1 /* ./src_f77/dlagv2.f -- translated by f2c (version 20030320).
2 You must link the resulting object file with the libraries:
3 -lf2c -lm (in that order)
4 */
5
6 #include <punc/vf2c.h>
7
8 /* Table of constant values */
9
10 static integer c__2 = 2;
11 static integer c__1 = 1;
12
dlagv2_(doublereal * a,integer * lda,doublereal * b,integer * ldb,doublereal * alphar,doublereal * alphai,doublereal * beta,doublereal * csl,doublereal * snl,doublereal * csr,doublereal * snr)13 /* Subroutine */ int dlagv2_(doublereal *a, integer *lda, doublereal *b,
14 integer *ldb, doublereal *alphar, doublereal *alphai, doublereal *
15 beta, doublereal *csl, doublereal *snl, doublereal *csr, doublereal *
16 snr)
17 {
18 /* System generated locals */
19 integer a_dim1, a_offset, b_dim1, b_offset;
20 doublereal d__1, d__2, d__3, d__4, d__5, d__6;
21
22 /* Local variables */
23 static doublereal r__, t, h1, h2, h3, wi, qq, rr, wr1, wr2, ulp;
24 extern /* Subroutine */ int drot_(integer *, doublereal *, integer *,
25 doublereal *, integer *, doublereal *, doublereal *), dlag2_(
26 doublereal *, integer *, doublereal *, integer *, doublereal *,
27 doublereal *, doublereal *, doublereal *, doublereal *,
28 doublereal *);
29 static doublereal anorm, bnorm, scale1, scale2;
30 extern /* Subroutine */ int dlasv2_(doublereal *, doublereal *,
31 doublereal *, doublereal *, doublereal *, doublereal *,
32 doublereal *, doublereal *, doublereal *);
33 extern doublereal dlapy2_(doublereal *, doublereal *);
34 static doublereal ascale, bscale;
35 extern doublereal dlamch_(char *, ftnlen);
36 static doublereal safmin;
37 extern /* Subroutine */ int dlartg_(doublereal *, doublereal *,
38 doublereal *, doublereal *, doublereal *);
39
40
41 /* -- LAPACK auxiliary routine (version 3.0) -- */
42 /* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., */
43 /* Courant Institute, Argonne National Lab, and Rice University */
44 /* June 30, 1999 */
45
46 /* .. Scalar Arguments .. */
47 /* .. */
48 /* .. Array Arguments .. */
49 /* .. */
50
51 /* Purpose */
52 /* ======= */
53
54 /* DLAGV2 computes the Generalized Schur factorization of a real 2-by-2 */
55 /* matrix pencil (A,B) where B is upper triangular. This routine */
56 /* computes orthogonal (rotation) matrices given by CSL, SNL and CSR, */
57 /* SNR such that */
58
59 /* 1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0 */
60 /* types), then */
61
62 /* [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ] */
63 /* [ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ] */
64
65 /* [ b11 b12 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ] */
66 /* [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ], */
67
68 /* 2) if the pencil (A,B) has a pair of complex conjugate eigenvalues, */
69 /* then */
70
71 /* [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ] */
72 /* [ a21 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ] */
73
74 /* [ b11 0 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ] */
75 /* [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ] */
76
77 /* where b11 >= b22 > 0. */
78
79
80 /* Arguments */
81 /* ========= */
82
83 /* A (input/output) DOUBLE PRECISION array, dimension (LDA, 2) */
84 /* On entry, the 2 x 2 matrix A. */
85 /* On exit, A is overwritten by the ``A-part'' of the */
86 /* generalized Schur form. */
87
88 /* LDA (input) INTEGER */
89 /* THe leading dimension of the array A. LDA >= 2. */
90
91 /* B (input/output) DOUBLE PRECISION array, dimension (LDB, 2) */
92 /* On entry, the upper triangular 2 x 2 matrix B. */
93 /* On exit, B is overwritten by the ``B-part'' of the */
94 /* generalized Schur form. */
95
96 /* LDB (input) INTEGER */
97 /* THe leading dimension of the array B. LDB >= 2. */
98
99 /* ALPHAR (output) DOUBLE PRECISION array, dimension (2) */
100 /* ALPHAI (output) DOUBLE PRECISION array, dimension (2) */
101 /* BETA (output) DOUBLE PRECISION array, dimension (2) */
102 /* (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the */
103 /* pencil (A,B), k=1,2, i = sqrt(-1). Note that BETA(k) may */
104 /* be zero. */
105
106 /* CSL (output) DOUBLE PRECISION */
107 /* The cosine of the left rotation matrix. */
108
109 /* SNL (output) DOUBLE PRECISION */
110 /* The sine of the left rotation matrix. */
111
112 /* CSR (output) DOUBLE PRECISION */
113 /* The cosine of the right rotation matrix. */
114
115 /* SNR (output) DOUBLE PRECISION */
116 /* The sine of the right rotation matrix. */
117
118 /* Further Details */
119 /* =============== */
120
121 /* Based on contributions by */
122 /* Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA */
123
124 /* ===================================================================== */
125
126 /* .. Parameters .. */
127 /* .. */
128 /* .. Local Scalars .. */
129 /* .. */
130 /* .. External Subroutines .. */
131 /* .. */
132 /* .. External Functions .. */
133 /* .. */
134 /* .. Intrinsic Functions .. */
135 /* .. */
136 /* .. Executable Statements .. */
137
138 /* Parameter adjustments */
139 a_dim1 = *lda;
140 a_offset = 1 + a_dim1;
141 a -= a_offset;
142 b_dim1 = *ldb;
143 b_offset = 1 + b_dim1;
144 b -= b_offset;
145 --alphar;
146 --alphai;
147 --beta;
148
149 /* Function Body */
150 safmin = dlamch_("S", (ftnlen)1);
151 ulp = dlamch_("P", (ftnlen)1);
152
153 /* Scale A */
154
155 /* Computing MAX */
156 d__5 = (d__1 = a[a_dim1 + 1], abs(d__1)) + (d__2 = a[a_dim1 + 2], abs(
157 d__2)), d__6 = (d__3 = a[(a_dim1 << 1) + 1], abs(d__3)) + (d__4 =
158 a[(a_dim1 << 1) + 2], abs(d__4)), d__5 = max(d__5,d__6);
159 anorm = max(d__5,safmin);
160 ascale = 1. / anorm;
161 a[a_dim1 + 1] = ascale * a[a_dim1 + 1];
162 a[(a_dim1 << 1) + 1] = ascale * a[(a_dim1 << 1) + 1];
163 a[a_dim1 + 2] = ascale * a[a_dim1 + 2];
164 a[(a_dim1 << 1) + 2] = ascale * a[(a_dim1 << 1) + 2];
165
166 /* Scale B */
167
168 /* Computing MAX */
169 d__4 = (d__3 = b[b_dim1 + 1], abs(d__3)), d__5 = (d__1 = b[(b_dim1 << 1)
170 + 1], abs(d__1)) + (d__2 = b[(b_dim1 << 1) + 2], abs(d__2)), d__4
171 = max(d__4,d__5);
172 bnorm = max(d__4,safmin);
173 bscale = 1. / bnorm;
174 b[b_dim1 + 1] = bscale * b[b_dim1 + 1];
175 b[(b_dim1 << 1) + 1] = bscale * b[(b_dim1 << 1) + 1];
176 b[(b_dim1 << 1) + 2] = bscale * b[(b_dim1 << 1) + 2];
177
178 /* Check if A can be deflated */
179
180 if ((d__1 = a[a_dim1 + 2], abs(d__1)) <= ulp) {
181 *csl = 1.;
182 *snl = 0.;
183 *csr = 1.;
184 *snr = 0.;
185 a[a_dim1 + 2] = 0.;
186 b[b_dim1 + 2] = 0.;
187
188 /* Check if B is singular */
189
190 } else if ((d__1 = b[b_dim1 + 1], abs(d__1)) <= ulp) {
191 dlartg_(&a[a_dim1 + 1], &a[a_dim1 + 2], csl, snl, &r__);
192 *csr = 1.;
193 *snr = 0.;
194 drot_(&c__2, &a[a_dim1 + 1], lda, &a[a_dim1 + 2], lda, csl, snl);
195 drot_(&c__2, &b[b_dim1 + 1], ldb, &b[b_dim1 + 2], ldb, csl, snl);
196 a[a_dim1 + 2] = 0.;
197 b[b_dim1 + 1] = 0.;
198 b[b_dim1 + 2] = 0.;
199
200 } else if ((d__1 = b[(b_dim1 << 1) + 2], abs(d__1)) <= ulp) {
201 dlartg_(&a[(a_dim1 << 1) + 2], &a[a_dim1 + 2], csr, snr, &t);
202 *snr = -(*snr);
203 drot_(&c__2, &a[a_dim1 + 1], &c__1, &a[(a_dim1 << 1) + 1], &c__1, csr,
204 snr);
205 drot_(&c__2, &b[b_dim1 + 1], &c__1, &b[(b_dim1 << 1) + 1], &c__1, csr,
206 snr);
207 *csl = 1.;
208 *snl = 0.;
209 a[a_dim1 + 2] = 0.;
210 b[b_dim1 + 2] = 0.;
211 b[(b_dim1 << 1) + 2] = 0.;
212
213 } else {
214
215 /* B is nonsingular, first compute the eigenvalues of (A,B) */
216
217 dlag2_(&a[a_offset], lda, &b[b_offset], ldb, &safmin, &scale1, &
218 scale2, &wr1, &wr2, &wi);
219
220 if (wi == 0.) {
221
222 /* two real eigenvalues, compute s*A-w*B */
223
224 h1 = scale1 * a[a_dim1 + 1] - wr1 * b[b_dim1 + 1];
225 h2 = scale1 * a[(a_dim1 << 1) + 1] - wr1 * b[(b_dim1 << 1) + 1];
226 h3 = scale1 * a[(a_dim1 << 1) + 2] - wr1 * b[(b_dim1 << 1) + 2];
227
228 rr = dlapy2_(&h1, &h2);
229 d__1 = scale1 * a[a_dim1 + 2];
230 qq = dlapy2_(&d__1, &h3);
231
232 if (rr > qq) {
233
234 /* find right rotation matrix to zero 1,1 element of */
235 /* (sA - wB) */
236
237 dlartg_(&h2, &h1, csr, snr, &t);
238
239 } else {
240
241 /* find right rotation matrix to zero 2,1 element of */
242 /* (sA - wB) */
243
244 d__1 = scale1 * a[a_dim1 + 2];
245 dlartg_(&h3, &d__1, csr, snr, &t);
246
247 }
248
249 *snr = -(*snr);
250 drot_(&c__2, &a[a_dim1 + 1], &c__1, &a[(a_dim1 << 1) + 1], &c__1,
251 csr, snr);
252 drot_(&c__2, &b[b_dim1 + 1], &c__1, &b[(b_dim1 << 1) + 1], &c__1,
253 csr, snr);
254
255 /* compute inf norms of A and B */
256
257 /* Computing MAX */
258 d__5 = (d__1 = a[a_dim1 + 1], abs(d__1)) + (d__2 = a[(a_dim1 << 1)
259 + 1], abs(d__2)), d__6 = (d__3 = a[a_dim1 + 2], abs(d__3)
260 ) + (d__4 = a[(a_dim1 << 1) + 2], abs(d__4));
261 h1 = max(d__5,d__6);
262 /* Computing MAX */
263 d__5 = (d__1 = b[b_dim1 + 1], abs(d__1)) + (d__2 = b[(b_dim1 << 1)
264 + 1], abs(d__2)), d__6 = (d__3 = b[b_dim1 + 2], abs(d__3)
265 ) + (d__4 = b[(b_dim1 << 1) + 2], abs(d__4));
266 h2 = max(d__5,d__6);
267
268 if (scale1 * h1 >= abs(wr1) * h2) {
269
270 /* find left rotation matrix Q to zero out B(2,1) */
271
272 dlartg_(&b[b_dim1 + 1], &b[b_dim1 + 2], csl, snl, &r__);
273
274 } else {
275
276 /* find left rotation matrix Q to zero out A(2,1) */
277
278 dlartg_(&a[a_dim1 + 1], &a[a_dim1 + 2], csl, snl, &r__);
279
280 }
281
282 drot_(&c__2, &a[a_dim1 + 1], lda, &a[a_dim1 + 2], lda, csl, snl);
283 drot_(&c__2, &b[b_dim1 + 1], ldb, &b[b_dim1 + 2], ldb, csl, snl);
284
285 a[a_dim1 + 2] = 0.;
286 b[b_dim1 + 2] = 0.;
287
288 } else {
289
290 /* a pair of complex conjugate eigenvalues */
291 /* first compute the SVD of the matrix B */
292
293 dlasv2_(&b[b_dim1 + 1], &b[(b_dim1 << 1) + 1], &b[(b_dim1 << 1) +
294 2], &r__, &t, snr, csr, snl, csl);
295
296 /* Form (A,B) := Q(A,B)Z' where Q is left rotation matrix and */
297 /* Z is right rotation matrix computed from DLASV2 */
298
299 drot_(&c__2, &a[a_dim1 + 1], lda, &a[a_dim1 + 2], lda, csl, snl);
300 drot_(&c__2, &b[b_dim1 + 1], ldb, &b[b_dim1 + 2], ldb, csl, snl);
301 drot_(&c__2, &a[a_dim1 + 1], &c__1, &a[(a_dim1 << 1) + 1], &c__1,
302 csr, snr);
303 drot_(&c__2, &b[b_dim1 + 1], &c__1, &b[(b_dim1 << 1) + 1], &c__1,
304 csr, snr);
305
306 b[b_dim1 + 2] = 0.;
307 b[(b_dim1 << 1) + 1] = 0.;
308
309 }
310
311 }
312
313 /* Unscaling */
314
315 a[a_dim1 + 1] = anorm * a[a_dim1 + 1];
316 a[a_dim1 + 2] = anorm * a[a_dim1 + 2];
317 a[(a_dim1 << 1) + 1] = anorm * a[(a_dim1 << 1) + 1];
318 a[(a_dim1 << 1) + 2] = anorm * a[(a_dim1 << 1) + 2];
319 b[b_dim1 + 1] = bnorm * b[b_dim1 + 1];
320 b[b_dim1 + 2] = bnorm * b[b_dim1 + 2];
321 b[(b_dim1 << 1) + 1] = bnorm * b[(b_dim1 << 1) + 1];
322 b[(b_dim1 << 1) + 2] = bnorm * b[(b_dim1 << 1) + 2];
323
324 if (wi == 0.) {
325 alphar[1] = a[a_dim1 + 1];
326 alphar[2] = a[(a_dim1 << 1) + 2];
327 alphai[1] = 0.;
328 alphai[2] = 0.;
329 beta[1] = b[b_dim1 + 1];
330 beta[2] = b[(b_dim1 << 1) + 2];
331 } else {
332 alphar[1] = anorm * wr1 / scale1 / bnorm;
333 alphai[1] = anorm * wi / scale1 / bnorm;
334 alphar[2] = alphar[1];
335 alphai[2] = -alphai[1];
336 beta[1] = 1.;
337 beta[2] = 1.;
338 }
339
340 /* L10: */
341
342 return 0;
343
344 /* End of DLAGV2 */
345
346 } /* dlagv2_ */
347
348