LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
ssytri2x.f
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1 *> \brief \b SSYTRI2X
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SSYTRI2X( UPLO, N, A, LDA, IPIV, WORK, NB, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER UPLO
25 * INTEGER INFO, LDA, N, NB
26 * ..
27 * .. Array Arguments ..
28 * INTEGER IPIV( * )
29 * REAL A( LDA, * ), WORK( N+NB+1,* )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> SSYTRI2X computes the inverse of a real symmetric indefinite matrix
39 *> A using the factorization A = U*D*U**T or A = L*D*L**T computed by
40 *> SSYTRF.
41 *> \endverbatim
42 *
43 * Arguments:
44 * ==========
45 *
46 *> \param[in] UPLO
47 *> \verbatim
48 *> UPLO is CHARACTER*1
49 *> Specifies whether the details of the factorization are stored
50 *> as an upper or lower triangular matrix.
51 *> = 'U': Upper triangular, form is A = U*D*U**T;
52 *> = 'L': Lower triangular, form is A = L*D*L**T.
53 *> \endverbatim
54 *>
55 *> \param[in] N
56 *> \verbatim
57 *> N is INTEGER
58 *> The order of the matrix A. N >= 0.
59 *> \endverbatim
60 *>
61 *> \param[in,out] A
62 *> \verbatim
63 *> A is REAL array, dimension (LDA,N)
64 *> On entry, the NNB diagonal matrix D and the multipliers
65 *> used to obtain the factor U or L as computed by SSYTRF.
66 *>
67 *> On exit, if INFO = 0, the (symmetric) inverse of the original
68 *> matrix. If UPLO = 'U', the upper triangular part of the
69 *> inverse is formed and the part of A below the diagonal is not
70 *> referenced; if UPLO = 'L' the lower triangular part of the
71 *> inverse is formed and the part of A above the diagonal is
72 *> not referenced.
73 *> \endverbatim
74 *>
75 *> \param[in] LDA
76 *> \verbatim
77 *> LDA is INTEGER
78 *> The leading dimension of the array A. LDA >= max(1,N).
79 *> \endverbatim
80 *>
81 *> \param[in] IPIV
82 *> \verbatim
83 *> IPIV is INTEGER array, dimension (N)
84 *> Details of the interchanges and the NNB structure of D
85 *> as determined by SSYTRF.
86 *> \endverbatim
87 *>
88 *> \param[out] WORK
89 *> \verbatim
90 *> WORK is REAL array, dimension (N+NNB+1,NNB+3)
91 *> \endverbatim
92 *>
93 *> \param[in] NB
94 *> \verbatim
95 *> NB is INTEGER
96 *> Block size
97 *> \endverbatim
98 *>
99 *> \param[out] INFO
100 *> \verbatim
101 *> INFO is INTEGER
102 *> = 0: successful exit
103 *> < 0: if INFO = -i, the i-th argument had an illegal value
104 *> > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
105 *> inverse could not be computed.
106 *> \endverbatim
107 *
108 * Authors:
109 * ========
110 *
111 *> \author Univ. of Tennessee
112 *> \author Univ. of California Berkeley
113 *> \author Univ. of Colorado Denver
114 *> \author NAG Ltd.
115 *
116 *> \date November 2011
117 *
118 *> \ingroup realSYcomputational
119 *
120 * =====================================================================
121  SUBROUTINE ssytri2x( UPLO, N, A, LDA, IPIV, WORK, NB, INFO )
122 *
123 * -- LAPACK computational routine (version 3.4.0) --
124 * -- LAPACK is a software package provided by Univ. of Tennessee, --
125 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
126 * November 2011
127 *
128 * .. Scalar Arguments ..
129  CHARACTER uplo
130  INTEGER info, lda, n, nb
131 * ..
132 * .. Array Arguments ..
133  INTEGER ipiv( * )
134  REAL a( lda, * ), work( n+nb+1,* )
135 * ..
136 *
137 * =====================================================================
138 *
139 * .. Parameters ..
140  REAL one, zero
141  parameter( one = 1.0e+0, zero = 0.0e+0 )
142 * ..
143 * .. Local Scalars ..
144  LOGICAL upper
145  INTEGER i, iinfo, ip, k, cut, nnb
146  INTEGER count
147  INTEGER j, u11, invd
148
149  REAL ak, akkp1, akp1, d, t
150  REAL u01_i_j, u01_ip1_j
151  REAL u11_i_j, u11_ip1_j
152 * ..
153 * .. External Functions ..
154  LOGICAL lsame
155  EXTERNAL lsame
156 * ..
157 * .. External Subroutines ..
158  EXTERNAL ssyconv, xerbla, strtri
159  EXTERNAL sgemm, strmm, ssyswapr
160 * ..
161 * .. Intrinsic Functions ..
162  INTRINSIC max
163 * ..
164 * .. Executable Statements ..
165 *
166 * Test the input parameters.
167 *
168  info = 0
169  upper = lsame( uplo, 'U' )
170  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
171  info = -1
172  ELSE IF( n.LT.0 ) THEN
173  info = -2
174  ELSE IF( lda.LT.max( 1, n ) ) THEN
175  info = -4
176  END IF
177 *
178 * Quick return if possible
179 *
180 *
181  IF( info.NE.0 ) THEN
182  CALL xerbla( 'SSYTRI2X', -info )
183  return
184  END IF
185  IF( n.EQ.0 )
186  \$ return
187 *
188 * Convert A
189 * Workspace got Non-diag elements of D
190 *
191  CALL ssyconv( uplo, 'C', n, a, lda, ipiv, work, iinfo )
192 *
193 * Check that the diagonal matrix D is nonsingular.
194 *
195  IF( upper ) THEN
196 *
197 * Upper triangular storage: examine D from bottom to top
198 *
199  DO info = n, 1, -1
200  IF( ipiv( info ).GT.0 .AND. a( info, info ).EQ.zero )
201  \$ return
202  END DO
203  ELSE
204 *
205 * Lower triangular storage: examine D from top to bottom.
206 *
207  DO info = 1, n
208  IF( ipiv( info ).GT.0 .AND. a( info, info ).EQ.zero )
209  \$ return
210  END DO
211  END IF
212  info = 0
213 *
214 * Splitting Workspace
215 * U01 is a block (N,NB+1)
216 * The first element of U01 is in WORK(1,1)
217 * U11 is a block (NB+1,NB+1)
218 * The first element of U11 is in WORK(N+1,1)
219  u11 = n
220 * INVD is a block (N,2)
221 * The first element of INVD is in WORK(1,INVD)
222  invd = nb+2
223
224  IF( upper ) THEN
225 *
226 * invA = P * inv(U**T)*inv(D)*inv(U)*P**T.
227 *
228  CALL strtri( uplo, 'U', n, a, lda, info )
229 *
230 * inv(D) and inv(D)*inv(U)
231 *
232  k=1
233  DO WHILE ( k .LE. n )
234  IF( ipiv( k ).GT.0 ) THEN
235 * 1 x 1 diagonal NNB
236  work(k,invd) = one / a( k, k )
237  work(k,invd+1) = 0
238  k=k+1
239  ELSE
240 * 2 x 2 diagonal NNB
241  t = work(k+1,1)
242  ak = a( k, k ) / t
243  akp1 = a( k+1, k+1 ) / t
244  akkp1 = work(k+1,1) / t
245  d = t*( ak*akp1-one )
246  work(k,invd) = akp1 / d
247  work(k+1,invd+1) = ak / d
248  work(k,invd+1) = -akkp1 / d
249  work(k+1,invd) = -akkp1 / d
250  k=k+2
251  END IF
252  END DO
253 *
254 * inv(U**T) = (inv(U))**T
255 *
256 * inv(U**T)*inv(D)*inv(U)
257 *
258  cut=n
259  DO WHILE (cut .GT. 0)
260  nnb=nb
261  IF (cut .LE. nnb) THEN
262  nnb=cut
263  ELSE
264  count = 0
265 * count negative elements,
266  DO i=cut+1-nnb,cut
267  IF (ipiv(i) .LT. 0) count=count+1
268  END DO
269 * need a even number for a clear cut
270  IF (mod(count,2) .EQ. 1) nnb=nnb+1
271  END IF
272
273  cut=cut-nnb
274 *
275 * U01 Block
276 *
277  DO i=1,cut
278  DO j=1,nnb
279  work(i,j)=a(i,cut+j)
280  END DO
281  END DO
282 *
283 * U11 Block
284 *
285  DO i=1,nnb
286  work(u11+i,i)=one
287  DO j=1,i-1
288  work(u11+i,j)=zero
289  END DO
290  DO j=i+1,nnb
291  work(u11+i,j)=a(cut+i,cut+j)
292  END DO
293  END DO
294 *
295 * invD*U01
296 *
297  i=1
298  DO WHILE (i .LE. cut)
299  IF (ipiv(i) > 0) THEN
300  DO j=1,nnb
301  work(i,j)=work(i,invd)*work(i,j)
302  END DO
303  i=i+1
304  ELSE
305  DO j=1,nnb
306  u01_i_j = work(i,j)
307  u01_ip1_j = work(i+1,j)
308  work(i,j)=work(i,invd)*u01_i_j+
309  \$ work(i,invd+1)*u01_ip1_j
310  work(i+1,j)=work(i+1,invd)*u01_i_j+
311  \$ work(i+1,invd+1)*u01_ip1_j
312  END DO
313  i=i+2
314  END IF
315  END DO
316 *
317 * invD1*U11
318 *
319  i=1
320  DO WHILE (i .LE. nnb)
321  IF (ipiv(cut+i) > 0) THEN
322  DO j=i,nnb
323  work(u11+i,j)=work(cut+i,invd)*work(u11+i,j)
324  END DO
325  i=i+1
326  ELSE
327  DO j=i,nnb
328  u11_i_j = work(u11+i,j)
329  u11_ip1_j = work(u11+i+1,j)
330  work(u11+i,j)=work(cut+i,invd)*work(u11+i,j) +
331  \$ work(cut+i,invd+1)*work(u11+i+1,j)
332  work(u11+i+1,j)=work(cut+i+1,invd)*u11_i_j+
333  \$ work(cut+i+1,invd+1)*u11_ip1_j
334  END DO
335  i=i+2
336  END IF
337  END DO
338 *
339 * U11**T*invD1*U11->U11
340 *
341  CALL strmm('L','U','T','U',nnb, nnb,
342  \$ one,a(cut+1,cut+1),lda,work(u11+1,1),n+nb+1)
343 *
344  DO i=1,nnb
345  DO j=i,nnb
346  a(cut+i,cut+j)=work(u11+i,j)
347  END DO
348  END DO
349 *
350 * U01**T*invD*U01->A(CUT+I,CUT+J)
351 *
352  CALL sgemm('T','N',nnb,nnb,cut,one,a(1,cut+1),lda,
353  \$ work,n+nb+1, zero, work(u11+1,1), n+nb+1)
354 *
355 * U11 = U11**T*invD1*U11 + U01**T*invD*U01
356 *
357  DO i=1,nnb
358  DO j=i,nnb
359  a(cut+i,cut+j)=a(cut+i,cut+j)+work(u11+i,j)
360  END DO
361  END DO
362 *
363 * U01 = U00**T*invD0*U01
364 *
365  CALL strmm('L',uplo,'T','U',cut, nnb,
366  \$ one,a,lda,work,n+nb+1)
367
368 *
369 * Update U01
370 *
371  DO i=1,cut
372  DO j=1,nnb
373  a(i,cut+j)=work(i,j)
374  END DO
375  END DO
376 *
377 * Next Block
378 *
379  END DO
380 *
381 * Apply PERMUTATIONS P and P**T: P * inv(U**T)*inv(D)*inv(U) *P**T
382 *
383  i=1
384  DO WHILE ( i .LE. n )
385  IF( ipiv(i) .GT. 0 ) THEN
386  ip=ipiv(i)
387  IF (i .LT. ip) CALL ssyswapr( uplo, n, a, lda, i ,ip )
388  IF (i .GT. ip) CALL ssyswapr( uplo, n, a, lda, ip ,i )
389  ELSE
390  ip=-ipiv(i)
391  i=i+1
392  IF ( (i-1) .LT. ip)
393  \$ CALL ssyswapr( uplo, n, a, lda, i-1 ,ip )
394  IF ( (i-1) .GT. ip)
395  \$ CALL ssyswapr( uplo, n, a, lda, ip ,i-1 )
396  ENDIF
397  i=i+1
398  END DO
399  ELSE
400 *
401 * LOWER...
402 *
403 * invA = P * inv(U**T)*inv(D)*inv(U)*P**T.
404 *
405  CALL strtri( uplo, 'U', n, a, lda, info )
406 *
407 * inv(D) and inv(D)*inv(U)
408 *
409  k=n
410  DO WHILE ( k .GE. 1 )
411  IF( ipiv( k ).GT.0 ) THEN
412 * 1 x 1 diagonal NNB
413  work(k,invd) = one / a( k, k )
414  work(k,invd+1) = 0
415  k=k-1
416  ELSE
417 * 2 x 2 diagonal NNB
418  t = work(k-1,1)
419  ak = a( k-1, k-1 ) / t
420  akp1 = a( k, k ) / t
421  akkp1 = work(k-1,1) / t
422  d = t*( ak*akp1-one )
423  work(k-1,invd) = akp1 / d
424  work(k,invd) = ak / d
425  work(k,invd+1) = -akkp1 / d
426  work(k-1,invd+1) = -akkp1 / d
427  k=k-2
428  END IF
429  END DO
430 *
431 * inv(U**T) = (inv(U))**T
432 *
433 * inv(U**T)*inv(D)*inv(U)
434 *
435  cut=0
436  DO WHILE (cut .LT. n)
437  nnb=nb
438  IF (cut + nnb .GT. n) THEN
439  nnb=n-cut
440  ELSE
441  count = 0
442 * count negative elements,
443  DO i=cut+1,cut+nnb
444  IF (ipiv(i) .LT. 0) count=count+1
445  END DO
446 * need a even number for a clear cut
447  IF (mod(count,2) .EQ. 1) nnb=nnb+1
448  END IF
449 * L21 Block
450  DO i=1,n-cut-nnb
451  DO j=1,nnb
452  work(i,j)=a(cut+nnb+i,cut+j)
453  END DO
454  END DO
455 * L11 Block
456  DO i=1,nnb
457  work(u11+i,i)=one
458  DO j=i+1,nnb
459  work(u11+i,j)=zero
460  END DO
461  DO j=1,i-1
462  work(u11+i,j)=a(cut+i,cut+j)
463  END DO
464  END DO
465 *
466 * invD*L21
467 *
468  i=n-cut-nnb
469  DO WHILE (i .GE. 1)
470  IF (ipiv(cut+nnb+i) > 0) THEN
471  DO j=1,nnb
472  work(i,j)=work(cut+nnb+i,invd)*work(i,j)
473  END DO
474  i=i-1
475  ELSE
476  DO j=1,nnb
477  u01_i_j = work(i,j)
478  u01_ip1_j = work(i-1,j)
479  work(i,j)=work(cut+nnb+i,invd)*u01_i_j+
480  \$ work(cut+nnb+i,invd+1)*u01_ip1_j
481  work(i-1,j)=work(cut+nnb+i-1,invd+1)*u01_i_j+
482  \$ work(cut+nnb+i-1,invd)*u01_ip1_j
483  END DO
484  i=i-2
485  END IF
486  END DO
487 *
488 * invD1*L11
489 *
490  i=nnb
491  DO WHILE (i .GE. 1)
492  IF (ipiv(cut+i) > 0) THEN
493  DO j=1,nnb
494  work(u11+i,j)=work(cut+i,invd)*work(u11+i,j)
495  END DO
496  i=i-1
497  ELSE
498  DO j=1,nnb
499  u11_i_j = work(u11+i,j)
500  u11_ip1_j = work(u11+i-1,j)
501  work(u11+i,j)=work(cut+i,invd)*work(u11+i,j) +
502  \$ work(cut+i,invd+1)*u11_ip1_j
503  work(u11+i-1,j)=work(cut+i-1,invd+1)*u11_i_j+
504  \$ work(cut+i-1,invd)*u11_ip1_j
505  END DO
506  i=i-2
507  END IF
508  END DO
509 *
510 * L11**T*invD1*L11->L11
511 *
512  CALL strmm('L',uplo,'T','U',nnb, nnb,
513  \$ one,a(cut+1,cut+1),lda,work(u11+1,1),n+nb+1)
514
515 *
516  DO i=1,nnb
517  DO j=1,i
518  a(cut+i,cut+j)=work(u11+i,j)
519  END DO
520  END DO
521 *
522  IF ( (cut+nnb) .LT. n ) THEN
523 *
524 * L21**T*invD2*L21->A(CUT+I,CUT+J)
525 *
526  CALL sgemm('T','N',nnb,nnb,n-nnb-cut,one,a(cut+nnb+1,cut+1)
527  \$ ,lda,work,n+nb+1, zero, work(u11+1,1), n+nb+1)
528
529 *
530 * L11 = L11**T*invD1*L11 + U01**T*invD*U01
531 *
532  DO i=1,nnb
533  DO j=1,i
534  a(cut+i,cut+j)=a(cut+i,cut+j)+work(u11+i,j)
535  END DO
536  END DO
537 *
538 * L01 = L22**T*invD2*L21
539 *
540  CALL strmm('L',uplo,'T','U', n-nnb-cut, nnb,
541  \$ one,a(cut+nnb+1,cut+nnb+1),lda,work,n+nb+1)
542 *
543 * Update L21
544 *
545  DO i=1,n-cut-nnb
546  DO j=1,nnb
547  a(cut+nnb+i,cut+j)=work(i,j)
548  END DO
549  END DO
550
551  ELSE
552 *
553 * L11 = L11**T*invD1*L11
554 *
555  DO i=1,nnb
556  DO j=1,i
557  a(cut+i,cut+j)=work(u11+i,j)
558  END DO
559  END DO
560  END IF
561 *
562 * Next Block
563 *
564  cut=cut+nnb
565  END DO
566 *
567 * Apply PERMUTATIONS P and P**T: P * inv(U**T)*inv(D)*inv(U) *P**T
568 *
569  i=n
570  DO WHILE ( i .GE. 1 )
571  IF( ipiv(i) .GT. 0 ) THEN
572  ip=ipiv(i)
573  IF (i .LT. ip) CALL ssyswapr( uplo, n, a, lda, i ,ip )
574  IF (i .GT. ip) CALL ssyswapr( uplo, n, a, lda, ip ,i )
575  ELSE
576  ip=-ipiv(i)
577  IF ( i .LT. ip) CALL ssyswapr( uplo, n, a, lda, i ,ip )
578  IF ( i .GT. ip) CALL ssyswapr( uplo, n, a, lda, ip ,i )
579  i=i-1
580  ENDIF
581  i=i-1
582  END DO
583  END IF
584 *
585  return
586 *
587 * End of SSYTRI2X
588 *
589  END
590