1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
|
/* dlanv2.f -- translated by f2c (version 20061008).
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"
#include "blaswrap.h"
/* Table of constant values */
static doublereal c_b4 = 1.;
/* Subroutine */ int dlanv2_(doublereal *a, doublereal *b, doublereal *c__,
doublereal *d__, doublereal *rt1r, doublereal *rt1i, doublereal *rt2r,
doublereal *rt2i, doublereal *cs, doublereal *sn)
{
/* System generated locals */
doublereal d__1, d__2;
/* Builtin functions */
double d_sign(doublereal *, doublereal *), sqrt(doublereal);
/* Local variables */
doublereal p, z__, aa, bb, cc, dd, cs1, sn1, sab, sac, eps, tau, temp,
scale, bcmax, bcmis, sigma;
extern doublereal dlapy2_(doublereal *, doublereal *), dlamch_(char *);
/* -- LAPACK driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric */
/* matrix in standard form: */
/* [ A B ] = [ CS -SN ] [ AA BB ] [ CS SN ] */
/* [ C D ] [ SN CS ] [ CC DD ] [-SN CS ] */
/* where either */
/* 1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or */
/* 2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex */
/* conjugate eigenvalues. */
/* Arguments */
/* ========= */
/* A (input/output) DOUBLE PRECISION */
/* B (input/output) DOUBLE PRECISION */
/* C (input/output) DOUBLE PRECISION */
/* D (input/output) DOUBLE PRECISION */
/* On entry, the elements of the input matrix. */
/* On exit, they are overwritten by the elements of the */
/* standardised Schur form. */
/* RT1R (output) DOUBLE PRECISION */
/* RT1I (output) DOUBLE PRECISION */
/* RT2R (output) DOUBLE PRECISION */
/* RT2I (output) DOUBLE PRECISION */
/* The real and imaginary parts of the eigenvalues. If the */
/* eigenvalues are a complex conjugate pair, RT1I > 0. */
/* CS (output) DOUBLE PRECISION */
/* SN (output) DOUBLE PRECISION */
/* Parameters of the rotation matrix. */
/* Further Details */
/* =============== */
/* Modified by V. Sima, Research Institute for Informatics, Bucharest, */
/* Romania, to reduce the risk of cancellation errors, */
/* when computing real eigenvalues, and to ensure, if possible, that */
/* abs(RT1R) >= abs(RT2R). */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
eps = dlamch_("P");
if (*c__ == 0.) {
*cs = 1.;
*sn = 0.;
goto L10;
} else if (*b == 0.) {
/* Swap rows and columns */
*cs = 0.;
*sn = 1.;
temp = *d__;
*d__ = *a;
*a = temp;
*b = -(*c__);
*c__ = 0.;
goto L10;
} else if (*a - *d__ == 0. && d_sign(&c_b4, b) != d_sign(&c_b4, c__)) {
*cs = 1.;
*sn = 0.;
goto L10;
} else {
temp = *a - *d__;
p = temp * .5;
/* Computing MAX */
d__1 = abs(*b), d__2 = abs(*c__);
bcmax = max(d__1,d__2);
/* Computing MIN */
d__1 = abs(*b), d__2 = abs(*c__);
bcmis = min(d__1,d__2) * d_sign(&c_b4, b) * d_sign(&c_b4, c__);
/* Computing MAX */
d__1 = abs(p);
scale = max(d__1,bcmax);
z__ = p / scale * p + bcmax / scale * bcmis;
/* If Z is of the order of the machine accuracy, postpone the */
/* decision on the nature of eigenvalues */
if (z__ >= eps * 4.) {
/* Real eigenvalues. Compute A and D. */
d__1 = sqrt(scale) * sqrt(z__);
z__ = p + d_sign(&d__1, &p);
*a = *d__ + z__;
*d__ -= bcmax / z__ * bcmis;
/* Compute B and the rotation matrix */
tau = dlapy2_(c__, &z__);
*cs = z__ / tau;
*sn = *c__ / tau;
*b -= *c__;
*c__ = 0.;
} else {
/* Complex eigenvalues, or real (almost) equal eigenvalues. */
/* Make diagonal elements equal. */
sigma = *b + *c__;
tau = dlapy2_(&sigma, &temp);
*cs = sqrt((abs(sigma) / tau + 1.) * .5);
*sn = -(p / (tau * *cs)) * d_sign(&c_b4, &sigma);
/* Compute [ AA BB ] = [ A B ] [ CS -SN ] */
/* [ CC DD ] [ C D ] [ SN CS ] */
aa = *a * *cs + *b * *sn;
bb = -(*a) * *sn + *b * *cs;
cc = *c__ * *cs + *d__ * *sn;
dd = -(*c__) * *sn + *d__ * *cs;
/* Compute [ A B ] = [ CS SN ] [ AA BB ] */
/* [ C D ] [-SN CS ] [ CC DD ] */
*a = aa * *cs + cc * *sn;
*b = bb * *cs + dd * *sn;
*c__ = -aa * *sn + cc * *cs;
*d__ = -bb * *sn + dd * *cs;
temp = (*a + *d__) * .5;
*a = temp;
*d__ = temp;
if (*c__ != 0.) {
if (*b != 0.) {
if (d_sign(&c_b4, b) == d_sign(&c_b4, c__)) {
/* Real eigenvalues: reduce to upper triangular form */
sab = sqrt((abs(*b)));
sac = sqrt((abs(*c__)));
d__1 = sab * sac;
p = d_sign(&d__1, c__);
tau = 1. / sqrt((d__1 = *b + *c__, abs(d__1)));
*a = temp + p;
*d__ = temp - p;
*b -= *c__;
*c__ = 0.;
cs1 = sab * tau;
sn1 = sac * tau;
temp = *cs * cs1 - *sn * sn1;
*sn = *cs * sn1 + *sn * cs1;
*cs = temp;
}
} else {
*b = -(*c__);
*c__ = 0.;
temp = *cs;
*cs = -(*sn);
*sn = temp;
}
}
}
}
L10:
/* Store eigenvalues in (RT1R,RT1I) and (RT2R,RT2I). */
*rt1r = *a;
*rt2r = *d__;
if (*c__ == 0.) {
*rt1i = 0.;
*rt2i = 0.;
} else {
*rt1i = sqrt((abs(*b))) * sqrt((abs(*c__)));
*rt2i = -(*rt1i);
}
return 0;
/* End of DLANV2 */
} /* dlanv2_ */
|
