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
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
|
/* slaed6.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"
/* Subroutine */ int slaed6_(integer *kniter, logical *orgati, real *rho,
real *d__, real *z__, real *finit, real *tau, integer *info)
{
/* System generated locals */
integer i__1;
real r__1, r__2, r__3, r__4;
/* Builtin functions */
double sqrt(doublereal), log(doublereal), pow_ri(real *, integer *);
/* Local variables */
real a, b, c__, f;
integer i__;
real fc, df, ddf, lbd, eta, ubd, eps, base;
integer iter;
real temp, temp1, temp2, temp3, temp4;
logical scale;
integer niter;
real small1, small2, sminv1, sminv2, dscale[3], sclfac;
extern doublereal slamch_(char *);
real zscale[3], erretm, sclinv;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* February 2007 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SLAED6 computes the positive or negative root (closest to the origin) */
/* of */
/* z(1) z(2) z(3) */
/* f(x) = rho + --------- + ---------- + --------- */
/* d(1)-x d(2)-x d(3)-x */
/* It is assumed that */
/* if ORGATI = .true. the root is between d(2) and d(3); */
/* otherwise it is between d(1) and d(2) */
/* This routine will be called by SLAED4 when necessary. In most cases, */
/* the root sought is the smallest in magnitude, though it might not be */
/* in some extremely rare situations. */
/* Arguments */
/* ========= */
/* KNITER (input) INTEGER */
/* Refer to SLAED4 for its significance. */
/* ORGATI (input) LOGICAL */
/* If ORGATI is true, the needed root is between d(2) and */
/* d(3); otherwise it is between d(1) and d(2). See */
/* SLAED4 for further details. */
/* RHO (input) REAL */
/* Refer to the equation f(x) above. */
/* D (input) REAL array, dimension (3) */
/* D satisfies d(1) < d(2) < d(3). */
/* Z (input) REAL array, dimension (3) */
/* Each of the elements in z must be positive. */
/* FINIT (input) REAL */
/* The value of f at 0. It is more accurate than the one */
/* evaluated inside this routine (if someone wants to do */
/* so). */
/* TAU (output) REAL */
/* The root of the equation f(x). */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* > 0: if INFO = 1, failure to converge */
/* Further Details */
/* =============== */
/* 30/06/99: Based on contributions by */
/* Ren-Cang Li, Computer Science Division, University of California */
/* at Berkeley, USA */
/* 10/02/03: This version has a few statements commented out for thread safety */
/* (machine parameters are computed on each entry). SJH. */
/* 05/10/06: Modified from a new version of Ren-Cang Li, use */
/* Gragg-Thornton-Warner cubic convergent scheme for better stability. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
--z__;
--d__;
/* Function Body */
*info = 0;
if (*orgati) {
lbd = d__[2];
ubd = d__[3];
} else {
lbd = d__[1];
ubd = d__[2];
}
if (*finit < 0.f) {
lbd = 0.f;
} else {
ubd = 0.f;
}
niter = 1;
*tau = 0.f;
if (*kniter == 2) {
if (*orgati) {
temp = (d__[3] - d__[2]) / 2.f;
c__ = *rho + z__[1] / (d__[1] - d__[2] - temp);
a = c__ * (d__[2] + d__[3]) + z__[2] + z__[3];
b = c__ * d__[2] * d__[3] + z__[2] * d__[3] + z__[3] * d__[2];
} else {
temp = (d__[1] - d__[2]) / 2.f;
c__ = *rho + z__[3] / (d__[3] - d__[2] - temp);
a = c__ * (d__[1] + d__[2]) + z__[1] + z__[2];
b = c__ * d__[1] * d__[2] + z__[1] * d__[2] + z__[2] * d__[1];
}
/* Computing MAX */
r__1 = dabs(a), r__2 = dabs(b), r__1 = max(r__1,r__2), r__2 = dabs(
c__);
temp = dmax(r__1,r__2);
a /= temp;
b /= temp;
c__ /= temp;
if (c__ == 0.f) {
*tau = b / a;
} else if (a <= 0.f) {
*tau = (a - sqrt((r__1 = a * a - b * 4.f * c__, dabs(r__1)))) / (
c__ * 2.f);
} else {
*tau = b * 2.f / (a + sqrt((r__1 = a * a - b * 4.f * c__, dabs(
r__1))));
}
if (*tau < lbd || *tau > ubd) {
*tau = (lbd + ubd) / 2.f;
}
if (d__[1] == *tau || d__[2] == *tau || d__[3] == *tau) {
*tau = 0.f;
} else {
temp = *finit + *tau * z__[1] / (d__[1] * (d__[1] - *tau)) + *tau
* z__[2] / (d__[2] * (d__[2] - *tau)) + *tau * z__[3] / (
d__[3] * (d__[3] - *tau));
if (temp <= 0.f) {
lbd = *tau;
} else {
ubd = *tau;
}
if (dabs(*finit) <= dabs(temp)) {
*tau = 0.f;
}
}
}
/* get machine parameters for possible scaling to avoid overflow */
/* modified by Sven: parameters SMALL1, SMINV1, SMALL2, */
/* SMINV2, EPS are not SAVEd anymore between one call to the */
/* others but recomputed at each call */
eps = slamch_("Epsilon");
base = slamch_("Base");
i__1 = (integer) (log(slamch_("SafMin")) / log(base) / 3.f);
small1 = pow_ri(&base, &i__1);
sminv1 = 1.f / small1;
small2 = small1 * small1;
sminv2 = sminv1 * sminv1;
/* Determine if scaling of inputs necessary to avoid overflow */
/* when computing 1/TEMP**3 */
if (*orgati) {
/* Computing MIN */
r__3 = (r__1 = d__[2] - *tau, dabs(r__1)), r__4 = (r__2 = d__[3] - *
tau, dabs(r__2));
temp = dmin(r__3,r__4);
} else {
/* Computing MIN */
r__3 = (r__1 = d__[1] - *tau, dabs(r__1)), r__4 = (r__2 = d__[2] - *
tau, dabs(r__2));
temp = dmin(r__3,r__4);
}
scale = FALSE_;
if (temp <= small1) {
scale = TRUE_;
if (temp <= small2) {
/* Scale up by power of radix nearest 1/SAFMIN**(2/3) */
sclfac = sminv2;
sclinv = small2;
} else {
/* Scale up by power of radix nearest 1/SAFMIN**(1/3) */
sclfac = sminv1;
sclinv = small1;
}
/* Scaling up safe because D, Z, TAU scaled elsewhere to be O(1) */
for (i__ = 1; i__ <= 3; ++i__) {
dscale[i__ - 1] = d__[i__] * sclfac;
zscale[i__ - 1] = z__[i__] * sclfac;
/* L10: */
}
*tau *= sclfac;
lbd *= sclfac;
ubd *= sclfac;
} else {
/* Copy D and Z to DSCALE and ZSCALE */
for (i__ = 1; i__ <= 3; ++i__) {
dscale[i__ - 1] = d__[i__];
zscale[i__ - 1] = z__[i__];
/* L20: */
}
}
fc = 0.f;
df = 0.f;
ddf = 0.f;
for (i__ = 1; i__ <= 3; ++i__) {
temp = 1.f / (dscale[i__ - 1] - *tau);
temp1 = zscale[i__ - 1] * temp;
temp2 = temp1 * temp;
temp3 = temp2 * temp;
fc += temp1 / dscale[i__ - 1];
df += temp2;
ddf += temp3;
/* L30: */
}
f = *finit + *tau * fc;
if (dabs(f) <= 0.f) {
goto L60;
}
if (f <= 0.f) {
lbd = *tau;
} else {
ubd = *tau;
}
/* Iteration begins -- Use Gragg-Thornton-Warner cubic convergent */
/* scheme */
/* It is not hard to see that */
/* 1) Iterations will go up monotonically */
/* if FINIT < 0; */
/* 2) Iterations will go down monotonically */
/* if FINIT > 0. */
iter = niter + 1;
for (niter = iter; niter <= 40; ++niter) {
if (*orgati) {
temp1 = dscale[1] - *tau;
temp2 = dscale[2] - *tau;
} else {
temp1 = dscale[0] - *tau;
temp2 = dscale[1] - *tau;
}
a = (temp1 + temp2) * f - temp1 * temp2 * df;
b = temp1 * temp2 * f;
c__ = f - (temp1 + temp2) * df + temp1 * temp2 * ddf;
/* Computing MAX */
r__1 = dabs(a), r__2 = dabs(b), r__1 = max(r__1,r__2), r__2 = dabs(
c__);
temp = dmax(r__1,r__2);
a /= temp;
b /= temp;
c__ /= temp;
if (c__ == 0.f) {
eta = b / a;
} else if (a <= 0.f) {
eta = (a - sqrt((r__1 = a * a - b * 4.f * c__, dabs(r__1)))) / (
c__ * 2.f);
} else {
eta = b * 2.f / (a + sqrt((r__1 = a * a - b * 4.f * c__, dabs(
r__1))));
}
if (f * eta >= 0.f) {
eta = -f / df;
}
*tau += eta;
if (*tau < lbd || *tau > ubd) {
*tau = (lbd + ubd) / 2.f;
}
fc = 0.f;
erretm = 0.f;
df = 0.f;
ddf = 0.f;
for (i__ = 1; i__ <= 3; ++i__) {
temp = 1.f / (dscale[i__ - 1] - *tau);
temp1 = zscale[i__ - 1] * temp;
temp2 = temp1 * temp;
temp3 = temp2 * temp;
temp4 = temp1 / dscale[i__ - 1];
fc += temp4;
erretm += dabs(temp4);
df += temp2;
ddf += temp3;
/* L40: */
}
f = *finit + *tau * fc;
erretm = (dabs(*finit) + dabs(*tau) * erretm) * 8.f + dabs(*tau) * df;
if (dabs(f) <= eps * erretm) {
goto L60;
}
if (f <= 0.f) {
lbd = *tau;
} else {
ubd = *tau;
}
/* L50: */
}
*info = 1;
L60:
/* Undo scaling */
if (scale) {
*tau *= sclinv;
}
return 0;
/* End of SLAED6 */
} /* slaed6_ */
|
