📄 dlagv2.c
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#include "f2c.h"
#include "netlib.h"
/* Table of constant values */
static integer c__2 = 2;
static integer c__1 = 1;
/* Subroutine */ void dlagv2_(a, lda, b, ldb, alphar, alphai, beta, csl, snl, csr, snr)
doublereal *a;
integer *lda;
doublereal *b;
integer *ldb;
doublereal *alphar, *alphai, *beta, *csl, *snl, *csr, *snr;
{
/* System generated locals */
doublereal d__1;
/* Local variables */
static doublereal r, t, anorm, bnorm, h1, h2, h3, scale1, scale2;
static doublereal ascale, bscale;
static doublereal wi, qq, rr, safmin;
static doublereal wr1, wr2, ulp;
/* -- LAPACK auxiliary routine (version 3.0) -- */
/* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., */
/* Courant Institute, Argonne National Lab, and Rice University */
/* June 30, 1999 */
/* Purpose */
/* ======= */
/* */
/* DLAGV2 computes the Generalized Schur factorization of a real 2-by-2 */
/* matrix pencil (A,B) where B is upper triangular. This routine */
/* computes orthogonal (rotation) matrices given by CSL, SNL and CSR, */
/* SNR such that */
/* */
/* 1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0 */
/* types), then */
/* */
/* [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ] */
/* [ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ] */
/* */
/* [ b11 b12 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ] */
/* [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ], */
/* */
/* 2) if the pencil (A,B) has a pair of complex conjugate eigenvalues, */
/* then */
/* */
/* [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ] */
/* [ a21 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ] */
/* */
/* [ b11 0 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ] */
/* [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ] */
/* */
/* where b11 >= b22 > 0. */
/* */
/* */
/* Arguments */
/* ========= */
/* */
/* A (input/output) DOUBLE PRECISION array, dimension (LDA, 2) */
/* On entry, the 2 x 2 matrix A. */
/* On exit, A is overwritten by the ``A-part'' of the */
/* generalized Schur form. */
/* */
/* LDA (input) INTEGER */
/* THe leading dimension of the array A. LDA >= 2. */
/* */
/* B (input/output) DOUBLE PRECISION array, dimension (LDB, 2) */
/* On entry, the upper triangular 2 x 2 matrix B. */
/* On exit, B is overwritten by the ``B-part'' of the */
/* generalized Schur form. */
/* */
/* LDB (input) INTEGER */
/* THe leading dimension of the array B. LDB >= 2. */
/* */
/* ALPHAR (output) DOUBLE PRECISION array, dimension (2) */
/* ALPHAI (output) DOUBLE PRECISION array, dimension (2) */
/* BETA (output) DOUBLE PRECISION array, dimension (2) */
/* (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the */
/* pencil (A,B), k=1,2, i = sqrt(-1). Note that BETA(k) may */
/* be zero. */
/* */
/* CSL (output) DOUBLE PRECISION */
/* The cosine of the left rotation matrix. */
/* */
/* SNL (output) DOUBLE PRECISION */
/* The sine of the left rotation matrix. */
/* */
/* CSR (output) DOUBLE PRECISION */
/* The cosine of the right rotation matrix. */
/* */
/* SNR (output) DOUBLE PRECISION */
/* The sine of the right rotation matrix. */
/* */
/* Further Details */
/* =============== */
/* */
/* Based on contributions by */
/* Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA */
/* */
/* ===================================================================== */
safmin = dlamch_("S");
ulp = dlamch_("P");
/* Scale A */
anorm = max(max(abs(a[0]) + abs(a[1]), abs(a[*lda]) + abs(a[*lda + 1])), safmin);
ascale = 1. / anorm;
a[0] *= ascale;
a[1] *= ascale;
a[*lda] *= ascale;
a[*lda + 1] *= ascale;
/* Scale B */
bnorm = max(max(abs(b[0]), abs(b[*ldb]) + abs(b[*ldb + 1])), safmin);
bscale = 1. / bnorm;
b[0] *= bscale;
b[*ldb] *= bscale;
b[*ldb + 1] *= bscale;
/* Check if A can be deflated */
if (abs(a[1]) <= ulp) {
*csl = 1.;
*snl = 0.;
*csr = 1.;
*snr = 0.;
a[1] = 0.;
b[1] = 0.;
/* Check if B is singular */
} else if (abs(b[0]) <= ulp) {
dlartg_(&a[0], &a[1], csl, snl, &r);
*csr = 1.;
*snr = 0.;
drot_(&c__2, &a[0], lda, &a[1], lda, csl, snl);
drot_(&c__2, &b[0], ldb, &b[1], ldb, csl, snl);
a[1] = 0.;
b[0] = 0.;
b[1] = 0.;
} else if (abs(b[*ldb + 1]) <= ulp) {
dlartg_(&a[*lda + 1], &a[1], csr, snr, &t);
*snr = -(*snr);
drot_(&c__2, &a[0], &c__1, &a[*lda], &c__1, csr, snr);
drot_(&c__2, &b[0], &c__1, &b[*ldb], &c__1, csr, snr);
*csl = 1.;
*snl = 0.;
a[1] = 0.;
b[1] = 0.;
b[*ldb + 1] = 0.;
} else {
/* B is nonsingular, first compute the eigenvalues of (A,B) */
dlag2_(a, lda, b, ldb, &safmin, &scale1, &scale2, &wr1, &wr2, &wi);
if (wi == 0.) {
/* two real eigenvalues, compute s*A-w*B */
h1 = scale1 * a[0] - wr1 * b[0];
h2 = scale1 * a[*lda ] - wr1 * b[*ldb ];
h3 = scale1 * a[*lda + 1] - wr1 * b[*ldb + 1];
rr = dlapy2_(&h1, &h2);
d__1 = scale1 * a[1];
qq = dlapy2_(&d__1, &h3);
if (rr > qq) {
/* find right rotation matrix to zero 1,1 element of (sA - wB) */
dlartg_(&h2, &h1, csr, snr, &t);
} else {
/* find right rotation matrix to zero 2,1 element of (sA - wB) */
d__1 = scale1 * a[1];
dlartg_(&h3, &d__1, csr, snr, &t);
}
*snr = -(*snr);
drot_(&c__2, &a[0], &c__1, &a[*lda], &c__1, csr, snr);
drot_(&c__2, &b[0], &c__1, &b[*ldb], &c__1, csr, snr);
/* compute inf norms of A and B */
h1 = max(abs(a[0]) + abs(a[*lda]),
abs(a[1]) + abs(a[*lda + 1]));
h2 = max(abs(b[0]) + abs(b[*ldb]),
abs(b[1]) + abs(b[*ldb + 1]));
if (scale1 * h1 >= abs(wr1) * h2) {
/* find left rotation matrix Q to zero out B(2,1) */
dlartg_(&b[0], &b[1], csl, snl, &r);
} else {
/* find left rotation matrix Q to zero out A(2,1) */
dlartg_(&a[0], &a[1], csl, snl, &r);
}
drot_(&c__2, &a[0], lda, &a[1], lda, csl, snl);
drot_(&c__2, &b[0], ldb, &b[1], ldb, csl, snl);
a[1] = 0.;
b[1] = 0.;
} else {
/* a pair of complex conjugate eigenvalues */
/* first compute the SVD of the matrix B */
dlasv2_(&b[0], &b[*ldb], &b[*ldb + 1], &r, &t, snr, csr, snl, csl);
/* Form (A,B) := Q(A,B)Z' where Q is left rotation matrix and */
/* Z is right rotation matrix computed from DLASV2 */
drot_(&c__2, &a[0], lda, &a[1], lda, csl, snl);
drot_(&c__2, &b[0], ldb, &b[1], ldb, csl, snl);
drot_(&c__2, &a[0], &c__1, &a[*lda], &c__1, csr, snr);
drot_(&c__2, &b[0], &c__1, &b[*ldb], &c__1, csr, snr);
b[1] = 0.;
b[*ldb] = 0.;
}
}
/* Unscaling */
a[0] *= anorm;
a[1] *= anorm;
a[*lda] *= anorm;
a[*lda + 1] *= anorm;
b[0] *= bnorm;
b[1] *= bnorm;
b[*ldb] *= bnorm;
b[*ldb + 1] *= bnorm;
if (wi == 0.) {
alphar[0] = a[0];
alphar[1] = a[*lda + 1];
alphai[0] = 0.;
alphai[1] = 0.;
beta[0] = b[0];
beta[1] = b[*ldb + 1];
} else {
alphar[0] = anorm * wr1 / scale1 / bnorm;
alphai[0] = anorm * wi / scale1 / bnorm;
alphar[1] = alphar[0];
alphai[1] = -alphai[0];
beta[0] = 1.;
beta[1] = 1.;
}
} /* dlagv2_ */
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