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
|
/* cpbstf.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 integer c__1 = 1;
static real c_b9 = -1.f;
/* Subroutine */ int cpbstf_(char *uplo, integer *n, integer *kd, complex *ab,
integer *ldab, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, i__1, i__2, i__3;
real r__1;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer j, m, km;
real ajj;
integer kld;
extern /* Subroutine */ int cher_(char *, integer *, real *, complex *,
integer *, complex *, integer *);
extern logical lsame_(char *, char *);
logical upper;
extern /* Subroutine */ int clacgv_(integer *, complex *, integer *),
csscal_(integer *, real *, complex *, integer *), xerbla_(char *,
integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CPBSTF computes a split Cholesky factorization of a complex */
/* Hermitian positive definite band matrix A. */
/* This routine is designed to be used in conjunction with CHBGST. */
/* The factorization has the form A = S**H*S where S is a band matrix */
/* of the same bandwidth as A and the following structure: */
/* S = ( U ) */
/* ( M L ) */
/* where U is upper triangular of order m = (n+kd)/2, and L is lower */
/* triangular of order n-m. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* KD (input) INTEGER */
/* The number of superdiagonals of the matrix A if UPLO = 'U', */
/* or the number of subdiagonals if UPLO = 'L'. KD >= 0. */
/* AB (input/output) COMPLEX array, dimension (LDAB,N) */
/* On entry, the upper or lower triangle of the Hermitian band */
/* matrix A, stored in the first kd+1 rows of the array. The */
/* j-th column of A is stored in the j-th column of the array AB */
/* as follows: */
/* if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; */
/* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). */
/* On exit, if INFO = 0, the factor S from the split Cholesky */
/* factorization A = S**H*S. See Further Details. */
/* LDAB (input) INTEGER */
/* The leading dimension of the array AB. LDAB >= KD+1. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: if INFO = i, the factorization could not be completed, */
/* because the updated element a(i,i) was negative; the */
/* matrix A is not positive definite. */
/* Further Details */
/* =============== */
/* The band storage scheme is illustrated by the following example, when */
/* N = 7, KD = 2: */
/* S = ( s11 s12 s13 ) */
/* ( s22 s23 s24 ) */
/* ( s33 s34 ) */
/* ( s44 ) */
/* ( s53 s54 s55 ) */
/* ( s64 s65 s66 ) */
/* ( s75 s76 s77 ) */
/* If UPLO = 'U', the array AB holds: */
/* on entry: on exit: */
/* * * a13 a24 a35 a46 a57 * * s13 s24 s53' s64' s75' */
/* * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54' s65' s76' */
/* a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 */
/* If UPLO = 'L', the array AB holds: */
/* on entry: on exit: */
/* a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77 */
/* a21 a32 a43 a54 a65 a76 * s12' s23' s34' s54 s65 s76 * */
/* a31 a42 a53 a64 a64 * * s13' s24' s53 s64 s75 * * */
/* Array elements marked * are not used by the routine; s12' denotes */
/* conjg(s12); the diagonal elements of S are real. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*kd < 0) {
*info = -3;
} else if (*ldab < *kd + 1) {
*info = -5;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CPBSTF", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Computing MAX */
i__1 = 1, i__2 = *ldab - 1;
kld = max(i__1,i__2);
/* Set the splitting point m. */
m = (*n + *kd) / 2;
if (upper) {
/* Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m). */
i__1 = m + 1;
for (j = *n; j >= i__1; --j) {
/* Compute s(j,j) and test for non-positive-definiteness. */
i__2 = *kd + 1 + j * ab_dim1;
ajj = ab[i__2].r;
if (ajj <= 0.f) {
i__2 = *kd + 1 + j * ab_dim1;
ab[i__2].r = ajj, ab[i__2].i = 0.f;
goto L50;
}
ajj = sqrt(ajj);
i__2 = *kd + 1 + j * ab_dim1;
ab[i__2].r = ajj, ab[i__2].i = 0.f;
/* Computing MIN */
i__2 = j - 1;
km = min(i__2,*kd);
/* Compute elements j-km:j-1 of the j-th column and update the */
/* the leading submatrix within the band. */
r__1 = 1.f / ajj;
csscal_(&km, &r__1, &ab[*kd + 1 - km + j * ab_dim1], &c__1);
cher_("Upper", &km, &c_b9, &ab[*kd + 1 - km + j * ab_dim1], &c__1,
&ab[*kd + 1 + (j - km) * ab_dim1], &kld);
/* L10: */
}
/* Factorize the updated submatrix A(1:m,1:m) as U**H*U. */
i__1 = m;
for (j = 1; j <= i__1; ++j) {
/* Compute s(j,j) and test for non-positive-definiteness. */
i__2 = *kd + 1 + j * ab_dim1;
ajj = ab[i__2].r;
if (ajj <= 0.f) {
i__2 = *kd + 1 + j * ab_dim1;
ab[i__2].r = ajj, ab[i__2].i = 0.f;
goto L50;
}
ajj = sqrt(ajj);
i__2 = *kd + 1 + j * ab_dim1;
ab[i__2].r = ajj, ab[i__2].i = 0.f;
/* Computing MIN */
i__2 = *kd, i__3 = m - j;
km = min(i__2,i__3);
/* Compute elements j+1:j+km of the j-th row and update the */
/* trailing submatrix within the band. */
if (km > 0) {
r__1 = 1.f / ajj;
csscal_(&km, &r__1, &ab[*kd + (j + 1) * ab_dim1], &kld);
clacgv_(&km, &ab[*kd + (j + 1) * ab_dim1], &kld);
cher_("Upper", &km, &c_b9, &ab[*kd + (j + 1) * ab_dim1], &kld,
&ab[*kd + 1 + (j + 1) * ab_dim1], &kld);
clacgv_(&km, &ab[*kd + (j + 1) * ab_dim1], &kld);
}
/* L20: */
}
} else {
/* Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m). */
i__1 = m + 1;
for (j = *n; j >= i__1; --j) {
/* Compute s(j,j) and test for non-positive-definiteness. */
i__2 = j * ab_dim1 + 1;
ajj = ab[i__2].r;
if (ajj <= 0.f) {
i__2 = j * ab_dim1 + 1;
ab[i__2].r = ajj, ab[i__2].i = 0.f;
goto L50;
}
ajj = sqrt(ajj);
i__2 = j * ab_dim1 + 1;
ab[i__2].r = ajj, ab[i__2].i = 0.f;
/* Computing MIN */
i__2 = j - 1;
km = min(i__2,*kd);
/* Compute elements j-km:j-1 of the j-th row and update the */
/* trailing submatrix within the band. */
r__1 = 1.f / ajj;
csscal_(&km, &r__1, &ab[km + 1 + (j - km) * ab_dim1], &kld);
clacgv_(&km, &ab[km + 1 + (j - km) * ab_dim1], &kld);
cher_("Lower", &km, &c_b9, &ab[km + 1 + (j - km) * ab_dim1], &kld,
&ab[(j - km) * ab_dim1 + 1], &kld);
clacgv_(&km, &ab[km + 1 + (j - km) * ab_dim1], &kld);
/* L30: */
}
/* Factorize the updated submatrix A(1:m,1:m) as U**H*U. */
i__1 = m;
for (j = 1; j <= i__1; ++j) {
/* Compute s(j,j) and test for non-positive-definiteness. */
i__2 = j * ab_dim1 + 1;
ajj = ab[i__2].r;
if (ajj <= 0.f) {
i__2 = j * ab_dim1 + 1;
ab[i__2].r = ajj, ab[i__2].i = 0.f;
goto L50;
}
ajj = sqrt(ajj);
i__2 = j * ab_dim1 + 1;
ab[i__2].r = ajj, ab[i__2].i = 0.f;
/* Computing MIN */
i__2 = *kd, i__3 = m - j;
km = min(i__2,i__3);
/* Compute elements j+1:j+km of the j-th column and update the */
/* trailing submatrix within the band. */
if (km > 0) {
r__1 = 1.f / ajj;
csscal_(&km, &r__1, &ab[j * ab_dim1 + 2], &c__1);
cher_("Lower", &km, &c_b9, &ab[j * ab_dim1 + 2], &c__1, &ab[(
j + 1) * ab_dim1 + 1], &kld);
}
/* L40: */
}
}
return 0;
L50:
*info = j;
return 0;
/* End of CPBSTF */
} /* cpbstf_ */
|
