LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
zlahef_rk.f
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1 *> \brief \b ZLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZLAHEF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW,
22 * INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER UPLO
26 * INTEGER INFO, KB, LDA, LDW, N, NB
27 * ..
28 * .. Array Arguments ..
29 * INTEGER IPIV( * )
30 * COMPLEX*16 A( LDA, * ), E( * ), W( LDW, * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *> ZLAHEF_RK computes a partial factorization of a complex Hermitian
39 *> matrix A using the bounded Bunch-Kaufman (rook) diagonal
40 *> pivoting method. The partial factorization has the form:
41 *>
42 *> A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
43 *> ( 0 U22 ) ( 0 D ) ( U12**H U22**H )
44 *>
45 *> A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) if UPLO = 'L',
46 *> ( L21 I ) ( 0 A22 ) ( 0 I )
47 *>
48 *> where the order of D is at most NB. The actual order is returned in
49 *> the argument KB, and is either NB or NB-1, or N if N <= NB.
50 *>
51 *> ZLAHEF_RK is an auxiliary routine called by ZHETRF_RK. It uses
52 *> blocked code (calling Level 3 BLAS) to update the submatrix
53 *> A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
54 *> \endverbatim
55 *
56 * Arguments:
57 * ==========
58 *
59 *> \param[in] UPLO
60 *> \verbatim
61 *> UPLO is CHARACTER*1
62 *> Specifies whether the upper or lower triangular part of the
63 *> Hermitian matrix A is stored:
64 *> = 'U': Upper triangular
65 *> = 'L': Lower triangular
66 *> \endverbatim
67 *>
68 *> \param[in] N
69 *> \verbatim
70 *> N is INTEGER
71 *> The order of the matrix A. N >= 0.
72 *> \endverbatim
73 *>
74 *> \param[in] NB
75 *> \verbatim
76 *> NB is INTEGER
77 *> The maximum number of columns of the matrix A that should be
78 *> factored. NB should be at least 2 to allow for 2-by-2 pivot
79 *> blocks.
80 *> \endverbatim
81 *>
82 *> \param[out] KB
83 *> \verbatim
84 *> KB is INTEGER
85 *> The number of columns of A that were actually factored.
86 *> KB is either NB-1 or NB, or N if N <= NB.
87 *> \endverbatim
88 *>
89 *> \param[in,out] A
90 *> \verbatim
91 *> A is COMPLEX*16 array, dimension (LDA,N)
92 *> On entry, the Hermitian matrix A.
93 *> If UPLO = 'U': the leading N-by-N upper triangular part
94 *> of A contains the upper triangular part of the matrix A,
95 *> and the strictly lower triangular part of A is not
96 *> referenced.
97 *>
98 *> If UPLO = 'L': the leading N-by-N lower triangular part
99 *> of A contains the lower triangular part of the matrix A,
100 *> and the strictly upper triangular part of A is not
101 *> referenced.
102 *>
103 *> On exit, contains:
104 *> a) ONLY diagonal elements of the Hermitian block diagonal
105 *> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
106 *> (superdiagonal (or subdiagonal) elements of D
107 *> are stored on exit in array E), and
108 *> b) If UPLO = 'U': factor U in the superdiagonal part of A.
109 *> If UPLO = 'L': factor L in the subdiagonal part of A.
110 *> \endverbatim
111 *>
112 *> \param[in] LDA
113 *> \verbatim
114 *> LDA is INTEGER
115 *> The leading dimension of the array A. LDA >= max(1,N).
116 *> \endverbatim
117 *>
118 *> \param[out] E
119 *> \verbatim
120 *> E is COMPLEX*16 array, dimension (N)
121 *> On exit, contains the superdiagonal (or subdiagonal)
122 *> elements of the Hermitian block diagonal matrix D
123 *> with 1-by-1 or 2-by-2 diagonal blocks, where
124 *> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
125 *> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
126 *>
127 *> NOTE: For 1-by-1 diagonal block D(k), where
128 *> 1 <= k <= N, the element E(k) is set to 0 in both
129 *> UPLO = 'U' or UPLO = 'L' cases.
130 *> \endverbatim
131 *>
132 *> \param[out] IPIV
133 *> \verbatim
134 *> IPIV is INTEGER array, dimension (N)
135 *> IPIV describes the permutation matrix P in the factorization
136 *> of matrix A as follows. The absolute value of IPIV(k)
137 *> represents the index of row and column that were
138 *> interchanged with the k-th row and column. The value of UPLO
139 *> describes the order in which the interchanges were applied.
140 *> Also, the sign of IPIV represents the block structure of
141 *> the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
142 *> diagonal blocks which correspond to 1 or 2 interchanges
143 *> at each factorization step.
144 *>
145 *> If UPLO = 'U',
146 *> ( in factorization order, k decreases from N to 1 ):
147 *> a) A single positive entry IPIV(k) > 0 means:
148 *> D(k,k) is a 1-by-1 diagonal block.
149 *> If IPIV(k) != k, rows and columns k and IPIV(k) were
150 *> interchanged in the submatrix A(1:N,N-KB+1:N);
151 *> If IPIV(k) = k, no interchange occurred.
152 *>
153 *>
154 *> b) A pair of consecutive negative entries
155 *> IPIV(k) < 0 and IPIV(k-1) < 0 means:
156 *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
157 *> (NOTE: negative entries in IPIV appear ONLY in pairs).
158 *> 1) If -IPIV(k) != k, rows and columns
159 *> k and -IPIV(k) were interchanged
160 *> in the matrix A(1:N,N-KB+1:N).
161 *> If -IPIV(k) = k, no interchange occurred.
162 *> 2) If -IPIV(k-1) != k-1, rows and columns
163 *> k-1 and -IPIV(k-1) were interchanged
164 *> in the submatrix A(1:N,N-KB+1:N).
165 *> If -IPIV(k-1) = k-1, no interchange occurred.
166 *>
167 *> c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.
168 *>
169 *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
170 *>
171 *> If UPLO = 'L',
172 *> ( in factorization order, k increases from 1 to N ):
173 *> a) A single positive entry IPIV(k) > 0 means:
174 *> D(k,k) is a 1-by-1 diagonal block.
175 *> If IPIV(k) != k, rows and columns k and IPIV(k) were
176 *> interchanged in the submatrix A(1:N,1:KB).
177 *> If IPIV(k) = k, no interchange occurred.
178 *>
179 *> b) A pair of consecutive negative entries
180 *> IPIV(k) < 0 and IPIV(k+1) < 0 means:
181 *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
182 *> (NOTE: negative entries in IPIV appear ONLY in pairs).
183 *> 1) If -IPIV(k) != k, rows and columns
184 *> k and -IPIV(k) were interchanged
185 *> in the submatrix A(1:N,1:KB).
186 *> If -IPIV(k) = k, no interchange occurred.
187 *> 2) If -IPIV(k+1) != k+1, rows and columns
188 *> k-1 and -IPIV(k-1) were interchanged
189 *> in the submatrix A(1:N,1:KB).
190 *> If -IPIV(k+1) = k+1, no interchange occurred.
191 *>
192 *> c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.
193 *>
194 *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
195 *> \endverbatim
196 *>
197 *> \param[out] W
198 *> \verbatim
199 *> W is COMPLEX*16 array, dimension (LDW,NB)
200 *> \endverbatim
201 *>
202 *> \param[in] LDW
203 *> \verbatim
204 *> LDW is INTEGER
205 *> The leading dimension of the array W. LDW >= max(1,N).
206 *> \endverbatim
207 *>
208 *> \param[out] INFO
209 *> \verbatim
210 *> INFO is INTEGER
211 *> = 0: successful exit
212 *>
213 *> < 0: If INFO = -k, the k-th argument had an illegal value
214 *>
215 *> > 0: If INFO = k, the matrix A is singular, because:
216 *> If UPLO = 'U': column k in the upper
217 *> triangular part of A contains all zeros.
218 *> If UPLO = 'L': column k in the lower
219 *> triangular part of A contains all zeros.
220 *>
221 *> Therefore D(k,k) is exactly zero, and superdiagonal
222 *> elements of column k of U (or subdiagonal elements of
223 *> column k of L ) are all zeros. The factorization has
224 *> been completed, but the block diagonal matrix D is
225 *> exactly singular, and division by zero will occur if
226 *> it is used to solve a system of equations.
227 *>
228 *> NOTE: INFO only stores the first occurrence of
229 *> a singularity, any subsequent occurrence of singularity
230 *> is not stored in INFO even though the factorization
231 *> always completes.
232 *> \endverbatim
233 *
234 * Authors:
235 * ========
236 *
237 *> \author Univ. of Tennessee
238 *> \author Univ. of California Berkeley
239 *> \author Univ. of Colorado Denver
240 *> \author NAG Ltd.
241 *
242 *> \ingroup complex16HEcomputational
243 *
244 *> \par Contributors:
245 * ==================
246 *>
247 *> \verbatim
248 *>
249 *> December 2016, Igor Kozachenko,
250 *> Computer Science Division,
251 *> University of California, Berkeley
252 *>
253 *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
254 *> School of Mathematics,
255 *> University of Manchester
256 *>
257 *> \endverbatim
258 *
259 * =====================================================================
260  SUBROUTINE zlahef_rk( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW,
261  $ INFO )
262 *
263 * -- LAPACK computational routine --
264 * -- LAPACK is a software package provided by Univ. of Tennessee, --
265 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
266 *
267 * .. Scalar Arguments ..
268  CHARACTER UPLO
269  INTEGER INFO, KB, LDA, LDW, N, NB
270 * ..
271 * .. Array Arguments ..
272  INTEGER IPIV( * )
273  COMPLEX*16 A( LDA, * ), W( LDW, * ), E( * )
274 * ..
275 *
276 * =====================================================================
277 *
278 * .. Parameters ..
279  DOUBLE PRECISION ZERO, ONE
280  parameter( zero = 0.0d+0, one = 1.0d+0 )
281  COMPLEX*16 CONE
282  parameter( cone = ( 1.0d+0, 0.0d+0 ) )
283  DOUBLE PRECISION EIGHT, SEVTEN
284  parameter( eight = 8.0d+0, sevten = 17.0d+0 )
285  COMPLEX*16 CZERO
286  parameter( czero = ( 0.0d+0, 0.0d+0 ) )
287 * ..
288 * .. Local Scalars ..
289  LOGICAL DONE
290  INTEGER IMAX, ITEMP, II, J, JB, JJ, JMAX, K, KK, KKW,
291  $ kp, kstep, kw, p
292  DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, DTEMP, R1, ROWMAX, T,
293  $ sfmin
294  COMPLEX*16 D11, D21, D22, Z
295 * ..
296 * .. External Functions ..
297  LOGICAL LSAME
298  INTEGER IZAMAX
299  DOUBLE PRECISION DLAMCH
300  EXTERNAL lsame, izamax, dlamch
301 * ..
302 * .. External Subroutines ..
303  EXTERNAL zcopy, zdscal, zgemm, zgemv, zlacgv, zswap
304 * ..
305 * .. Intrinsic Functions ..
306  INTRINSIC abs, dble, dconjg, dimag, max, min, sqrt
307 * ..
308 * .. Statement Functions ..
309  DOUBLE PRECISION CABS1
310 * ..
311 * .. Statement Function definitions ..
312  cabs1( z ) = abs( dble( z ) ) + abs( dimag( z ) )
313 * ..
314 * .. Executable Statements ..
315 *
316  info = 0
317 *
318 * Initialize ALPHA for use in choosing pivot block size.
319 *
320  alpha = ( one+sqrt( sevten ) ) / eight
321 *
322 * Compute machine safe minimum
323 *
324  sfmin = dlamch( 'S' )
325 *
326  IF( lsame( uplo, 'U' ) ) THEN
327 *
328 * Factorize the trailing columns of A using the upper triangle
329 * of A and working backwards, and compute the matrix W = U12*D
330 * for use in updating A11 (note that conjg(W) is actually stored)
331 * Initialize the first entry of array E, where superdiagonal
332 * elements of D are stored
333 *
334  e( 1 ) = czero
335 *
336 * K is the main loop index, decreasing from N in steps of 1 or 2
337 *
338  k = n
339  10 CONTINUE
340 *
341 * KW is the column of W which corresponds to column K of A
342 *
343  kw = nb + k - n
344 *
345 * Exit from loop
346 *
347  IF( ( k.LE.n-nb+1 .AND. nb.LT.n ) .OR. k.LT.1 )
348  $ GO TO 30
349 *
350  kstep = 1
351  p = k
352 *
353 * Copy column K of A to column KW of W and update it
354 *
355  IF( k.GT.1 )
356  $ CALL zcopy( k-1, a( 1, k ), 1, w( 1, kw ), 1 )
357  w( k, kw ) = dble( a( k, k ) )
358  IF( k.LT.n ) THEN
359  CALL zgemv( 'No transpose', k, n-k, -cone, a( 1, k+1 ), lda,
360  $ w( k, kw+1 ), ldw, cone, w( 1, kw ), 1 )
361  w( k, kw ) = dble( w( k, kw ) )
362  END IF
363 *
364 * Determine rows and columns to be interchanged and whether
365 * a 1-by-1 or 2-by-2 pivot block will be used
366 *
367  absakk = abs( dble( w( k, kw ) ) )
368 *
369 * IMAX is the row-index of the largest off-diagonal element in
370 * column K, and COLMAX is its absolute value.
371 * Determine both COLMAX and IMAX.
372 *
373  IF( k.GT.1 ) THEN
374  imax = izamax( k-1, w( 1, kw ), 1 )
375  colmax = cabs1( w( imax, kw ) )
376  ELSE
377  colmax = zero
378  END IF
379 *
380  IF( max( absakk, colmax ).EQ.zero ) THEN
381 *
382 * Column K is zero or underflow: set INFO and continue
383 *
384  IF( info.EQ.0 )
385  $ info = k
386  kp = k
387  a( k, k ) = dble( w( k, kw ) )
388  IF( k.GT.1 )
389  $ CALL zcopy( k-1, w( 1, kw ), 1, a( 1, k ), 1 )
390 *
391 * Set E( K ) to zero
392 *
393  IF( k.GT.1 )
394  $ e( k ) = czero
395 *
396  ELSE
397 *
398 * ============================================================
399 *
400 * BEGIN pivot search
401 *
402 * Case(1)
403 * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
404 * (used to handle NaN and Inf)
405  IF( .NOT.( absakk.LT.alpha*colmax ) ) THEN
406 *
407 * no interchange, use 1-by-1 pivot block
408 *
409  kp = k
410 *
411  ELSE
412 *
413 * Lop until pivot found
414 *
415  done = .false.
416 *
417  12 CONTINUE
418 *
419 * BEGIN pivot search loop body
420 *
421 *
422 * Copy column IMAX to column KW-1 of W and update it
423 *
424  IF( imax.GT.1 )
425  $ CALL zcopy( imax-1, a( 1, imax ), 1, w( 1, kw-1 ),
426  $ 1 )
427  w( imax, kw-1 ) = dble( a( imax, imax ) )
428 *
429  CALL zcopy( k-imax, a( imax, imax+1 ), lda,
430  $ w( imax+1, kw-1 ), 1 )
431  CALL zlacgv( k-imax, w( imax+1, kw-1 ), 1 )
432 *
433  IF( k.LT.n ) THEN
434  CALL zgemv( 'No transpose', k, n-k, -cone,
435  $ a( 1, k+1 ), lda, w( imax, kw+1 ), ldw,
436  $ cone, w( 1, kw-1 ), 1 )
437  w( imax, kw-1 ) = dble( w( imax, kw-1 ) )
438  END IF
439 *
440 * JMAX is the column-index of the largest off-diagonal
441 * element in row IMAX, and ROWMAX is its absolute value.
442 * Determine both ROWMAX and JMAX.
443 *
444  IF( imax.NE.k ) THEN
445  jmax = imax + izamax( k-imax, w( imax+1, kw-1 ),
446  $ 1 )
447  rowmax = cabs1( w( jmax, kw-1 ) )
448  ELSE
449  rowmax = zero
450  END IF
451 *
452  IF( imax.GT.1 ) THEN
453  itemp = izamax( imax-1, w( 1, kw-1 ), 1 )
454  dtemp = cabs1( w( itemp, kw-1 ) )
455  IF( dtemp.GT.rowmax ) THEN
456  rowmax = dtemp
457  jmax = itemp
458  END IF
459  END IF
460 *
461 * Case(2)
462 * Equivalent to testing for
463 * ABS( DBLE( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX
464 * (used to handle NaN and Inf)
465 *
466  IF( .NOT.( abs( dble( w( imax,kw-1 ) ) )
467  $ .LT.alpha*rowmax ) ) THEN
468 *
469 * interchange rows and columns K and IMAX,
470 * use 1-by-1 pivot block
471 *
472  kp = imax
473 *
474 * copy column KW-1 of W to column KW of W
475 *
476  CALL zcopy( k, w( 1, kw-1 ), 1, w( 1, kw ), 1 )
477 *
478  done = .true.
479 *
480 * Case(3)
481 * Equivalent to testing for ROWMAX.EQ.COLMAX,
482 * (used to handle NaN and Inf)
483 *
484  ELSE IF( ( p.EQ.jmax ) .OR. ( rowmax.LE.colmax ) )
485  $ THEN
486 *
487 * interchange rows and columns K-1 and IMAX,
488 * use 2-by-2 pivot block
489 *
490  kp = imax
491  kstep = 2
492  done = .true.
493 *
494 * Case(4)
495  ELSE
496 *
497 * Pivot not found: set params and repeat
498 *
499  p = imax
500  colmax = rowmax
501  imax = jmax
502 *
503 * Copy updated JMAXth (next IMAXth) column to Kth of W
504 *
505  CALL zcopy( k, w( 1, kw-1 ), 1, w( 1, kw ), 1 )
506 *
507  END IF
508 *
509 *
510 * END pivot search loop body
511 *
512  IF( .NOT.done ) GOTO 12
513 *
514  END IF
515 *
516 * END pivot search
517 *
518 * ============================================================
519 *
520 * KK is the column of A where pivoting step stopped
521 *
522  kk = k - kstep + 1
523 *
524 * KKW is the column of W which corresponds to column KK of A
525 *
526  kkw = nb + kk - n
527 *
528 * Interchange rows and columns P and K.
529 * Updated column P is already stored in column KW of W.
530 *
531  IF( ( kstep.EQ.2 ) .AND. ( p.NE.k ) ) THEN
532 *
533 * Copy non-updated column K to column P of submatrix A
534 * at step K. No need to copy element into columns
535 * K and K-1 of A for 2-by-2 pivot, since these columns
536 * will be later overwritten.
537 *
538  a( p, p ) = dble( a( k, k ) )
539  CALL zcopy( k-1-p, a( p+1, k ), 1, a( p, p+1 ),
540  $ lda )
541  CALL zlacgv( k-1-p, a( p, p+1 ), lda )
542  IF( p.GT.1 )
543  $ CALL zcopy( p-1, a( 1, k ), 1, a( 1, p ), 1 )
544 *
545 * Interchange rows K and P in the last K+1 to N columns of A
546 * (columns K and K-1 of A for 2-by-2 pivot will be
547 * later overwritten). Interchange rows K and P
548 * in last KKW to NB columns of W.
549 *
550  IF( k.LT.n )
551  $ CALL zswap( n-k, a( k, k+1 ), lda, a( p, k+1 ),
552  $ lda )
553  CALL zswap( n-kk+1, w( k, kkw ), ldw, w( p, kkw ),
554  $ ldw )
555  END IF
556 *
557 * Interchange rows and columns KP and KK.
558 * Updated column KP is already stored in column KKW of W.
559 *
560  IF( kp.NE.kk ) THEN
561 *
562 * Copy non-updated column KK to column KP of submatrix A
563 * at step K. No need to copy element into column K
564 * (or K and K-1 for 2-by-2 pivot) of A, since these columns
565 * will be later overwritten.
566 *
567  a( kp, kp ) = dble( a( kk, kk ) )
568  CALL zcopy( kk-1-kp, a( kp+1, kk ), 1, a( kp, kp+1 ),
569  $ lda )
570  CALL zlacgv( kk-1-kp, a( kp, kp+1 ), lda )
571  IF( kp.GT.1 )
572  $ CALL zcopy( kp-1, a( 1, kk ), 1, a( 1, kp ), 1 )
573 *
574 * Interchange rows KK and KP in last K+1 to N columns of A
575 * (columns K (or K and K-1 for 2-by-2 pivot) of A will be
576 * later overwritten). Interchange rows KK and KP
577 * in last KKW to NB columns of W.
578 *
579  IF( k.LT.n )
580  $ CALL zswap( n-k, a( kk, k+1 ), lda, a( kp, k+1 ),
581  $ lda )
582  CALL zswap( n-kk+1, w( kk, kkw ), ldw, w( kp, kkw ),
583  $ ldw )
584  END IF
585 *
586  IF( kstep.EQ.1 ) THEN
587 *
588 * 1-by-1 pivot block D(k): column kw of W now holds
589 *
590 * W(kw) = U(k)*D(k),
591 *
592 * where U(k) is the k-th column of U
593 *
594 * (1) Store subdiag. elements of column U(k)
595 * and 1-by-1 block D(k) in column k of A.
596 * (NOTE: Diagonal element U(k,k) is a UNIT element
597 * and not stored)
598 * A(k,k) := D(k,k) = W(k,kw)
599 * A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k)
600 *
601 * (NOTE: No need to use for Hermitian matrix
602 * A( K, K ) = DBLE( W( K, K) ) to separately copy diagonal
603 * element D(k,k) from W (potentially saves only one load))
604  CALL zcopy( k, w( 1, kw ), 1, a( 1, k ), 1 )
605  IF( k.GT.1 ) THEN
606 *
607 * (NOTE: No need to check if A(k,k) is NOT ZERO,
608 * since that was ensured earlier in pivot search:
609 * case A(k,k) = 0 falls into 2x2 pivot case(3))
610 *
611 * Handle division by a small number
612 *
613  t = dble( a( k, k ) )
614  IF( abs( t ).GE.sfmin ) THEN
615  r1 = one / t
616  CALL zdscal( k-1, r1, a( 1, k ), 1 )
617  ELSE
618  DO 14 ii = 1, k-1
619  a( ii, k ) = a( ii, k ) / t
620  14 CONTINUE
621  END IF
622 *
623 * (2) Conjugate column W(kw)
624 *
625  CALL zlacgv( k-1, w( 1, kw ), 1 )
626 *
627 * Store the superdiagonal element of D in array E
628 *
629  e( k ) = czero
630 *
631  END IF
632 *
633  ELSE
634 *
635 * 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold
636 *
637 * ( W(kw-1) W(kw) ) = ( U(k-1) U(k) )*D(k)
638 *
639 * where U(k) and U(k-1) are the k-th and (k-1)-th columns
640 * of U
641 *
642 * (1) Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2
643 * block D(k-1:k,k-1:k) in columns k-1 and k of A.
644 * (NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT
645 * block and not stored)
646 * A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw)
647 * A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) =
648 * = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) )
649 *
650  IF( k.GT.2 ) THEN
651 *
652 * Factor out the columns of the inverse of 2-by-2 pivot
653 * block D, so that each column contains 1, to reduce the
654 * number of FLOPS when we multiply panel
655 * ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1).
656 *
657 * D**(-1) = ( d11 cj(d21) )**(-1) =
658 * ( d21 d22 )
659 *
660 * = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) =
661 * ( (-d21) ( d11 ) )
662 *
663 * = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) *
664 *
665 * * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) =
666 * ( ( -1 ) ( d11/conj(d21) ) )
667 *
668 * = 1/(|d21|**2) * 1/(D22*D11-1) *
669 *
670 * * ( d21*( D11 ) conj(d21)*( -1 ) ) =
671 * ( ( -1 ) ( D22 ) )
672 *
673 * = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) =
674 * ( ( -1 ) ( D22 ) )
675 *
676 * = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) =
677 * ( ( -1 ) ( D22 ) )
678 *
679 * Handle division by a small number. (NOTE: order of
680 * operations is important)
681 *
682 * = ( T*(( D11 )/conj(D21)) T*(( -1 )/D21 ) )
683 * ( (( -1 ) ) (( D22 ) ) ),
684 *
685 * where D11 = d22/d21,
686 * D22 = d11/conj(d21),
687 * D21 = d21,
688 * T = 1/(D22*D11-1).
689 *
690 * (NOTE: No need to check for division by ZERO,
691 * since that was ensured earlier in pivot search:
692 * (a) d21 != 0 in 2x2 pivot case(4),
693 * since |d21| should be larger than |d11| and |d22|;
694 * (b) (D22*D11 - 1) != 0, since from (a),
695 * both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.)
696 *
697  d21 = w( k-1, kw )
698  d11 = w( k, kw ) / dconjg( d21 )
699  d22 = w( k-1, kw-1 ) / d21
700  t = one / ( dble( d11*d22 )-one )
701 *
702 * Update elements in columns A(k-1) and A(k) as
703 * dot products of rows of ( W(kw-1) W(kw) ) and columns
704 * of D**(-1)
705 *
706  DO 20 j = 1, k - 2
707  a( j, k-1 ) = t*( ( d11*w( j, kw-1 )-w( j, kw ) ) /
708  $ d21 )
709  a( j, k ) = t*( ( d22*w( j, kw )-w( j, kw-1 ) ) /
710  $ dconjg( d21 ) )
711  20 CONTINUE
712  END IF
713 *
714 * Copy diagonal elements of D(K) to A,
715 * copy superdiagonal element of D(K) to E(K) and
716 * ZERO out superdiagonal entry of A
717 *
718  a( k-1, k-1 ) = w( k-1, kw-1 )
719  a( k-1, k ) = czero
720  a( k, k ) = w( k, kw )
721  e( k ) = w( k-1, kw )
722  e( k-1 ) = czero
723 *
724 * (2) Conjugate columns W(kw) and W(kw-1)
725 *
726  CALL zlacgv( k-1, w( 1, kw ), 1 )
727  CALL zlacgv( k-2, w( 1, kw-1 ), 1 )
728 *
729  END IF
730 *
731 * End column K is nonsingular
732 *
733  END IF
734 *
735 * Store details of the interchanges in IPIV
736 *
737  IF( kstep.EQ.1 ) THEN
738  ipiv( k ) = kp
739  ELSE
740  ipiv( k ) = -p
741  ipiv( k-1 ) = -kp
742  END IF
743 *
744 * Decrease K and return to the start of the main loop
745 *
746  k = k - kstep
747  GO TO 10
748 *
749  30 CONTINUE
750 *
751 * Update the upper triangle of A11 (= A(1:k,1:k)) as
752 *
753 * A11 := A11 - U12*D*U12**H = A11 - U12*W**H
754 *
755 * computing blocks of NB columns at a time (note that conjg(W) is
756 * actually stored)
757 *
758  DO 50 j = ( ( k-1 ) / nb )*nb + 1, 1, -nb
759  jb = min( nb, k-j+1 )
760 *
761 * Update the upper triangle of the diagonal block
762 *
763  DO 40 jj = j, j + jb - 1
764  a( jj, jj ) = dble( a( jj, jj ) )
765  CALL zgemv( 'No transpose', jj-j+1, n-k, -cone,
766  $ a( j, k+1 ), lda, w( jj, kw+1 ), ldw, cone,
767  $ a( j, jj ), 1 )
768  a( jj, jj ) = dble( a( jj, jj ) )
769  40 CONTINUE
770 *
771 * Update the rectangular superdiagonal block
772 *
773  IF( j.GE.2 )
774  $ CALL zgemm( 'No transpose', 'Transpose', j-1, jb, n-k,
775  $ -cone, a( 1, k+1 ), lda, w( j, kw+1 ), ldw,
776  $ cone, a( 1, j ), lda )
777  50 CONTINUE
778 *
779 * Set KB to the number of columns factorized
780 *
781  kb = n - k
782 *
783  ELSE
784 *
785 * Factorize the leading columns of A using the lower triangle
786 * of A and working forwards, and compute the matrix W = L21*D
787 * for use in updating A22 (note that conjg(W) is actually stored)
788 *
789 * Initialize the unused last entry of the subdiagonal array E.
790 *
791  e( n ) = czero
792 *
793 * K is the main loop index, increasing from 1 in steps of 1 or 2
794 *
795  k = 1
796  70 CONTINUE
797 *
798 * Exit from loop
799 *
800  IF( ( k.GE.nb .AND. nb.LT.n ) .OR. k.GT.n )
801  $ GO TO 90
802 *
803  kstep = 1
804  p = k
805 *
806 * Copy column K of A to column K of W and update column K of W
807 *
808  w( k, k ) = dble( a( k, k ) )
809  IF( k.LT.n )
810  $ CALL zcopy( n-k, a( k+1, k ), 1, w( k+1, k ), 1 )
811  IF( k.GT.1 ) THEN
812  CALL zgemv( 'No transpose', n-k+1, k-1, -cone, a( k, 1 ),
813  $ lda, w( k, 1 ), ldw, cone, w( k, k ), 1 )
814  w( k, k ) = dble( w( k, k ) )
815  END IF
816 *
817 * Determine rows and columns to be interchanged and whether
818 * a 1-by-1 or 2-by-2 pivot block will be used
819 *
820  absakk = abs( dble( w( k, k ) ) )
821 *
822 * IMAX is the row-index of the largest off-diagonal element in
823 * column K, and COLMAX is its absolute value.
824 * Determine both COLMAX and IMAX.
825 *
826  IF( k.LT.n ) THEN
827  imax = k + izamax( n-k, w( k+1, k ), 1 )
828  colmax = cabs1( w( imax, k ) )
829  ELSE
830  colmax = zero
831  END IF
832 *
833  IF( max( absakk, colmax ).EQ.zero ) THEN
834 *
835 * Column K is zero or underflow: set INFO and continue
836 *
837  IF( info.EQ.0 )
838  $ info = k
839  kp = k
840  a( k, k ) = dble( w( k, k ) )
841  IF( k.LT.n )
842  $ CALL zcopy( n-k, w( k+1, k ), 1, a( k+1, k ), 1 )
843 *
844 * Set E( K ) to zero
845 *
846  IF( k.LT.n )
847  $ e( k ) = czero
848 *
849  ELSE
850 *
851 * ============================================================
852 *
853 * BEGIN pivot search
854 *
855 * Case(1)
856 * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
857 * (used to handle NaN and Inf)
858 *
859  IF( .NOT.( absakk.LT.alpha*colmax ) ) THEN
860 *
861 * no interchange, use 1-by-1 pivot block
862 *
863  kp = k
864 *
865  ELSE
866 *
867  done = .false.
868 *
869 * Loop until pivot found
870 *
871  72 CONTINUE
872 *
873 * BEGIN pivot search loop body
874 *
875 *
876 * Copy column IMAX to column k+1 of W and update it
877 *
878  CALL zcopy( imax-k, a( imax, k ), lda, w( k, k+1 ), 1)
879  CALL zlacgv( imax-k, w( k, k+1 ), 1 )
880  w( imax, k+1 ) = dble( a( imax, imax ) )
881 *
882  IF( imax.LT.n )
883  $ CALL zcopy( n-imax, a( imax+1, imax ), 1,
884  $ w( imax+1, k+1 ), 1 )
885 *
886  IF( k.GT.1 ) THEN
887  CALL zgemv( 'No transpose', n-k+1, k-1, -cone,
888  $ a( k, 1 ), lda, w( imax, 1 ), ldw,
889  $ cone, w( k, k+1 ), 1 )
890  w( imax, k+1 ) = dble( w( imax, k+1 ) )
891  END IF
892 *
893 * JMAX is the column-index of the largest off-diagonal
894 * element in row IMAX, and ROWMAX is its absolute value.
895 * Determine both ROWMAX and JMAX.
896 *
897  IF( imax.NE.k ) THEN
898  jmax = k - 1 + izamax( imax-k, w( k, k+1 ), 1 )
899  rowmax = cabs1( w( jmax, k+1 ) )
900  ELSE
901  rowmax = zero
902  END IF
903 *
904  IF( imax.LT.n ) THEN
905  itemp = imax + izamax( n-imax, w( imax+1, k+1 ), 1)
906  dtemp = cabs1( w( itemp, k+1 ) )
907  IF( dtemp.GT.rowmax ) THEN
908  rowmax = dtemp
909  jmax = itemp
910  END IF
911  END IF
912 *
913 * Case(2)
914 * Equivalent to testing for
915 * ABS( DBLE( W( IMAX,K+1 ) ) ).GE.ALPHA*ROWMAX
916 * (used to handle NaN and Inf)
917 *
918  IF( .NOT.( abs( dble( w( imax,k+1 ) ) )
919  $ .LT.alpha*rowmax ) ) THEN
920 *
921 * interchange rows and columns K and IMAX,
922 * use 1-by-1 pivot block
923 *
924  kp = imax
925 *
926 * copy column K+1 of W to column K of W
927 *
928  CALL zcopy( n-k+1, w( k, k+1 ), 1, w( k, k ), 1 )
929 *
930  done = .true.
931 *
932 * Case(3)
933 * Equivalent to testing for ROWMAX.EQ.COLMAX,
934 * (used to handle NaN and Inf)
935 *
936  ELSE IF( ( p.EQ.jmax ) .OR. ( rowmax.LE.colmax ) )
937  $ THEN
938 *
939 * interchange rows and columns K+1 and IMAX,
940 * use 2-by-2 pivot block
941 *
942  kp = imax
943  kstep = 2
944  done = .true.
945 *
946 * Case(4)
947  ELSE
948 *
949 * Pivot not found: set params and repeat
950 *
951  p = imax
952  colmax = rowmax
953  imax = jmax
954 *
955 * Copy updated JMAXth (next IMAXth) column to Kth of W
956 *
957  CALL zcopy( n-k+1, w( k, k+1 ), 1, w( k, k ), 1 )
958 *
959  END IF
960 *
961 *
962 * End pivot search loop body
963 *
964  IF( .NOT.done ) GOTO 72
965 *
966  END IF
967 *
968 * END pivot search
969 *
970 * ============================================================
971 *
972 * KK is the column of A where pivoting step stopped
973 *
974  kk = k + kstep - 1
975 *
976 * Interchange rows and columns P and K (only for 2-by-2 pivot).
977 * Updated column P is already stored in column K of W.
978 *
979  IF( ( kstep.EQ.2 ) .AND. ( p.NE.k ) ) THEN
980 *
981 * Copy non-updated column KK-1 to column P of submatrix A
982 * at step K. No need to copy element into columns
983 * K and K+1 of A for 2-by-2 pivot, since these columns
984 * will be later overwritten.
985 *
986  a( p, p ) = dble( a( k, k ) )
987  CALL zcopy( p-k-1, a( k+1, k ), 1, a( p, k+1 ), lda )
988  CALL zlacgv( p-k-1, a( p, k+1 ), lda )
989  IF( p.LT.n )
990  $ CALL zcopy( n-p, a( p+1, k ), 1, a( p+1, p ), 1 )
991 *
992 * Interchange rows K and P in first K-1 columns of A
993 * (columns K and K+1 of A for 2-by-2 pivot will be
994 * later overwritten). Interchange rows K and P
995 * in first KK columns of W.
996 *
997  IF( k.GT.1 )
998  $ CALL zswap( k-1, a( k, 1 ), lda, a( p, 1 ), lda )
999  CALL zswap( kk, w( k, 1 ), ldw, w( p, 1 ), ldw )
1000  END IF
1001 *
1002 * Interchange rows and columns KP and KK.
1003 * Updated column KP is already stored in column KK of W.
1004 *
1005  IF( kp.NE.kk ) THEN
1006 *
1007 * Copy non-updated column KK to column KP of submatrix A
1008 * at step K. No need to copy element into column K
1009 * (or K and K+1 for 2-by-2 pivot) of A, since these columns
1010 * will be later overwritten.
1011 *
1012  a( kp, kp ) = dble( a( kk, kk ) )
1013  CALL zcopy( kp-kk-1, a( kk+1, kk ), 1, a( kp, kk+1 ),
1014  $ lda )
1015  CALL zlacgv( kp-kk-1, a( kp, kk+1 ), lda )
1016  IF( kp.LT.n )
1017  $ CALL zcopy( n-kp, a( kp+1, kk ), 1, a( kp+1, kp ), 1 )
1018 *
1019 * Interchange rows KK and KP in first K-1 columns of A
1020 * (column K (or K and K+1 for 2-by-2 pivot) of A will be
1021 * later overwritten). Interchange rows KK and KP
1022 * in first KK columns of W.
1023 *
1024  IF( k.GT.1 )
1025  $ CALL zswap( k-1, a( kk, 1 ), lda, a( kp, 1 ), lda )
1026  CALL zswap( kk, w( kk, 1 ), ldw, w( kp, 1 ), ldw )
1027  END IF
1028 *
1029  IF( kstep.EQ.1 ) THEN
1030 *
1031 * 1-by-1 pivot block D(k): column k of W now holds
1032 *
1033 * W(k) = L(k)*D(k),
1034 *
1035 * where L(k) is the k-th column of L
1036 *
1037 * (1) Store subdiag. elements of column L(k)
1038 * and 1-by-1 block D(k) in column k of A.
1039 * (NOTE: Diagonal element L(k,k) is a UNIT element
1040 * and not stored)
1041 * A(k,k) := D(k,k) = W(k,k)
1042 * A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k)
1043 *
1044 * (NOTE: No need to use for Hermitian matrix
1045 * A( K, K ) = DBLE( W( K, K) ) to separately copy diagonal
1046 * element D(k,k) from W (potentially saves only one load))
1047  CALL zcopy( n-k+1, w( k, k ), 1, a( k, k ), 1 )
1048  IF( k.LT.n ) THEN
1049 *
1050 * (NOTE: No need to check if A(k,k) is NOT ZERO,
1051 * since that was ensured earlier in pivot search:
1052 * case A(k,k) = 0 falls into 2x2 pivot case(3))
1053 *
1054 * Handle division by a small number
1055 *
1056  t = dble( a( k, k ) )
1057  IF( abs( t ).GE.sfmin ) THEN
1058  r1 = one / t
1059  CALL zdscal( n-k, r1, a( k+1, k ), 1 )
1060  ELSE
1061  DO 74 ii = k + 1, n
1062  a( ii, k ) = a( ii, k ) / t
1063  74 CONTINUE
1064  END IF
1065 *
1066 * (2) Conjugate column W(k)
1067 *
1068  CALL zlacgv( n-k, w( k+1, k ), 1 )
1069 *
1070 * Store the subdiagonal element of D in array E
1071 *
1072  e( k ) = czero
1073 *
1074  END IF
1075 *
1076  ELSE
1077 *
1078 * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
1079 *
1080 * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
1081 *
1082 * where L(k) and L(k+1) are the k-th and (k+1)-th columns
1083 * of L
1084 *
1085 * (1) Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2
1086 * block D(k:k+1,k:k+1) in columns k and k+1 of A.
1087 * NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT
1088 * block and not stored.
1089 * A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1)
1090 * A(k+2:N,k:k+1) := L(k+2:N,k:k+1) =
1091 * = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) )
1092 *
1093  IF( k.LT.n-1 ) THEN
1094 *
1095 * Factor out the columns of the inverse of 2-by-2 pivot
1096 * block D, so that each column contains 1, to reduce the
1097 * number of FLOPS when we multiply panel
1098 * ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1).
1099 *
1100 * D**(-1) = ( d11 cj(d21) )**(-1) =
1101 * ( d21 d22 )
1102 *
1103 * = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) =
1104 * ( (-d21) ( d11 ) )
1105 *
1106 * = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) *
1107 *
1108 * * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) =
1109 * ( ( -1 ) ( d11/conj(d21) ) )
1110 *
1111 * = 1/(|d21|**2) * 1/(D22*D11-1) *
1112 *
1113 * * ( d21*( D11 ) conj(d21)*( -1 ) ) =
1114 * ( ( -1 ) ( D22 ) )
1115 *
1116 * = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) =
1117 * ( ( -1 ) ( D22 ) )
1118 *
1119 * = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) =
1120 * ( ( -1 ) ( D22 ) )
1121 *
1122 * Handle division by a small number. (NOTE: order of
1123 * operations is important)
1124 *
1125 * = ( T*(( D11 )/conj(D21)) T*(( -1 )/D21 ) )
1126 * ( (( -1 ) ) (( D22 ) ) ),
1127 *
1128 * where D11 = d22/d21,
1129 * D22 = d11/conj(d21),
1130 * D21 = d21,
1131 * T = 1/(D22*D11-1).
1132 *
1133 * (NOTE: No need to check for division by ZERO,
1134 * since that was ensured earlier in pivot search:
1135 * (a) d21 != 0 in 2x2 pivot case(4),
1136 * since |d21| should be larger than |d11| and |d22|;
1137 * (b) (D22*D11 - 1) != 0, since from (a),
1138 * both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.)
1139 *
1140  d21 = w( k+1, k )
1141  d11 = w( k+1, k+1 ) / d21
1142  d22 = w( k, k ) / dconjg( d21 )
1143  t = one / ( dble( d11*d22 )-one )
1144 *
1145 * Update elements in columns A(k) and A(k+1) as
1146 * dot products of rows of ( W(k) W(k+1) ) and columns
1147 * of D**(-1)
1148 *
1149  DO 80 j = k + 2, n
1150  a( j, k ) = t*( ( d11*w( j, k )-w( j, k+1 ) ) /
1151  $ dconjg( d21 ) )
1152  a( j, k+1 ) = t*( ( d22*w( j, k+1 )-w( j, k ) ) /
1153  $ d21 )
1154  80 CONTINUE
1155  END IF
1156 *
1157 * Copy diagonal elements of D(K) to A,
1158 * copy subdiagonal element of D(K) to E(K) and
1159 * ZERO out subdiagonal entry of A
1160 *
1161  a( k, k ) = w( k, k )
1162  a( k+1, k ) = czero
1163  a( k+1, k+1 ) = w( k+1, k+1 )
1164  e( k ) = w( k+1, k )
1165  e( k+1 ) = czero
1166 *
1167 * (2) Conjugate columns W(k) and W(k+1)
1168 *
1169  CALL zlacgv( n-k, w( k+1, k ), 1 )
1170  CALL zlacgv( n-k-1, w( k+2, k+1 ), 1 )
1171 *
1172  END IF
1173 *
1174 * End column K is nonsingular
1175 *
1176  END IF
1177 *
1178 * Store details of the interchanges in IPIV
1179 *
1180  IF( kstep.EQ.1 ) THEN
1181  ipiv( k ) = kp
1182  ELSE
1183  ipiv( k ) = -p
1184  ipiv( k+1 ) = -kp
1185  END IF
1186 *
1187 * Increase K and return to the start of the main loop
1188 *
1189  k = k + kstep
1190  GO TO 70
1191 *
1192  90 CONTINUE
1193 *
1194 * Update the lower triangle of A22 (= A(k:n,k:n)) as
1195 *
1196 * A22 := A22 - L21*D*L21**H = A22 - L21*W**H
1197 *
1198 * computing blocks of NB columns at a time (note that conjg(W) is
1199 * actually stored)
1200 *
1201  DO 110 j = k, n, nb
1202  jb = min( nb, n-j+1 )
1203 *
1204 * Update the lower triangle of the diagonal block
1205 *
1206  DO 100 jj = j, j + jb - 1
1207  a( jj, jj ) = dble( a( jj, jj ) )
1208  CALL zgemv( 'No transpose', j+jb-jj, k-1, -cone,
1209  $ a( jj, 1 ), lda, w( jj, 1 ), ldw, cone,
1210  $ a( jj, jj ), 1 )
1211  a( jj, jj ) = dble( a( jj, jj ) )
1212  100 CONTINUE
1213 *
1214 * Update the rectangular subdiagonal block
1215 *
1216  IF( j+jb.LE.n )
1217  $ CALL zgemm( 'No transpose', 'Transpose', n-j-jb+1, jb,
1218  $ k-1, -cone, a( j+jb, 1 ), lda, w( j, 1 ),
1219  $ ldw, cone, a( j+jb, j ), lda )
1220  110 CONTINUE
1221 *
1222 * Set KB to the number of columns factorized
1223 *
1224  kb = k - 1
1225 *
1226  END IF
1227  RETURN
1228 *
1229 * End of ZLAHEF_RK
1230 *
1231  END
subroutine zswap(N, ZX, INCX, ZY, INCY)
ZSWAP
Definition: zswap.f:81
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:78
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:158
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:187
subroutine zlahef_rk(UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW, INFO)
ZLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bun...
Definition: zlahef_rk.f:262
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:74