LAPACK  3.6.1
LAPACK: Linear Algebra PACKage
zlavhe.f
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1 *> \brief \b ZLAVHE
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 * Definition:
9 * ===========
10 *
11 * SUBROUTINE ZLAVHE( UPLO, TRANS, DIAG, N, NRHS, A, LDA, IPIV, B,
12 * LDB, INFO )
13 *
14 * .. Scalar Arguments ..
15 * CHARACTER DIAG, TRANS, UPLO
16 * INTEGER INFO, LDA, LDB, N, NRHS
17 * ..
18 * .. Array Arguments ..
19 * INTEGER IPIV( * )
20 * COMPLEX*16 A( LDA, * ), B( LDB, * )
21 * ..
22 *
23 *
24 *> \par Purpose:
25 * =============
26 *>
27 *> \verbatim
28 *>
29 *> ZLAVHE performs one of the matrix-vector operations
30 *> x := A*x or x := A^H*x,
31 *> where x is an N element vector and A is one of the factors
32 *> from the block U*D*U' or L*D*L' factorization computed by ZHETRF.
33 *>
34 *> If TRANS = 'N', multiplies by U or U * D (or L or L * D)
35 *> If TRANS = 'C', multiplies by U' or D * U' (or L' or D * L')
36 *> \endverbatim
37 *
38 * Arguments:
39 * ==========
40 *
41 *> \param[in] UPLO
42 *> \verbatim
43 *> UPLO is CHARACTER*1
44 *> Specifies whether the factor stored in A is upper or lower
45 *> triangular.
46 *> = 'U': Upper triangular
47 *> = 'L': Lower triangular
48 *> \endverbatim
49 *>
50 *> \param[in] TRANS
51 *> \verbatim
52 *> TRANS is CHARACTER*1
53 *> Specifies the operation to be performed:
54 *> = 'N': x := A*x
55 *> = 'C': x := A'*x
56 *> \endverbatim
57 *>
58 *> \param[in] DIAG
59 *> \verbatim
60 *> DIAG is CHARACTER*1
61 *> Specifies whether or not the diagonal blocks are unit
62 *> matrices. If the diagonal blocks are assumed to be unit,
63 *> then A = U or A = L, otherwise A = U*D or A = L*D.
64 *> = 'U': Diagonal blocks are assumed to be unit matrices.
65 *> = 'N': Diagonal blocks are assumed to be non-unit matrices.
66 *> \endverbatim
67 *>
68 *> \param[in] N
69 *> \verbatim
70 *> N is INTEGER
71 *> The number of rows and columns of the matrix A. N >= 0.
72 *> \endverbatim
73 *>
74 *> \param[in] NRHS
75 *> \verbatim
76 *> NRHS is INTEGER
77 *> The number of right hand sides, i.e., the number of vectors
78 *> x to be multiplied by A. NRHS >= 0.
79 *> \endverbatim
80 *>
81 *> \param[in] A
82 *> \verbatim
83 *> A is COMPLEX*16 array, dimension (LDA,N)
84 *> The block diagonal matrix D and the multipliers used to
85 *> obtain the factor U or L as computed by ZHETRF.
86 *> Stored as a 2-D triangular matrix.
87 *> \endverbatim
88 *>
89 *> \param[in] LDA
90 *> \verbatim
91 *> LDA is INTEGER
92 *> The leading dimension of the array A. LDA >= max(1,N).
93 *> \endverbatim
94 *>
95 *> \param[in] IPIV
96 *> \verbatim
97 *> IPIV is INTEGER array, dimension (N)
98 *> Details of the interchanges and the block structure of D,
99 *> as determined by ZHETRF.
100 *>
101 *> If UPLO = 'U':
102 *> If IPIV(k) > 0, then rows and columns k and IPIV(k)
103 *> were interchanged and D(k,k) is a 1-by-1 diagonal block.
104 *> (If IPIV( k ) = k, no interchange was done).
105 *>
106 *> If IPIV(k) = IPIV(k-1) < 0, then rows and
107 *> columns k-1 and -IPIV(k) were interchanged,
108 *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
109 *>
110 *> If UPLO = 'L':
111 *> If IPIV(k) > 0, then rows and columns k and IPIV(k)
112 *> were interchanged and D(k,k) is a 1-by-1 diagonal block.
113 *> (If IPIV( k ) = k, no interchange was done).
114 *>
115 *> If IPIV(k) = IPIV(k+1) < 0, then rows and
116 *> columns k+1 and -IPIV(k) were interchanged,
117 *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
118 *> \endverbatim
119 *>
120 *> \param[in,out] B
121 *> \verbatim
122 *> B is COMPLEX*16 array, dimension (LDB,NRHS)
123 *> On entry, B contains NRHS vectors of length N.
124 *> On exit, B is overwritten with the product A * B.
125 *> \endverbatim
126 *>
127 *> \param[in] LDB
128 *> \verbatim
129 *> LDB is INTEGER
130 *> The leading dimension of the array B. LDB >= max(1,N).
131 *> \endverbatim
132 *>
133 *> \param[out] INFO
134 *> \verbatim
135 *> INFO is INTEGER
136 *> = 0: successful exit
137 *> < 0: if INFO = -k, the k-th argument had an illegal value
138 *> \endverbatim
139 *
140 * Authors:
141 * ========
142 *
143 *> \author Univ. of Tennessee
144 *> \author Univ. of California Berkeley
145 *> \author Univ. of Colorado Denver
146 *> \author NAG Ltd.
147 *
148 *> \date November 2013
149 *
150 *> \ingroup complex16_lin
151 *
152 * =====================================================================
153  SUBROUTINE zlavhe( UPLO, TRANS, DIAG, N, NRHS, A, LDA, IPIV, B,
154  $ ldb, info )
155 *
156 * -- LAPACK test routine (version 3.5.0) --
157 * -- LAPACK is a software package provided by Univ. of Tennessee, --
158 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
159 * November 2013
160 *
161 * .. Scalar Arguments ..
162  CHARACTER DIAG, TRANS, UPLO
163  INTEGER INFO, LDA, LDB, N, NRHS
164 * ..
165 * .. Array Arguments ..
166  INTEGER IPIV( * )
167  COMPLEX*16 A( lda, * ), B( ldb, * )
168 * ..
169 *
170 * =====================================================================
171 *
172 * .. Parameters ..
173  COMPLEX*16 ONE
174  parameter ( one = ( 1.0d+0, 0.0d+0 ) )
175 * ..
176 * .. Local Scalars ..
177  LOGICAL NOUNIT
178  INTEGER J, K, KP
179  COMPLEX*16 D11, D12, D21, D22, T1, T2
180 * ..
181 * .. External Functions ..
182  LOGICAL LSAME
183  EXTERNAL lsame
184 * ..
185 * .. External Subroutines ..
186  EXTERNAL xerbla, zgemv, zgeru, zlacgv, zscal, zswap
187 * ..
188 * .. Intrinsic Functions ..
189  INTRINSIC abs, dconjg, max
190 * ..
191 * .. Executable Statements ..
192 *
193 * Test the input parameters.
194 *
195  info = 0
196  IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
197  info = -1
198  ELSE IF( .NOT.lsame( trans, 'N' ) .AND. .NOT.lsame( trans, 'C' ) )
199  $ THEN
200  info = -2
201  ELSE IF( .NOT.lsame( diag, 'U' ) .AND. .NOT.lsame( diag, 'N' ) )
202  $ THEN
203  info = -3
204  ELSE IF( n.LT.0 ) THEN
205  info = -4
206  ELSE IF( lda.LT.max( 1, n ) ) THEN
207  info = -6
208  ELSE IF( ldb.LT.max( 1, n ) ) THEN
209  info = -9
210  END IF
211  IF( info.NE.0 ) THEN
212  CALL xerbla( 'ZLAVHE ', -info )
213  RETURN
214  END IF
215 *
216 * Quick return if possible.
217 *
218  IF( n.EQ.0 )
219  $ RETURN
220 *
221  nounit = lsame( diag, 'N' )
222 *------------------------------------------
223 *
224 * Compute B := A * B (No transpose)
225 *
226 *------------------------------------------
227  IF( lsame( trans, 'N' ) ) THEN
228 *
229 * Compute B := U*B
230 * where U = P(m)*inv(U(m))* ... *P(1)*inv(U(1))
231 *
232  IF( lsame( uplo, 'U' ) ) THEN
233 *
234 * Loop forward applying the transformations.
235 *
236  k = 1
237  10 CONTINUE
238  IF( k.GT.n )
239  $ GO TO 30
240  IF( ipiv( k ).GT.0 ) THEN
241 *
242 * 1 x 1 pivot block
243 *
244 * Multiply by the diagonal element if forming U * D.
245 *
246  IF( nounit )
247  $ CALL zscal( nrhs, a( k, k ), b( k, 1 ), ldb )
248 *
249 * Multiply by P(K) * inv(U(K)) if K > 1.
250 *
251  IF( k.GT.1 ) THEN
252 *
253 * Apply the transformation.
254 *
255  CALL zgeru( k-1, nrhs, one, a( 1, k ), 1, b( k, 1 ),
256  $ ldb, b( 1, 1 ), ldb )
257 *
258 * Interchange if P(K) != I.
259 *
260  kp = ipiv( k )
261  IF( kp.NE.k )
262  $ CALL zswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
263  END IF
264  k = k + 1
265  ELSE
266 *
267 * 2 x 2 pivot block
268 *
269 * Multiply by the diagonal block if forming U * D.
270 *
271  IF( nounit ) THEN
272  d11 = a( k, k )
273  d22 = a( k+1, k+1 )
274  d12 = a( k, k+1 )
275  d21 = dconjg( d12 )
276  DO 20 j = 1, nrhs
277  t1 = b( k, j )
278  t2 = b( k+1, j )
279  b( k, j ) = d11*t1 + d12*t2
280  b( k+1, j ) = d21*t1 + d22*t2
281  20 CONTINUE
282  END IF
283 *
284 * Multiply by P(K) * inv(U(K)) if K > 1.
285 *
286  IF( k.GT.1 ) THEN
287 *
288 * Apply the transformations.
289 *
290  CALL zgeru( k-1, nrhs, one, a( 1, k ), 1, b( k, 1 ),
291  $ ldb, b( 1, 1 ), ldb )
292  CALL zgeru( k-1, nrhs, one, a( 1, k+1 ), 1,
293  $ b( k+1, 1 ), ldb, b( 1, 1 ), ldb )
294 *
295 * Interchange if P(K) != I.
296 *
297  kp = abs( ipiv( k ) )
298  IF( kp.NE.k )
299  $ CALL zswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
300  END IF
301  k = k + 2
302  END IF
303  GO TO 10
304  30 CONTINUE
305 *
306 * Compute B := L*B
307 * where L = P(1)*inv(L(1))* ... *P(m)*inv(L(m)) .
308 *
309  ELSE
310 *
311 * Loop backward applying the transformations to B.
312 *
313  k = n
314  40 CONTINUE
315  IF( k.LT.1 )
316  $ GO TO 60
317 *
318 * Test the pivot index. If greater than zero, a 1 x 1
319 * pivot was used, otherwise a 2 x 2 pivot was used.
320 *
321  IF( ipiv( k ).GT.0 ) THEN
322 *
323 * 1 x 1 pivot block:
324 *
325 * Multiply by the diagonal element if forming L * D.
326 *
327  IF( nounit )
328  $ CALL zscal( nrhs, a( k, k ), b( k, 1 ), ldb )
329 *
330 * Multiply by P(K) * inv(L(K)) if K < N.
331 *
332  IF( k.NE.n ) THEN
333  kp = ipiv( k )
334 *
335 * Apply the transformation.
336 *
337  CALL zgeru( n-k, nrhs, one, a( k+1, k ), 1, b( k, 1 ),
338  $ ldb, b( k+1, 1 ), ldb )
339 *
340 * Interchange if a permutation was applied at the
341 * K-th step of the factorization.
342 *
343  IF( kp.NE.k )
344  $ CALL zswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
345  END IF
346  k = k - 1
347 *
348  ELSE
349 *
350 * 2 x 2 pivot block:
351 *
352 * Multiply by the diagonal block if forming L * D.
353 *
354  IF( nounit ) THEN
355  d11 = a( k-1, k-1 )
356  d22 = a( k, k )
357  d21 = a( k, k-1 )
358  d12 = dconjg( d21 )
359  DO 50 j = 1, nrhs
360  t1 = b( k-1, j )
361  t2 = b( k, j )
362  b( k-1, j ) = d11*t1 + d12*t2
363  b( k, j ) = d21*t1 + d22*t2
364  50 CONTINUE
365  END IF
366 *
367 * Multiply by P(K) * inv(L(K)) if K < N.
368 *
369  IF( k.NE.n ) THEN
370 *
371 * Apply the transformation.
372 *
373  CALL zgeru( n-k, nrhs, one, a( k+1, k ), 1, b( k, 1 ),
374  $ ldb, b( k+1, 1 ), ldb )
375  CALL zgeru( n-k, nrhs, one, a( k+1, k-1 ), 1,
376  $ b( k-1, 1 ), ldb, b( k+1, 1 ), ldb )
377 *
378 * Interchange if a permutation was applied at the
379 * K-th step of the factorization.
380 *
381  kp = abs( ipiv( k ) )
382  IF( kp.NE.k )
383  $ CALL zswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
384  END IF
385  k = k - 2
386  END IF
387  GO TO 40
388  60 CONTINUE
389  END IF
390 *--------------------------------------------------
391 *
392 * Compute B := A^H * B (conjugate transpose)
393 *
394 *--------------------------------------------------
395  ELSE
396 *
397 * Form B := U^H*B
398 * where U = P(m)*inv(U(m))* ... *P(1)*inv(U(1))
399 * and U^H = inv(U^H(1))*P(1)* ... *inv(U^H(m))*P(m)
400 *
401  IF( lsame( uplo, 'U' ) ) THEN
402 *
403 * Loop backward applying the transformations.
404 *
405  k = n
406  70 CONTINUE
407  IF( k.LT.1 )
408  $ GO TO 90
409 *
410 * 1 x 1 pivot block.
411 *
412  IF( ipiv( k ).GT.0 ) THEN
413  IF( k.GT.1 ) THEN
414 *
415 * Interchange if P(K) != I.
416 *
417  kp = ipiv( k )
418  IF( kp.NE.k )
419  $ CALL zswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
420 *
421 * Apply the transformation
422 * y = y - B' conjg(x),
423 * where x is a column of A and y is a row of B.
424 *
425  CALL zlacgv( nrhs, b( k, 1 ), ldb )
426  CALL zgemv( 'Conjugate', k-1, nrhs, one, b, ldb,
427  $ a( 1, k ), 1, one, b( k, 1 ), ldb )
428  CALL zlacgv( nrhs, b( k, 1 ), ldb )
429  END IF
430  IF( nounit )
431  $ CALL zscal( nrhs, a( k, k ), b( k, 1 ), ldb )
432  k = k - 1
433 *
434 * 2 x 2 pivot block.
435 *
436  ELSE
437  IF( k.GT.2 ) THEN
438 *
439 * Interchange if P(K) != I.
440 *
441  kp = abs( ipiv( k ) )
442  IF( kp.NE.k-1 )
443  $ CALL zswap( nrhs, b( k-1, 1 ), ldb, b( kp, 1 ),
444  $ ldb )
445 *
446 * Apply the transformations
447 * y = y - B' conjg(x),
448 * where x is a block column of A and y is a block
449 * row of B.
450 *
451  CALL zlacgv( nrhs, b( k, 1 ), ldb )
452  CALL zgemv( 'Conjugate', k-2, nrhs, one, b, ldb,
453  $ a( 1, k ), 1, one, b( k, 1 ), ldb )
454  CALL zlacgv( nrhs, b( k, 1 ), ldb )
455 *
456  CALL zlacgv( nrhs, b( k-1, 1 ), ldb )
457  CALL zgemv( 'Conjugate', k-2, nrhs, one, b, ldb,
458  $ a( 1, k-1 ), 1, one, b( k-1, 1 ), ldb )
459  CALL zlacgv( nrhs, b( k-1, 1 ), ldb )
460  END IF
461 *
462 * Multiply by the diagonal block if non-unit.
463 *
464  IF( nounit ) THEN
465  d11 = a( k-1, k-1 )
466  d22 = a( k, k )
467  d12 = a( k-1, k )
468  d21 = dconjg( d12 )
469  DO 80 j = 1, nrhs
470  t1 = b( k-1, j )
471  t2 = b( k, j )
472  b( k-1, j ) = d11*t1 + d12*t2
473  b( k, j ) = d21*t1 + d22*t2
474  80 CONTINUE
475  END IF
476  k = k - 2
477  END IF
478  GO TO 70
479  90 CONTINUE
480 *
481 * Form B := L^H*B
482 * where L = P(1)*inv(L(1))* ... *P(m)*inv(L(m))
483 * and L^H = inv(L^H(m))*P(m)* ... *inv(L^H(1))*P(1)
484 *
485  ELSE
486 *
487 * Loop forward applying the L-transformations.
488 *
489  k = 1
490  100 CONTINUE
491  IF( k.GT.n )
492  $ GO TO 120
493 *
494 * 1 x 1 pivot block
495 *
496  IF( ipiv( k ).GT.0 ) THEN
497  IF( k.LT.n ) THEN
498 *
499 * Interchange if P(K) != I.
500 *
501  kp = ipiv( k )
502  IF( kp.NE.k )
503  $ CALL zswap( nrhs, b( k, 1 ), ldb, b( kp, 1 ), ldb )
504 *
505 * Apply the transformation
506 *
507  CALL zlacgv( nrhs, b( k, 1 ), ldb )
508  CALL zgemv( 'Conjugate', n-k, nrhs, one, b( k+1, 1 ),
509  $ ldb, a( k+1, k ), 1, one, b( k, 1 ), ldb )
510  CALL zlacgv( nrhs, b( k, 1 ), ldb )
511  END IF
512  IF( nounit )
513  $ CALL zscal( nrhs, a( k, k ), b( k, 1 ), ldb )
514  k = k + 1
515 *
516 * 2 x 2 pivot block.
517 *
518  ELSE
519  IF( k.LT.n-1 ) THEN
520 *
521 * Interchange if P(K) != I.
522 *
523  kp = abs( ipiv( k ) )
524  IF( kp.NE.k+1 )
525  $ CALL zswap( nrhs, b( k+1, 1 ), ldb, b( kp, 1 ),
526  $ ldb )
527 *
528 * Apply the transformation
529 *
530  CALL zlacgv( nrhs, b( k+1, 1 ), ldb )
531  CALL zgemv( 'Conjugate', n-k-1, nrhs, one,
532  $ b( k+2, 1 ), ldb, a( k+2, k+1 ), 1, one,
533  $ b( k+1, 1 ), ldb )
534  CALL zlacgv( nrhs, b( k+1, 1 ), ldb )
535 *
536  CALL zlacgv( nrhs, b( k, 1 ), ldb )
537  CALL zgemv( 'Conjugate', n-k-1, nrhs, one,
538  $ b( k+2, 1 ), ldb, a( k+2, k ), 1, one,
539  $ b( k, 1 ), ldb )
540  CALL zlacgv( nrhs, b( k, 1 ), ldb )
541  END IF
542 *
543 * Multiply by the diagonal block if non-unit.
544 *
545  IF( nounit ) THEN
546  d11 = a( k, k )
547  d22 = a( k+1, k+1 )
548  d21 = a( k+1, k )
549  d12 = dconjg( d21 )
550  DO 110 j = 1, nrhs
551  t1 = b( k, j )
552  t2 = b( k+1, j )
553  b( k, j ) = d11*t1 + d12*t2
554  b( k+1, j ) = d21*t1 + d22*t2
555  110 CONTINUE
556  END IF
557  k = k + 2
558  END IF
559  GO TO 100
560  120 CONTINUE
561  END IF
562 *
563  END IF
564  RETURN
565 *
566 * End of ZLAVHE
567 *
568  END
zgemv
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:160
zswap
subroutine zswap(N, ZX, INCX, ZY, INCY)
ZSWAP
Definition: zswap.f:52
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
zlavhe
subroutine zlavhe(UPLO, TRANS, DIAG, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
ZLAVHE
Definition: zlavhe.f:155
zgeru
subroutine zgeru(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
ZGERU
Definition: zgeru.f:132
zscal
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:54
zlacgv
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:76