LAPACK  3.9.0
LAPACK: Linear Algebra PACKage
zhetf2_rk.f
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1 *> \brief \b ZHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
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13 *> [ZIP]</a>
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZHETF2_RK( UPLO, N, A, LDA, E, IPIV, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER UPLO
25 * INTEGER INFO, LDA, N
26 * ..
27 * .. Array Arguments ..
28 * INTEGER IPIV( * )
29 * COMPLEX*16 A( LDA, * ), E ( * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *> ZHETF2_RK computes the factorization of a complex Hermitian matrix A
38 *> using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
39 *>
40 *> A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
41 *>
42 *> where U (or L) is unit upper (or lower) triangular matrix,
43 *> U**H (or L**H) is the conjugate of U (or L), P is a permutation
44 *> matrix, P**T is the transpose of P, and D is Hermitian and block
45 *> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
46 *>
47 *> This is the unblocked version of the algorithm, calling Level 2 BLAS.
48 *> For more information see Further Details section.
49 *> \endverbatim
50 *
51 * Arguments:
52 * ==========
53 *
54 *> \param[in] UPLO
55 *> \verbatim
56 *> UPLO is CHARACTER*1
57 *> Specifies whether the upper or lower triangular part of the
58 *> Hermitian matrix A is stored:
59 *> = 'U': Upper triangular
60 *> = 'L': Lower triangular
61 *> \endverbatim
62 *>
63 *> \param[in] N
64 *> \verbatim
65 *> N is INTEGER
66 *> The order of the matrix A. N >= 0.
67 *> \endverbatim
68 *>
69 *> \param[in,out] A
70 *> \verbatim
71 *> A is COMPLEX*16 array, dimension (LDA,N)
72 *> On entry, the Hermitian matrix A.
73 *> If UPLO = 'U': the leading N-by-N upper triangular part
74 *> of A contains the upper triangular part of the matrix A,
75 *> and the strictly lower triangular part of A is not
76 *> referenced.
77 *>
78 *> If UPLO = 'L': the leading N-by-N lower triangular part
79 *> of A contains the lower triangular part of the matrix A,
80 *> and the strictly upper triangular part of A is not
81 *> referenced.
82 *>
83 *> On exit, contains:
84 *> a) ONLY diagonal elements of the Hermitian block diagonal
85 *> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
86 *> (superdiagonal (or subdiagonal) elements of D
87 *> are stored on exit in array E), and
88 *> b) If UPLO = 'U': factor U in the superdiagonal part of A.
89 *> If UPLO = 'L': factor L in the subdiagonal part of A.
90 *> \endverbatim
91 *>
92 *> \param[in] LDA
93 *> \verbatim
94 *> LDA is INTEGER
95 *> The leading dimension of the array A. LDA >= max(1,N).
96 *> \endverbatim
97 *>
98 *> \param[out] E
99 *> \verbatim
100 *> E is COMPLEX*16 array, dimension (N)
101 *> On exit, contains the superdiagonal (or subdiagonal)
102 *> elements of the Hermitian block diagonal matrix D
103 *> with 1-by-1 or 2-by-2 diagonal blocks, where
104 *> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
105 *> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
106 *>
107 *> NOTE: For 1-by-1 diagonal block D(k), where
108 *> 1 <= k <= N, the element E(k) is set to 0 in both
109 *> UPLO = 'U' or UPLO = 'L' cases.
110 *> \endverbatim
111 *>
112 *> \param[out] IPIV
113 *> \verbatim
114 *> IPIV is INTEGER array, dimension (N)
115 *> IPIV describes the permutation matrix P in the factorization
116 *> of matrix A as follows. The absolute value of IPIV(k)
117 *> represents the index of row and column that were
118 *> interchanged with the k-th row and column. The value of UPLO
119 *> describes the order in which the interchanges were applied.
120 *> Also, the sign of IPIV represents the block structure of
121 *> the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
122 *> diagonal blocks which correspond to 1 or 2 interchanges
123 *> at each factorization step. For more info see Further
124 *> Details section.
125 *>
126 *> If UPLO = 'U',
127 *> ( in factorization order, k decreases from N to 1 ):
128 *> a) A single positive entry IPIV(k) > 0 means:
129 *> D(k,k) is a 1-by-1 diagonal block.
130 *> If IPIV(k) != k, rows and columns k and IPIV(k) were
131 *> interchanged in the matrix A(1:N,1:N);
132 *> If IPIV(k) = k, no interchange occurred.
133 *>
134 *> b) A pair of consecutive negative entries
135 *> IPIV(k) < 0 and IPIV(k-1) < 0 means:
136 *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
137 *> (NOTE: negative entries in IPIV appear ONLY in pairs).
138 *> 1) If -IPIV(k) != k, rows and columns
139 *> k and -IPIV(k) were interchanged
140 *> in the matrix A(1:N,1:N).
141 *> If -IPIV(k) = k, no interchange occurred.
142 *> 2) If -IPIV(k-1) != k-1, rows and columns
143 *> k-1 and -IPIV(k-1) were interchanged
144 *> in the matrix A(1:N,1:N).
145 *> If -IPIV(k-1) = k-1, no interchange occurred.
146 *>
147 *> c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
148 *>
149 *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
150 *>
151 *> If UPLO = 'L',
152 *> ( in factorization order, k increases from 1 to N ):
153 *> a) A single positive entry IPIV(k) > 0 means:
154 *> D(k,k) is a 1-by-1 diagonal block.
155 *> If IPIV(k) != k, rows and columns k and IPIV(k) were
156 *> interchanged in the matrix A(1:N,1:N).
157 *> If IPIV(k) = k, no interchange occurred.
158 *>
159 *> b) A pair of consecutive negative entries
160 *> IPIV(k) < 0 and IPIV(k+1) < 0 means:
161 *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
162 *> (NOTE: negative entries in IPIV appear ONLY in pairs).
163 *> 1) If -IPIV(k) != k, rows and columns
164 *> k and -IPIV(k) were interchanged
165 *> in the matrix A(1:N,1:N).
166 *> If -IPIV(k) = k, no interchange occurred.
167 *> 2) If -IPIV(k+1) != k+1, rows and columns
168 *> k-1 and -IPIV(k-1) were interchanged
169 *> in the matrix A(1:N,1:N).
170 *> If -IPIV(k+1) = k+1, no interchange occurred.
171 *>
172 *> c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
173 *>
174 *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
175 *> \endverbatim
176 *>
177 *> \param[out] INFO
178 *> \verbatim
179 *> INFO is INTEGER
180 *> = 0: successful exit
181 *>
182 *> < 0: If INFO = -k, the k-th argument had an illegal value
183 *>
184 *> > 0: If INFO = k, the matrix A is singular, because:
185 *> If UPLO = 'U': column k in the upper
186 *> triangular part of A contains all zeros.
187 *> If UPLO = 'L': column k in the lower
188 *> triangular part of A contains all zeros.
189 *>
190 *> Therefore D(k,k) is exactly zero, and superdiagonal
191 *> elements of column k of U (or subdiagonal elements of
192 *> column k of L ) are all zeros. The factorization has
193 *> been completed, but the block diagonal matrix D is
194 *> exactly singular, and division by zero will occur if
195 *> it is used to solve a system of equations.
196 *>
197 *> NOTE: INFO only stores the first occurrence of
198 *> a singularity, any subsequent occurrence of singularity
199 *> is not stored in INFO even though the factorization
200 *> always completes.
201 *> \endverbatim
202 *
203 * Authors:
204 * ========
205 *
206 *> \author Univ. of Tennessee
207 *> \author Univ. of California Berkeley
208 *> \author Univ. of Colorado Denver
209 *> \author NAG Ltd.
210 *
211 *> \date December 2016
212 *
213 *> \ingroup complex16HEcomputational
214 *
215 *> \par Further Details:
216 * =====================
217 *>
218 *> \verbatim
219 *> TODO: put further details
220 *> \endverbatim
221 *
222 *> \par Contributors:
223 * ==================
224 *>
225 *> \verbatim
226 *>
227 *> December 2016, Igor Kozachenko,
228 *> Computer Science Division,
229 *> University of California, Berkeley
230 *>
231 *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
232 *> School of Mathematics,
233 *> University of Manchester
234 *>
235 *> 01-01-96 - Based on modifications by
236 *> J. Lewis, Boeing Computer Services Company
237 *> A. Petitet, Computer Science Dept.,
238 *> Univ. of Tenn., Knoxville abd , USA
239 *> \endverbatim
240 *
241 * =====================================================================
242  SUBROUTINE zhetf2_rk( UPLO, N, A, LDA, E, IPIV, INFO )
243 *
244 * -- LAPACK computational routine (version 3.7.0) --
245 * -- LAPACK is a software package provided by Univ. of Tennessee, --
246 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
247 * December 2016
248 *
249 * .. Scalar Arguments ..
250  CHARACTER UPLO
251  INTEGER INFO, LDA, N
252 * ..
253 * .. Array Arguments ..
254  INTEGER IPIV( * )
255  COMPLEX*16 A( LDA, * ), E( * )
256 * ..
257 *
258 * ======================================================================
259 *
260 * .. Parameters ..
261  DOUBLE PRECISION ZERO, ONE
262  parameter( zero = 0.0d+0, one = 1.0d+0 )
263  DOUBLE PRECISION EIGHT, SEVTEN
264  parameter( eight = 8.0d+0, sevten = 17.0d+0 )
265  COMPLEX*16 CZERO
266  parameter( czero = ( 0.0d+0, 0.0d+0 ) )
267 * ..
268 * .. Local Scalars ..
269  LOGICAL DONE, UPPER
270  INTEGER I, II, IMAX, ITEMP, J, JMAX, K, KK, KP, KSTEP,
271  $ P
272  DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D, D11, D22, R1, DTEMP,
273  $ ROWMAX, TT, SFMIN
274  COMPLEX*16 D12, D21, T, WK, WKM1, WKP1, Z
275 * ..
276 * .. External Functions ..
277 *
278  LOGICAL LSAME
279  INTEGER IZAMAX
280  DOUBLE PRECISION DLAMCH, DLAPY2
281  EXTERNAL lsame, izamax, dlamch, dlapy2
282 * ..
283 * .. External Subroutines ..
284  EXTERNAL xerbla, zdscal, zher, zswap
285 * ..
286 * .. Intrinsic Functions ..
287  INTRINSIC abs, dble, dcmplx, dconjg, dimag, max, sqrt
288 * ..
289 * .. Statement Functions ..
290  DOUBLE PRECISION CABS1
291 * ..
292 * .. Statement Function definitions ..
293  cabs1( z ) = abs( dble( z ) ) + abs( dimag( z ) )
294 * ..
295 * .. Executable Statements ..
296 *
297 * Test the input parameters.
298 *
299  info = 0
300  upper = lsame( uplo, 'U' )
301  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
302  info = -1
303  ELSE IF( n.LT.0 ) THEN
304  info = -2
305  ELSE IF( lda.LT.max( 1, n ) ) THEN
306  info = -4
307  END IF
308  IF( info.NE.0 ) THEN
309  CALL xerbla( 'ZHETF2_RK', -info )
310  RETURN
311  END IF
312 *
313 * Initialize ALPHA for use in choosing pivot block size.
314 *
315  alpha = ( one+sqrt( sevten ) ) / eight
316 *
317 * Compute machine safe minimum
318 *
319  sfmin = dlamch( 'S' )
320 *
321  IF( upper ) THEN
322 *
323 * Factorize A as U*D*U**H using the upper triangle of A
324 *
325 * Initialize the first entry of array E, where superdiagonal
326 * elements of D are stored
327 *
328  e( 1 ) = czero
329 *
330 * K is the main loop index, decreasing from N to 1 in steps of
331 * 1 or 2
332 *
333  k = n
334  10 CONTINUE
335 *
336 * If K < 1, exit from loop
337 *
338  IF( k.LT.1 )
339  $ GO TO 34
340  kstep = 1
341  p = k
342 *
343 * Determine rows and columns to be interchanged and whether
344 * a 1-by-1 or 2-by-2 pivot block will be used
345 *
346  absakk = abs( dble( a( k, k ) ) )
347 *
348 * IMAX is the row-index of the largest off-diagonal element in
349 * column K, and COLMAX is its absolute value.
350 * Determine both COLMAX and IMAX.
351 *
352  IF( k.GT.1 ) THEN
353  imax = izamax( k-1, a( 1, k ), 1 )
354  colmax = cabs1( a( imax, k ) )
355  ELSE
356  colmax = zero
357  END IF
358 *
359  IF( ( max( absakk, colmax ).EQ.zero ) ) THEN
360 *
361 * Column K is zero or underflow: set INFO and continue
362 *
363  IF( info.EQ.0 )
364  $ info = k
365  kp = k
366  a( k, k ) = dble( a( k, k ) )
367 *
368 * Set E( K ) to zero
369 *
370  IF( k.GT.1 )
371  $ e( k ) = czero
372 *
373  ELSE
374 *
375 * ============================================================
376 *
377 * BEGIN pivot search
378 *
379 * Case(1)
380 * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
381 * (used to handle NaN and Inf)
382 *
383  IF( .NOT.( absakk.LT.alpha*colmax ) ) THEN
384 *
385 * no interchange, use 1-by-1 pivot block
386 *
387  kp = k
388 *
389  ELSE
390 *
391  done = .false.
392 *
393 * Loop until pivot found
394 *
395  12 CONTINUE
396 *
397 * BEGIN pivot search loop body
398 *
399 *
400 * JMAX is the column-index of the largest off-diagonal
401 * element in row IMAX, and ROWMAX is its absolute value.
402 * Determine both ROWMAX and JMAX.
403 *
404  IF( imax.NE.k ) THEN
405  jmax = imax + izamax( k-imax, a( imax, imax+1 ),
406  $ lda )
407  rowmax = cabs1( a( imax, jmax ) )
408  ELSE
409  rowmax = zero
410  END IF
411 *
412  IF( imax.GT.1 ) THEN
413  itemp = izamax( imax-1, a( 1, imax ), 1 )
414  dtemp = cabs1( a( itemp, imax ) )
415  IF( dtemp.GT.rowmax ) THEN
416  rowmax = dtemp
417  jmax = itemp
418  END IF
419  END IF
420 *
421 * Case(2)
422 * Equivalent to testing for
423 * ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX
424 * (used to handle NaN and Inf)
425 *
426  IF( .NOT.( abs( dble( a( imax, imax ) ) )
427  $ .LT.alpha*rowmax ) ) THEN
428 *
429 * interchange rows and columns K and IMAX,
430 * use 1-by-1 pivot block
431 *
432  kp = imax
433  done = .true.
434 *
435 * Case(3)
436 * Equivalent to testing for ROWMAX.EQ.COLMAX,
437 * (used to handle NaN and Inf)
438 *
439  ELSE IF( ( p.EQ.jmax ) .OR. ( rowmax.LE.colmax ) )
440  $ THEN
441 *
442 * interchange rows and columns K-1 and IMAX,
443 * use 2-by-2 pivot block
444 *
445  kp = imax
446  kstep = 2
447  done = .true.
448 *
449 * Case(4)
450  ELSE
451 *
452 * Pivot not found: set params and repeat
453 *
454  p = imax
455  colmax = rowmax
456  imax = jmax
457  END IF
458 *
459 * END pivot search loop body
460 *
461  IF( .NOT.done ) GOTO 12
462 *
463  END IF
464 *
465 * END pivot search
466 *
467 * ============================================================
468 *
469 * KK is the column of A where pivoting step stopped
470 *
471  kk = k - kstep + 1
472 *
473 * For only a 2x2 pivot, interchange rows and columns K and P
474 * in the leading submatrix A(1:k,1:k)
475 *
476  IF( ( kstep.EQ.2 ) .AND. ( p.NE.k ) ) THEN
477 * (1) Swap columnar parts
478  IF( p.GT.1 )
479  $ CALL zswap( p-1, a( 1, k ), 1, a( 1, p ), 1 )
480 * (2) Swap and conjugate middle parts
481  DO 14 j = p + 1, k - 1
482  t = dconjg( a( j, k ) )
483  a( j, k ) = dconjg( a( p, j ) )
484  a( p, j ) = t
485  14 CONTINUE
486 * (3) Swap and conjugate corner elements at row-col interserction
487  a( p, k ) = dconjg( a( p, k ) )
488 * (4) Swap diagonal elements at row-col intersection
489  r1 = dble( a( k, k ) )
490  a( k, k ) = dble( a( p, p ) )
491  a( p, p ) = r1
492 *
493 * Convert upper triangle of A into U form by applying
494 * the interchanges in columns k+1:N.
495 *
496  IF( k.LT.n )
497  $ CALL zswap( n-k, a( k, k+1 ), lda, a( p, k+1 ), lda )
498 *
499  END IF
500 *
501 * For both 1x1 and 2x2 pivots, interchange rows and
502 * columns KK and KP in the leading submatrix A(1:k,1:k)
503 *
504  IF( kp.NE.kk ) THEN
505 * (1) Swap columnar parts
506  IF( kp.GT.1 )
507  $ CALL zswap( kp-1, a( 1, kk ), 1, a( 1, kp ), 1 )
508 * (2) Swap and conjugate middle parts
509  DO 15 j = kp + 1, kk - 1
510  t = dconjg( a( j, kk ) )
511  a( j, kk ) = dconjg( a( kp, j ) )
512  a( kp, j ) = t
513  15 CONTINUE
514 * (3) Swap and conjugate corner elements at row-col interserction
515  a( kp, kk ) = dconjg( a( kp, kk ) )
516 * (4) Swap diagonal elements at row-col intersection
517  r1 = dble( a( kk, kk ) )
518  a( kk, kk ) = dble( a( kp, kp ) )
519  a( kp, kp ) = r1
520 *
521  IF( kstep.EQ.2 ) THEN
522 * (*) Make sure that diagonal element of pivot is real
523  a( k, k ) = dble( a( k, k ) )
524 * (5) Swap row elements
525  t = a( k-1, k )
526  a( k-1, k ) = a( kp, k )
527  a( kp, k ) = t
528  END IF
529 *
530 * Convert upper triangle of A into U form by applying
531 * the interchanges in columns k+1:N.
532 *
533  IF( k.LT.n )
534  $ CALL zswap( n-k, a( kk, k+1 ), lda, a( kp, k+1 ),
535  $ lda )
536 *
537  ELSE
538 * (*) Make sure that diagonal element of pivot is real
539  a( k, k ) = dble( a( k, k ) )
540  IF( kstep.EQ.2 )
541  $ a( k-1, k-1 ) = dble( a( k-1, k-1 ) )
542  END IF
543 *
544 * Update the leading submatrix
545 *
546  IF( kstep.EQ.1 ) THEN
547 *
548 * 1-by-1 pivot block D(k): column k now holds
549 *
550 * W(k) = U(k)*D(k)
551 *
552 * where U(k) is the k-th column of U
553 *
554  IF( k.GT.1 ) THEN
555 *
556 * Perform a rank-1 update of A(1:k-1,1:k-1) and
557 * store U(k) in column k
558 *
559  IF( abs( dble( a( k, k ) ) ).GE.sfmin ) THEN
560 *
561 * Perform a rank-1 update of A(1:k-1,1:k-1) as
562 * A := A - U(k)*D(k)*U(k)**T
563 * = A - W(k)*1/D(k)*W(k)**T
564 *
565  d11 = one / dble( a( k, k ) )
566  CALL zher( uplo, k-1, -d11, a( 1, k ), 1, a, lda )
567 *
568 * Store U(k) in column k
569 *
570  CALL zdscal( k-1, d11, a( 1, k ), 1 )
571  ELSE
572 *
573 * Store L(k) in column K
574 *
575  d11 = dble( a( k, k ) )
576  DO 16 ii = 1, k - 1
577  a( ii, k ) = a( ii, k ) / d11
578  16 CONTINUE
579 *
580 * Perform a rank-1 update of A(k+1:n,k+1:n) as
581 * A := A - U(k)*D(k)*U(k)**T
582 * = A - W(k)*(1/D(k))*W(k)**T
583 * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
584 *
585  CALL zher( uplo, k-1, -d11, a( 1, k ), 1, a, lda )
586  END IF
587 *
588 * Store the superdiagonal element of D in array E
589 *
590  e( k ) = czero
591 *
592  END IF
593 *
594  ELSE
595 *
596 * 2-by-2 pivot block D(k): columns k and k-1 now hold
597 *
598 * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
599 *
600 * where U(k) and U(k-1) are the k-th and (k-1)-th columns
601 * of U
602 *
603 * Perform a rank-2 update of A(1:k-2,1:k-2) as
604 *
605 * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
606 * = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T
607 *
608 * and store L(k) and L(k+1) in columns k and k+1
609 *
610  IF( k.GT.2 ) THEN
611 * D = |A12|
612  d = dlapy2( dble( a( k-1, k ) ),
613  $ dimag( a( k-1, k ) ) )
614  d11 = a( k, k ) / d
615  d22 = a( k-1, k-1 ) / d
616  d12 = a( k-1, k ) / d
617  tt = one / ( d11*d22-one )
618 *
619  DO 30 j = k - 2, 1, -1
620 *
621 * Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
622 *
623  wkm1 = tt*( d11*a( j, k-1 )-dconjg( d12 )*
624  $ a( j, k ) )
625  wk = tt*( d22*a( j, k )-d12*a( j, k-1 ) )
626 *
627 * Perform a rank-2 update of A(1:k-2,1:k-2)
628 *
629  DO 20 i = j, 1, -1
630  a( i, j ) = a( i, j ) -
631  $ ( a( i, k ) / d )*dconjg( wk ) -
632  $ ( a( i, k-1 ) / d )*dconjg( wkm1 )
633  20 CONTINUE
634 *
635 * Store U(k) and U(k-1) in cols k and k-1 for row J
636 *
637  a( j, k ) = wk / d
638  a( j, k-1 ) = wkm1 / d
639 * (*) Make sure that diagonal element of pivot is real
640  a( j, j ) = dcmplx( dble( a( j, j ) ), zero )
641 *
642  30 CONTINUE
643 *
644  END IF
645 *
646 * Copy superdiagonal elements of D(K) to E(K) and
647 * ZERO out superdiagonal entry of A
648 *
649  e( k ) = a( k-1, k )
650  e( k-1 ) = czero
651  a( k-1, k ) = czero
652 *
653  END IF
654 *
655 * End column K is nonsingular
656 *
657  END IF
658 *
659 * Store details of the interchanges in IPIV
660 *
661  IF( kstep.EQ.1 ) THEN
662  ipiv( k ) = kp
663  ELSE
664  ipiv( k ) = -p
665  ipiv( k-1 ) = -kp
666  END IF
667 *
668 * Decrease K and return to the start of the main loop
669 *
670  k = k - kstep
671  GO TO 10
672 *
673  34 CONTINUE
674 *
675  ELSE
676 *
677 * Factorize A as L*D*L**H using the lower triangle of A
678 *
679 * Initialize the unused last entry of the subdiagonal array E.
680 *
681  e( n ) = czero
682 *
683 * K is the main loop index, increasing from 1 to N in steps of
684 * 1 or 2
685 *
686  k = 1
687  40 CONTINUE
688 *
689 * If K > N, exit from loop
690 *
691  IF( k.GT.n )
692  $ GO TO 64
693  kstep = 1
694  p = k
695 *
696 * Determine rows and columns to be interchanged and whether
697 * a 1-by-1 or 2-by-2 pivot block will be used
698 *
699  absakk = abs( dble( a( k, k ) ) )
700 *
701 * IMAX is the row-index of the largest off-diagonal element in
702 * column K, and COLMAX is its absolute value.
703 * Determine both COLMAX and IMAX.
704 *
705  IF( k.LT.n ) THEN
706  imax = k + izamax( n-k, a( k+1, k ), 1 )
707  colmax = cabs1( a( imax, k ) )
708  ELSE
709  colmax = zero
710  END IF
711 *
712  IF( max( absakk, colmax ).EQ.zero ) THEN
713 *
714 * Column K is zero or underflow: set INFO and continue
715 *
716  IF( info.EQ.0 )
717  $ info = k
718  kp = k
719  a( k, k ) = dble( a( k, k ) )
720 *
721 * Set E( K ) to zero
722 *
723  IF( k.LT.n )
724  $ e( k ) = czero
725 *
726  ELSE
727 *
728 * ============================================================
729 *
730 * BEGIN pivot search
731 *
732 * Case(1)
733 * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
734 * (used to handle NaN and Inf)
735 *
736  IF( .NOT.( absakk.LT.alpha*colmax ) ) THEN
737 *
738 * no interchange, use 1-by-1 pivot block
739 *
740  kp = k
741 *
742  ELSE
743 *
744  done = .false.
745 *
746 * Loop until pivot found
747 *
748  42 CONTINUE
749 *
750 * BEGIN pivot search loop body
751 *
752 *
753 * JMAX is the column-index of the largest off-diagonal
754 * element in row IMAX, and ROWMAX is its absolute value.
755 * Determine both ROWMAX and JMAX.
756 *
757  IF( imax.NE.k ) THEN
758  jmax = k - 1 + izamax( imax-k, a( imax, k ), lda )
759  rowmax = cabs1( a( imax, jmax ) )
760  ELSE
761  rowmax = zero
762  END IF
763 *
764  IF( imax.LT.n ) THEN
765  itemp = imax + izamax( n-imax, a( imax+1, imax ),
766  $ 1 )
767  dtemp = cabs1( a( itemp, imax ) )
768  IF( dtemp.GT.rowmax ) THEN
769  rowmax = dtemp
770  jmax = itemp
771  END IF
772  END IF
773 *
774 * Case(2)
775 * Equivalent to testing for
776 * ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX
777 * (used to handle NaN and Inf)
778 *
779  IF( .NOT.( abs( dble( a( imax, imax ) ) )
780  $ .LT.alpha*rowmax ) ) THEN
781 *
782 * interchange rows and columns K and IMAX,
783 * use 1-by-1 pivot block
784 *
785  kp = imax
786  done = .true.
787 *
788 * Case(3)
789 * Equivalent to testing for ROWMAX.EQ.COLMAX,
790 * (used to handle NaN and Inf)
791 *
792  ELSE IF( ( p.EQ.jmax ) .OR. ( rowmax.LE.colmax ) )
793  $ THEN
794 *
795 * interchange rows and columns K+1 and IMAX,
796 * use 2-by-2 pivot block
797 *
798  kp = imax
799  kstep = 2
800  done = .true.
801 *
802 * Case(4)
803  ELSE
804 *
805 * Pivot not found: set params and repeat
806 *
807  p = imax
808  colmax = rowmax
809  imax = jmax
810  END IF
811 *
812 *
813 * END pivot search loop body
814 *
815  IF( .NOT.done ) GOTO 42
816 *
817  END IF
818 *
819 * END pivot search
820 *
821 * ============================================================
822 *
823 * KK is the column of A where pivoting step stopped
824 *
825  kk = k + kstep - 1
826 *
827 * For only a 2x2 pivot, interchange rows and columns K and P
828 * in the trailing submatrix A(k:n,k:n)
829 *
830  IF( ( kstep.EQ.2 ) .AND. ( p.NE.k ) ) THEN
831 * (1) Swap columnar parts
832  IF( p.LT.n )
833  $ CALL zswap( n-p, a( p+1, k ), 1, a( p+1, p ), 1 )
834 * (2) Swap and conjugate middle parts
835  DO 44 j = k + 1, p - 1
836  t = dconjg( a( j, k ) )
837  a( j, k ) = dconjg( a( p, j ) )
838  a( p, j ) = t
839  44 CONTINUE
840 * (3) Swap and conjugate corner elements at row-col interserction
841  a( p, k ) = dconjg( a( p, k ) )
842 * (4) Swap diagonal elements at row-col intersection
843  r1 = dble( a( k, k ) )
844  a( k, k ) = dble( a( p, p ) )
845  a( p, p ) = r1
846 *
847 * Convert lower triangle of A into L form by applying
848 * the interchanges in columns 1:k-1.
849 *
850  IF ( k.GT.1 )
851  $ CALL zswap( k-1, a( k, 1 ), lda, a( p, 1 ), lda )
852 *
853  END IF
854 *
855 * For both 1x1 and 2x2 pivots, interchange rows and
856 * columns KK and KP in the trailing submatrix A(k:n,k:n)
857 *
858  IF( kp.NE.kk ) THEN
859 * (1) Swap columnar parts
860  IF( kp.LT.n )
861  $ CALL zswap( n-kp, a( kp+1, kk ), 1, a( kp+1, kp ), 1 )
862 * (2) Swap and conjugate middle parts
863  DO 45 j = kk + 1, kp - 1
864  t = dconjg( a( j, kk ) )
865  a( j, kk ) = dconjg( a( kp, j ) )
866  a( kp, j ) = t
867  45 CONTINUE
868 * (3) Swap and conjugate corner elements at row-col interserction
869  a( kp, kk ) = dconjg( a( kp, kk ) )
870 * (4) Swap diagonal elements at row-col intersection
871  r1 = dble( a( kk, kk ) )
872  a( kk, kk ) = dble( a( kp, kp ) )
873  a( kp, kp ) = r1
874 *
875  IF( kstep.EQ.2 ) THEN
876 * (*) Make sure that diagonal element of pivot is real
877  a( k, k ) = dble( a( k, k ) )
878 * (5) Swap row elements
879  t = a( k+1, k )
880  a( k+1, k ) = a( kp, k )
881  a( kp, k ) = t
882  END IF
883 *
884 * Convert lower triangle of A into L form by applying
885 * the interchanges in columns 1:k-1.
886 *
887  IF ( k.GT.1 )
888  $ CALL zswap( k-1, a( kk, 1 ), lda, a( kp, 1 ), lda )
889 *
890  ELSE
891 * (*) Make sure that diagonal element of pivot is real
892  a( k, k ) = dble( a( k, k ) )
893  IF( kstep.EQ.2 )
894  $ a( k+1, k+1 ) = dble( a( k+1, k+1 ) )
895  END IF
896 *
897 * Update the trailing submatrix
898 *
899  IF( kstep.EQ.1 ) THEN
900 *
901 * 1-by-1 pivot block D(k): column k of A now holds
902 *
903 * W(k) = L(k)*D(k),
904 *
905 * where L(k) is the k-th column of L
906 *
907  IF( k.LT.n ) THEN
908 *
909 * Perform a rank-1 update of A(k+1:n,k+1:n) and
910 * store L(k) in column k
911 *
912 * Handle division by a small number
913 *
914  IF( abs( dble( a( k, k ) ) ).GE.sfmin ) THEN
915 *
916 * Perform a rank-1 update of A(k+1:n,k+1:n) as
917 * A := A - L(k)*D(k)*L(k)**T
918 * = A - W(k)*(1/D(k))*W(k)**T
919 *
920  d11 = one / dble( a( k, k ) )
921  CALL zher( uplo, n-k, -d11, a( k+1, k ), 1,
922  $ a( k+1, k+1 ), lda )
923 *
924 * Store L(k) in column k
925 *
926  CALL zdscal( n-k, d11, a( k+1, k ), 1 )
927  ELSE
928 *
929 * Store L(k) in column k
930 *
931  d11 = dble( a( k, k ) )
932  DO 46 ii = k + 1, n
933  a( ii, k ) = a( ii, k ) / d11
934  46 CONTINUE
935 *
936 * Perform a rank-1 update of A(k+1:n,k+1:n) as
937 * A := A - L(k)*D(k)*L(k)**T
938 * = A - W(k)*(1/D(k))*W(k)**T
939 * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
940 *
941  CALL zher( uplo, n-k, -d11, a( k+1, k ), 1,
942  $ a( k+1, k+1 ), lda )
943  END IF
944 *
945 * Store the subdiagonal element of D in array E
946 *
947  e( k ) = czero
948 *
949  END IF
950 *
951  ELSE
952 *
953 * 2-by-2 pivot block D(k): columns k and k+1 now hold
954 *
955 * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
956 *
957 * where L(k) and L(k+1) are the k-th and (k+1)-th columns
958 * of L
959 *
960 *
961 * Perform a rank-2 update of A(k+2:n,k+2:n) as
962 *
963 * A := A - ( L(k) L(k+1) ) * D(k) * ( L(k) L(k+1) )**T
964 * = A - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T
965 *
966 * and store L(k) and L(k+1) in columns k and k+1
967 *
968  IF( k.LT.n-1 ) THEN
969 * D = |A21|
970  d = dlapy2( dble( a( k+1, k ) ),
971  $ dimag( a( k+1, k ) ) )
972  d11 = dble( a( k+1, k+1 ) ) / d
973  d22 = dble( a( k, k ) ) / d
974  d21 = a( k+1, k ) / d
975  tt = one / ( d11*d22-one )
976 *
977  DO 60 j = k + 2, n
978 *
979 * Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
980 *
981  wk = tt*( d11*a( j, k )-d21*a( j, k+1 ) )
982  wkp1 = tt*( d22*a( j, k+1 )-dconjg( d21 )*
983  $ a( j, k ) )
984 *
985 * Perform a rank-2 update of A(k+2:n,k+2:n)
986 *
987  DO 50 i = j, n
988  a( i, j ) = a( i, j ) -
989  $ ( a( i, k ) / d )*dconjg( wk ) -
990  $ ( a( i, k+1 ) / d )*dconjg( wkp1 )
991  50 CONTINUE
992 *
993 * Store L(k) and L(k+1) in cols k and k+1 for row J
994 *
995  a( j, k ) = wk / d
996  a( j, k+1 ) = wkp1 / d
997 * (*) Make sure that diagonal element of pivot is real
998  a( j, j ) = dcmplx( dble( a( j, j ) ), zero )
999 *
1000  60 CONTINUE
1001 *
1002  END IF
1003 *
1004 * Copy subdiagonal elements of D(K) to E(K) and
1005 * ZERO out subdiagonal entry of A
1006 *
1007  e( k ) = a( k+1, k )
1008  e( k+1 ) = czero
1009  a( k+1, k ) = czero
1010 *
1011  END IF
1012 *
1013 * End column K is nonsingular
1014 *
1015  END IF
1016 *
1017 * Store details of the interchanges in IPIV
1018 *
1019  IF( kstep.EQ.1 ) THEN
1020  ipiv( k ) = kp
1021  ELSE
1022  ipiv( k ) = -p
1023  ipiv( k+1 ) = -kp
1024  END IF
1025 *
1026 * Increase K and return to the start of the main loop
1027 *
1028  k = k + kstep
1029  GO TO 40
1030 *
1031  64 CONTINUE
1032 *
1033  END IF
1034 *
1035  RETURN
1036 *
1037 * End of ZHETF2_RK
1038 *
1039  END
zdscal
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:80
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
zswap
subroutine zswap(N, ZX, INCX, ZY, INCY)
ZSWAP
Definition: zswap.f:83
zher
subroutine zher(UPLO, N, ALPHA, X, INCX, A, LDA)
ZHER
Definition: zher.f:137
zhetf2_rk
subroutine zhetf2_rk(UPLO, N, A, LDA, E, IPIV, INFO)
ZHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch...
Definition: zhetf2_rk.f:243