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
|
/* slaqps.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_b8 = -1.f;
static real c_b9 = 1.f;
static real c_b16 = 0.f;
/* Subroutine */ int slaqps_(integer *m, integer *n, integer *offset, integer
*nb, integer *kb, real *a, integer *lda, integer *jpvt, real *tau,
real *vn1, real *vn2, real *auxv, real *f, integer *ldf)
{
/* System generated locals */
integer a_dim1, a_offset, f_dim1, f_offset, i__1, i__2;
real r__1, r__2;
/* Builtin functions */
double sqrt(doublereal);
integer i_nint(real *);
/* Local variables */
integer j, k, rk;
real akk;
integer pvt;
real temp, temp2;
extern doublereal snrm2_(integer *, real *, integer *);
real tol3z;
extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *,
integer *, real *, real *, integer *, real *, integer *, real *,
real *, integer *);
integer itemp;
extern /* Subroutine */ int sgemv_(char *, integer *, integer *, real *,
real *, integer *, real *, integer *, real *, real *, integer *), sswap_(integer *, real *, integer *, real *, integer *);
extern doublereal slamch_(char *);
integer lsticc;
extern integer isamax_(integer *, real *, integer *);
extern /* Subroutine */ int slarfp_(integer *, real *, real *, integer *,
real *);
integer lastrk;
/* -- LAPACK auxiliary routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SLAQPS computes a step of QR factorization with column pivoting */
/* of a real M-by-N matrix A by using Blas-3. It tries to factorize */
/* NB columns from A starting from the row OFFSET+1, and updates all */
/* of the matrix with Blas-3 xGEMM. */
/* In some cases, due to catastrophic cancellations, it cannot */
/* factorize NB columns. Hence, the actual number of factorized */
/* columns is returned in KB. */
/* Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized. */
/* Arguments */
/* ========= */
/* M (input) INTEGER */
/* The number of rows of the matrix A. M >= 0. */
/* N (input) INTEGER */
/* The number of columns of the matrix A. N >= 0 */
/* OFFSET (input) INTEGER */
/* The number of rows of A that have been factorized in */
/* previous steps. */
/* NB (input) INTEGER */
/* The number of columns to factorize. */
/* KB (output) INTEGER */
/* The number of columns actually factorized. */
/* A (input/output) REAL array, dimension (LDA,N) */
/* On entry, the M-by-N matrix A. */
/* On exit, block A(OFFSET+1:M,1:KB) is the triangular */
/* factor obtained and block A(1:OFFSET,1:N) has been */
/* accordingly pivoted, but no factorized. */
/* The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has */
/* been updated. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,M). */
/* JPVT (input/output) INTEGER array, dimension (N) */
/* JPVT(I) = K <==> Column K of the full matrix A has been */
/* permuted into position I in AP. */
/* TAU (output) REAL array, dimension (KB) */
/* The scalar factors of the elementary reflectors. */
/* VN1 (input/output) REAL array, dimension (N) */
/* The vector with the partial column norms. */
/* VN2 (input/output) REAL array, dimension (N) */
/* The vector with the exact column norms. */
/* AUXV (input/output) REAL array, dimension (NB) */
/* Auxiliar vector. */
/* F (input/output) REAL array, dimension (LDF,NB) */
/* Matrix F' = L*Y'*A. */
/* LDF (input) INTEGER */
/* The leading dimension of the array F. LDF >= max(1,N). */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain */
/* X. Sun, Computer Science Dept., Duke University, USA */
/* Partial column norm updating strategy modified by */
/* Z. Drmac and Z. Bujanovic, Dept. of Mathematics, */
/* University of Zagreb, Croatia. */
/* June 2006. */
/* For more details see LAPACK Working Note 176. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--jpvt;
--tau;
--vn1;
--vn2;
--auxv;
f_dim1 = *ldf;
f_offset = 1 + f_dim1;
f -= f_offset;
/* Function Body */
/* Computing MIN */
i__1 = *m, i__2 = *n + *offset;
lastrk = min(i__1,i__2);
lsticc = 0;
k = 0;
tol3z = sqrt(slamch_("Epsilon"));
/* Beginning of while loop. */
L10:
if (k < *nb && lsticc == 0) {
++k;
rk = *offset + k;
/* Determine ith pivot column and swap if necessary */
i__1 = *n - k + 1;
pvt = k - 1 + isamax_(&i__1, &vn1[k], &c__1);
if (pvt != k) {
sswap_(m, &a[pvt * a_dim1 + 1], &c__1, &a[k * a_dim1 + 1], &c__1);
i__1 = k - 1;
sswap_(&i__1, &f[pvt + f_dim1], ldf, &f[k + f_dim1], ldf);
itemp = jpvt[pvt];
jpvt[pvt] = jpvt[k];
jpvt[k] = itemp;
vn1[pvt] = vn1[k];
vn2[pvt] = vn2[k];
}
/* Apply previous Householder reflectors to column K: */
/* A(RK:M,K) := A(RK:M,K) - A(RK:M,1:K-1)*F(K,1:K-1)'. */
if (k > 1) {
i__1 = *m - rk + 1;
i__2 = k - 1;
sgemv_("No transpose", &i__1, &i__2, &c_b8, &a[rk + a_dim1], lda,
&f[k + f_dim1], ldf, &c_b9, &a[rk + k * a_dim1], &c__1);
}
/* Generate elementary reflector H(k). */
if (rk < *m) {
i__1 = *m - rk + 1;
slarfp_(&i__1, &a[rk + k * a_dim1], &a[rk + 1 + k * a_dim1], &
c__1, &tau[k]);
} else {
slarfp_(&c__1, &a[rk + k * a_dim1], &a[rk + k * a_dim1], &c__1, &
tau[k]);
}
akk = a[rk + k * a_dim1];
a[rk + k * a_dim1] = 1.f;
/* Compute Kth column of F: */
/* Compute F(K+1:N,K) := tau(K)*A(RK:M,K+1:N)'*A(RK:M,K). */
if (k < *n) {
i__1 = *m - rk + 1;
i__2 = *n - k;
sgemv_("Transpose", &i__1, &i__2, &tau[k], &a[rk + (k + 1) *
a_dim1], lda, &a[rk + k * a_dim1], &c__1, &c_b16, &f[k +
1 + k * f_dim1], &c__1);
}
/* Padding F(1:K,K) with zeros. */
i__1 = k;
for (j = 1; j <= i__1; ++j) {
f[j + k * f_dim1] = 0.f;
/* L20: */
}
/* Incremental updating of F: */
/* F(1:N,K) := F(1:N,K) - tau(K)*F(1:N,1:K-1)*A(RK:M,1:K-1)' */
/* *A(RK:M,K). */
if (k > 1) {
i__1 = *m - rk + 1;
i__2 = k - 1;
r__1 = -tau[k];
sgemv_("Transpose", &i__1, &i__2, &r__1, &a[rk + a_dim1], lda, &a[
rk + k * a_dim1], &c__1, &c_b16, &auxv[1], &c__1);
i__1 = k - 1;
sgemv_("No transpose", n, &i__1, &c_b9, &f[f_dim1 + 1], ldf, &
auxv[1], &c__1, &c_b9, &f[k * f_dim1 + 1], &c__1);
}
/* Update the current row of A: */
/* A(RK,K+1:N) := A(RK,K+1:N) - A(RK,1:K)*F(K+1:N,1:K)'. */
if (k < *n) {
i__1 = *n - k;
sgemv_("No transpose", &i__1, &k, &c_b8, &f[k + 1 + f_dim1], ldf,
&a[rk + a_dim1], lda, &c_b9, &a[rk + (k + 1) * a_dim1],
lda);
}
/* Update partial column norms. */
if (rk < lastrk) {
i__1 = *n;
for (j = k + 1; j <= i__1; ++j) {
if (vn1[j] != 0.f) {
/* NOTE: The following 4 lines follow from the analysis in */
/* Lapack Working Note 176. */
temp = (r__1 = a[rk + j * a_dim1], dabs(r__1)) / vn1[j];
/* Computing MAX */
r__1 = 0.f, r__2 = (temp + 1.f) * (1.f - temp);
temp = dmax(r__1,r__2);
/* Computing 2nd power */
r__1 = vn1[j] / vn2[j];
temp2 = temp * (r__1 * r__1);
if (temp2 <= tol3z) {
vn2[j] = (real) lsticc;
lsticc = j;
} else {
vn1[j] *= sqrt(temp);
}
}
/* L30: */
}
}
a[rk + k * a_dim1] = akk;
/* End of while loop. */
goto L10;
}
*kb = k;
rk = *offset + *kb;
/* Apply the block reflector to the rest of the matrix: */
/* A(OFFSET+KB+1:M,KB+1:N) := A(OFFSET+KB+1:M,KB+1:N) - */
/* A(OFFSET+KB+1:M,1:KB)*F(KB+1:N,1:KB)'. */
/* Computing MIN */
i__1 = *n, i__2 = *m - *offset;
if (*kb < min(i__1,i__2)) {
i__1 = *m - rk;
i__2 = *n - *kb;
sgemm_("No transpose", "Transpose", &i__1, &i__2, kb, &c_b8, &a[rk +
1 + a_dim1], lda, &f[*kb + 1 + f_dim1], ldf, &c_b9, &a[rk + 1
+ (*kb + 1) * a_dim1], lda);
}
/* Recomputation of difficult columns. */
L40:
if (lsticc > 0) {
itemp = i_nint(&vn2[lsticc]);
i__1 = *m - rk;
vn1[lsticc] = snrm2_(&i__1, &a[rk + 1 + lsticc * a_dim1], &c__1);
/* NOTE: The computation of VN1( LSTICC ) relies on the fact that */
/* SNRM2 does not fail on vectors with norm below the value of */
/* SQRT(DLAMCH('S')) */
vn2[lsticc] = vn1[lsticc];
lsticc = itemp;
goto L40;
}
return 0;
/* End of SLAQPS */
} /* slaqps_ */
|
