
/* If smaller pivot < SMINI, use SMINI */
if (abs(ur22) < smini) {
ur22 = smini;
*info = 1;
}
if (rswap[icmax - 1]) {
br1 = b[b_dim1 + 2];
br2 = b[b_dim1 + 1];
} else {
br1 = b[b_dim1 + 1];
br2 = b[b_dim1 + 2];
}
br2 -= lr21 * br1;
/* Computing MAX */
d__2 = (d__1 = br1 * (ur22 * ur11r), abs(d__1)), d__3 = abs(br2);
bbnd = max(d__2,d__3);
if (bbnd > 1. && abs(ur22) < 1.) {
if (bbnd >= bignum * abs(ur22)) {
*scale = 1. / bbnd;
}
}
xr2 = br2 * *scale / ur22;
xr1 = *scale * br1 * ur11r - xr2 * (ur11r * ur12);
if (zswap[icmax - 1]) {
x[x_dim1 + 1] = xr2;
x[x_dim1 + 2] = xr1;
} else {
x[x_dim1 + 1] = xr1;
x[x_dim1 + 2] = xr2;
}
/* Computing MAX */
d__1 = abs(xr1), d__2 = abs(xr2);
*xnorm = max(d__1,d__2);
/* Further scaling if norm(A) norm(X) > overflow */
if (*xnorm > 1. && cmax > 1.) {
if (*xnorm > bignum / cmax) {
temp = cmax / bignum;
x[x_dim1 + 1] = temp * x[x_dim1 + 1];
x[x_dim1 + 2] = temp * x[x_dim1 + 2];
*xnorm = temp * *xnorm;
*scale = temp * *scale;
}
}
} else {
/* Complex 2x2 system (w is complex)
Find the largest element in C */
ci[0] = -(*wi) * *d1;
ci[1] = 0.;
ci[2] = 0.;
ci[3] = -(*wi) * *d2;
cmax = 0.;
icmax = 0;
for (j = 1; j <= 4; ++j) {
if ((d__1 = crv[j - 1], abs(d__1)) + (d__2 = civ[j - 1], abs(
d__2)) > cmax) {
cmax = (d__1 = crv[j - 1], abs(d__1)) + (d__2 = civ[j - 1]
, abs(d__2));
icmax = j;
}
/* L20: */
}
/* If norm(C) < SMINI, use SMINI*identity. */
if (cmax < smini) {
/* Computing MAX */
d__5 = (d__1 = b[b_dim1 + 1], abs(d__1)) + (d__2 = b[(b_dim1
<< 1) + 1], abs(d__2)), d__6 = (d__3 = b[b_dim1 + 2],
abs(d__3)) + (d__4 = b[(b_dim1 << 1) + 2], abs(d__4));
bnorm = max(d__5,d__6);
if (smini < 1. && bnorm > 1.) {
if (bnorm > bignum * smini) {
*scale = 1. / bnorm;
}
}
temp = *scale / smini;
x[x_dim1 + 1] = temp * b[b_dim1 + 1];
x[x_dim1 + 2] = temp * b[b_dim1 + 2];
x[(x_dim1 << 1) + 1] = temp * b[(b_dim1 << 1) + 1];
x[(x_dim1 << 1) + 2] = temp * b[(b_dim1 << 1) + 2];
*xnorm = temp * bnorm;
*info = 1;
return 0;
}
/* Gaussian elimination with complete pivoting. */
ur11 = crv[icmax - 1];
ui11 = civ[icmax - 1];
cr21 = crv[ipivot[(icmax << 2) - 3] - 1];
ci21 = civ[ipivot[(icmax << 2) - 3] - 1];
ur12 = crv[ipivot[(icmax << 2) - 2] - 1];
ui12 = civ[ipivot[(icmax << 2) - 2] - 1];
cr22 = crv[ipivot[(icmax << 2) - 1] - 1];
ci22 = civ[ipivot[(icmax << 2) - 1] - 1];
if (icmax == 1 || icmax == 4) {
/* Code when off-diagonals of pivoted C are real */
if (abs(ur11) > abs(ui11)) {
temp = ui11 / ur11;
/* Computing 2nd power */
d__1 = temp;
ur11r = 1. / (ur11 * (d__1 * d__1 + 1.));
ui11r = -temp * ur11r;
} else {
temp = ur11 / ui11;
/* Computing 2nd power */
d__1 = temp;
ui11r = -1. / (ui11 * (d__1 * d__1 + 1.));
ur11r = -temp * ui11r;
}
lr21 = cr21 * ur11r;
li21 = cr21 * ui11r;
ur12s = ur12 * ur11r;
ui12s = ur12 * ui11r;
ur22 = cr22 - ur12 * lr21;
ui22 = ci22 - ur12 * li21;
} else {
/* Code when diagonals of pivoted C are real */
ur11r = 1. / ur11;
ui11r = 0.;
lr21 = cr21 * ur11r;
li21 = ci21 * ur11r;
ur12s = ur12 * ur11r;
ui12s = ui12 * ur11r;
ur22 = cr22 - ur12 * lr21 + ui12 * li21;
ui22 = -ur12 * li21 - ui12 * lr21;
}
u22abs = abs(ur22) + abs(ui22);
/* If smaller pivot < SMINI, use SMINI */
if (u22abs < smini) {
ur22 = smini;
ui22 = 0.;
*info = 1;
}
if (rswap[icmax - 1]) {
br2 = b[b_dim1 + 1];
br1 = b[b_dim1 + 2];
bi2 = b[(b_dim1 << 1) + 1];
bi1 = b[(b_dim1 << 1) + 2];
} else {
br1 = b[b_dim1 + 1];
br2 = b[b_dim1 + 2];
bi1 = b[(b_dim1 << 1) + 1];
bi2 = b[(b_dim1 << 1) + 2];
}
br2 = br2 - lr21 * br1 + li21 * bi1;
bi2 = bi2 - li21 * br1 - lr21 * bi1;
/* Computing MAX */
d__1 = (abs(br1) + abs(bi1)) * (u22abs * (abs(ur11r) + abs(ui11r))
), d__2 = abs(br2) + abs(bi2);
bbnd = max(d__1,d__2);
if (bbnd > 1. && u22abs < 1.) {
if (bbnd >= bignum * u22abs) {
*scale = 1. / bbnd;
br1 = *scale * br1;
bi1 = *scale * bi1;
br2 = *scale * br2;
bi2 = *scale * bi2;
}
}
igraphdladiv_(&br2, &bi2, &ur22, &ui22, &xr2, &xi2);
xr1 = ur11r * br1 - ui11r * bi1 - ur12s * xr2 + ui12s * xi2;
xi1 = ui11r * br1 + ur11r * bi1 - ui12s * xr2 - ur12s * xi2;
if (zswap[icmax - 1]) {
x[x_dim1 + 1] = xr2;
x[x_dim1 + 2] = xr1;
x[(x_dim1 << 1) + 1] = xi2;
x[(x_dim1 << 1) + 2] = xi1;
} else {
x[x_dim1 + 1] = xr1;
x[x_dim1 + 2] = xr2;
x[(x_dim1 << 1) + 1] = xi1;
x[(x_dim1 << 1) + 2] = xi2;
}
/* Computing MAX */
d__1 = abs(xr1) + abs(xi1), d__2 = abs(xr2) + abs(xi2);
*xnorm = max(d__1,d__2);
/* Further scaling if norm(A) norm(X) > overflow */
if (*xnorm > 1. && cmax > 1.) {
if (*xnorm > bignum / cmax) {
temp = cmax / bignum;
x[x_dim1 + 1] = temp * x[x_dim1 + 1];
x[x_dim1 + 2] = temp * x[x_dim1 + 2];
x[(x_dim1 << 1) + 1] = temp * x[(x_dim1 << 1) + 1];
x[(x_dim1 << 1) + 2] = temp * x[(x_dim1 << 1) + 2];
*xnorm = temp * *xnorm;
*scale = temp * *scale;
}
}
}
}
return 0;
/* End of DLALN2 */
} /* igraphdlaln2_ */
#undef crv
#undef civ
#undef cr
#undef ci