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references/igraph-1.0.1/vendor/lapack/dgebal.c
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/* -- translated by f2c (version 20240504).
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"
/* Table of constant values */
static integer c__1 = 1;
/* > \brief \b DGEBAL
=========== DOCUMENTATION ===========
Online html documentation available at
http://www.netlib.org/lapack/explore-html/
> \htmlonly
> Download DGEBAL + dependencies
> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebal.
f">
> [TGZ]</a>
> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebal.
f">
> [ZIP]</a>
> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebal.
f">
> [TXT]</a>
> \endhtmlonly
Definition:
===========
SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
CHARACTER JOB
INTEGER IHI, ILO, INFO, LDA, N
DOUBLE PRECISION A( LDA, * ), SCALE( * )
> \par Purpose:
=============
>
> \verbatim
>
> DGEBAL balances a general real matrix A. This involves, first,
> permuting A by a similarity transformation to isolate eigenvalues
> in the first 1 to ILO-1 and last IHI+1 to N elements on the
> diagonal; and second, applying a diagonal similarity transformation
> to rows and columns ILO to IHI to make the rows and columns as
> close in norm as possible. Both steps are optional.
>
> Balancing may reduce the 1-norm of the matrix, and improve the
> accuracy of the computed eigenvalues and/or eigenvectors.
> \endverbatim
Arguments:
==========
> \param[in] JOB
> \verbatim
> JOB is CHARACTER*1
> Specifies the operations to be performed on A:
> = 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0
> for i = 1,...,N;
> = 'P': permute only;
> = 'S': scale only;
> = 'B': both permute and scale.
> \endverbatim
>
> \param[in] N
> \verbatim
> N is INTEGER
> The order of the matrix A. N >= 0.
> \endverbatim
>
> \param[in,out] A
> \verbatim
> A is DOUBLE array, dimension (LDA,N)
> On entry, the input matrix A.
> On exit, A is overwritten by the balanced matrix.
> If JOB = 'N', A is not referenced.
> See Further Details.
> \endverbatim
>
> \param[in] LDA
> \verbatim
> LDA is INTEGER
> The leading dimension of the array A. LDA >= max(1,N).
> \endverbatim
>
> \param[out] ILO
> \verbatim
> ILO is INTEGER
> \endverbatim
> \param[out] IHI
> \verbatim
> IHI is INTEGER
> ILO and IHI are set to integers such that on exit
> A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
> If JOB = 'N' or 'S', ILO = 1 and IHI = N.
> \endverbatim
>
> \param[out] SCALE
> \verbatim
> SCALE is DOUBLE array, dimension (N)
> Details of the permutations and scaling factors applied to
> A. If P(j) is the index of the row and column interchanged
> with row and column j and D(j) is the scaling factor
> applied to row and column j, then
> SCALE(j) = P(j) for j = 1,...,ILO-1
> = D(j) for j = ILO,...,IHI
> = P(j) for j = IHI+1,...,N.
> The order in which the interchanges are made is N to IHI+1,
> then 1 to ILO-1.
> \endverbatim
>
> \param[out] INFO
> \verbatim
> INFO is INTEGER
> = 0: successful exit.
> < 0: if INFO = -i, the i-th argument had an illegal value.
> \endverbatim
Authors:
========
> \author Univ. of Tennessee
> \author Univ. of California Berkeley
> \author Univ. of Colorado Denver
> \author NAG Ltd.
> \date November 2013
> \ingroup doubleGEcomputational
> \par Further Details:
=====================
>
> \verbatim
>
> The permutations consist of row and column interchanges which put
> the matrix in the form
>
> ( T1 X Y )
> P A P = ( 0 B Z )
> ( 0 0 T2 )
>
> where T1 and T2 are upper triangular matrices whose eigenvalues lie
> along the diagonal. The column indices ILO and IHI mark the starting
> and ending columns of the submatrix B. Balancing consists of applying
> a diagonal similarity transformation inv(D) * B * D to make the
> 1-norms of each row of B and its corresponding column nearly equal.
> The output matrix is
>
> ( T1 X*D Y )
> ( 0 inv(D)*B*D inv(D)*Z ).
> ( 0 0 T2 )
>
> Information about the permutations P and the diagonal matrix D is
> returned in the vector SCALE.
>
> This subroutine is based on the EISPACK routine BALANC.
>
> Modified by Tzu-Yi Chen, Computer Science Division, University of
> California at Berkeley, USA
> \endverbatim
>
=====================================================================
Subroutine */ int igraphdgebal_(char *job, integer *n, doublereal *a, integer *
lda, integer *ilo, integer *ihi, doublereal *scale, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2;
doublereal d__1, d__2;
/* Local variables */
doublereal c__, f, g;
integer i__, j, k, l, m;
doublereal r__, s, ca, ra;
integer ica, ira, iexc;
extern doublereal igraphdnrm2_(integer *, doublereal *, integer *);
extern /* Subroutine */ int igraphdscal_(integer *, doublereal *, doublereal *,
integer *);
extern logical igraphlsame_(char *, char *);
extern /* Subroutine */ int igraphdswap_(integer *, doublereal *, integer *,
doublereal *, integer *);
doublereal sfmin1, sfmin2, sfmax1, sfmax2;
extern doublereal igraphdlamch_(char *);
extern integer igraphidamax_(integer *, doublereal *, integer *);
extern logical igraphdisnan_(doublereal *);
extern /* Subroutine */ int igraphxerbla_(char *, integer *, ftnlen);
logical noconv;
/* -- LAPACK computational routine (version 3.5.0) --
-- LAPACK is a software package provided by Univ. of Tennessee, --
-- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
November 2013
=====================================================================
Test the input parameters
Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--scale;
/* Function Body */
*info = 0;
if (! igraphlsame_(job, "N") && ! igraphlsame_(job, "P") && ! igraphlsame_(job, "S")
&& ! igraphlsame_(job, "B")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*n)) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
igraphxerbla_("DGEBAL", &i__1, (ftnlen)6);
return 0;
}
k = 1;
l = *n;
if (*n == 0) {
goto L210;
}
if (igraphlsame_(job, "N")) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
scale[i__] = 1.;
/* L10: */
}
goto L210;
}
if (igraphlsame_(job, "S")) {
goto L120;
}
/* Permutation to isolate eigenvalues if possible */
goto L50;
/* Row and column exchange. */
L20:
scale[m] = (doublereal) j;
if (j == m) {
goto L30;
}
igraphdswap_(&l, &a[j * a_dim1 + 1], &c__1, &a[m * a_dim1 + 1], &c__1);
i__1 = *n - k + 1;
igraphdswap_(&i__1, &a[j + k * a_dim1], lda, &a[m + k * a_dim1], lda);
L30:
switch (iexc) {
case 1: goto L40;
case 2: goto L80;
}
/* Search for rows isolating an eigenvalue and push them down. */
L40:
if (l == 1) {
goto L210;
}
--l;
L50:
for (j = l; j >= 1; --j) {
i__1 = l;
for (i__ = 1; i__ <= i__1; ++i__) {
if (i__ == j) {
goto L60;
}
if (a[j + i__ * a_dim1] != 0.) {
goto L70;
}
L60:
;
}
m = l;
iexc = 1;
goto L20;
L70:
;
}
goto L90;
/* Search for columns isolating an eigenvalue and push them left. */
L80:
++k;
L90:
i__1 = l;
for (j = k; j <= i__1; ++j) {
i__2 = l;
for (i__ = k; i__ <= i__2; ++i__) {
if (i__ == j) {
goto L100;
}
if (a[i__ + j * a_dim1] != 0.) {
goto L110;
}
L100:
;
}
m = k;
iexc = 2;
goto L20;
L110:
;
}
L120:
i__1 = l;
for (i__ = k; i__ <= i__1; ++i__) {
scale[i__] = 1.;
/* L130: */
}
if (igraphlsame_(job, "P")) {
goto L210;
}
/* Balance the submatrix in rows K to L.
Iterative loop for norm reduction */
sfmin1 = igraphdlamch_("S") / igraphdlamch_("P");
sfmax1 = 1. / sfmin1;
sfmin2 = sfmin1 * 2.;
sfmax2 = 1. / sfmin2;
L140:
noconv = FALSE_;
i__1 = l;
for (i__ = k; i__ <= i__1; ++i__) {
i__2 = l - k + 1;
c__ = igraphdnrm2_(&i__2, &a[k + i__ * a_dim1], &c__1);
i__2 = l - k + 1;
r__ = igraphdnrm2_(&i__2, &a[i__ + k * a_dim1], lda);
ica = igraphidamax_(&l, &a[i__ * a_dim1 + 1], &c__1);
ca = (d__1 = a[ica + i__ * a_dim1], abs(d__1));
i__2 = *n - k + 1;
ira = igraphidamax_(&i__2, &a[i__ + k * a_dim1], lda);
ra = (d__1 = a[i__ + (ira + k - 1) * a_dim1], abs(d__1));
/* Guard against zero C or R due to underflow. */
if (c__ == 0. || r__ == 0.) {
goto L200;
}
g = r__ / 2.;
f = 1.;
s = c__ + r__;
L160:
/* Computing MAX */
d__1 = max(f,c__);
/* Computing MIN */
d__2 = min(r__,g);
if (c__ >= g || max(d__1,ca) >= sfmax2 || min(d__2,ra) <= sfmin2) {
goto L170;
}
d__1 = c__ + f + ca + r__ + g + ra;
if (igraphdisnan_(&d__1)) {
/* Exit if NaN to avoid infinite loop */
*info = -3;
i__2 = -(*info);
igraphxerbla_("DGEBAL", &i__2, (ftnlen)6);
return 0;
}
f *= 2.;
c__ *= 2.;
ca *= 2.;
r__ /= 2.;
g /= 2.;
ra /= 2.;
goto L160;
L170:
g = c__ / 2.;
L180:
/* Computing MIN */
d__1 = min(f,c__), d__1 = min(d__1,g);
if (g < r__ || max(r__,ra) >= sfmax2 || min(d__1,ca) <= sfmin2) {
goto L190;
}
f /= 2.;
c__ /= 2.;
g /= 2.;
ca /= 2.;
r__ *= 2.;
ra *= 2.;
goto L180;
/* Now balance. */
L190:
if (c__ + r__ >= s * .95) {
goto L200;
}
if (f < 1. && scale[i__] < 1.) {
if (f * scale[i__] <= sfmin1) {
goto L200;
}
}
if (f > 1. && scale[i__] > 1.) {
if (scale[i__] >= sfmax1 / f) {
goto L200;
}
}
g = 1. / f;
scale[i__] *= f;
noconv = TRUE_;
i__2 = *n - k + 1;
igraphdscal_(&i__2, &g, &a[i__ + k * a_dim1], lda);
igraphdscal_(&l, &f, &a[i__ * a_dim1 + 1], &c__1);
L200:
;
}
if (noconv) {
goto L140;
}
L210:
*ilo = k;
*ihi = l;
return 0;
/* End of DGEBAL */
} /* igraphdgebal_ */