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zsytf2.f
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1 *> \brief \b ZSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download ZSYTF2 + dependencies
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11 *> [TGZ]</a>
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13 *> [ZIP]</a>
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZSYTF2( UPLO, N, A, LDA, 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, * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> ZSYTF2 computes the factorization of a complex symmetric matrix A
39 *> using the Bunch-Kaufman diagonal pivoting method:
40 *>
41 *> A = U*D*U**T or A = L*D*L**T
42 *>
43 *> where U (or L) is a product of permutation and unit upper (lower)
44 *> triangular matrices, U**T is the transpose of U, and D is symmetric and
45 *> block 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 *> \endverbatim
49 *
50 * Arguments:
51 * ==========
52 *
53 *> \param[in] UPLO
54 *> \verbatim
55 *> UPLO is CHARACTER*1
56 *> Specifies whether the upper or lower triangular part of the
57 *> symmetric matrix A is stored:
58 *> = 'U': Upper triangular
59 *> = 'L': Lower triangular
60 *> \endverbatim
61 *>
62 *> \param[in] N
63 *> \verbatim
64 *> N is INTEGER
65 *> The order of the matrix A. N >= 0.
66 *> \endverbatim
67 *>
68 *> \param[in,out] A
69 *> \verbatim
70 *> A is COMPLEX*16 array, dimension (LDA,N)
71 *> On entry, the symmetric matrix A. If UPLO = 'U', the leading
72 *> n-by-n upper triangular part of A contains the upper
73 *> triangular part of the matrix A, and the strictly lower
74 *> triangular part of A is not referenced. If UPLO = 'L', the
75 *> leading n-by-n lower triangular part of A contains the lower
76 *> triangular part of the matrix A, and the strictly upper
77 *> triangular part of A is not referenced.
78 *>
79 *> On exit, the block diagonal matrix D and the multipliers used
80 *> to obtain the factor U or L (see below for further details).
81 *> \endverbatim
82 *>
83 *> \param[in] LDA
84 *> \verbatim
85 *> LDA is INTEGER
86 *> The leading dimension of the array A. LDA >= max(1,N).
87 *> \endverbatim
88 *>
89 *> \param[out] IPIV
90 *> \verbatim
91 *> IPIV is INTEGER array, dimension (N)
92 *> Details of the interchanges and the block structure of D.
93 *>
94 *> If UPLO = 'U':
95 *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
96 *> interchanged and D(k,k) is a 1-by-1 diagonal block.
97 *>
98 *> If IPIV(k) = IPIV(k-1) < 0, then rows and columns
99 *> k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
100 *> is a 2-by-2 diagonal block.
101 *>
102 *> If UPLO = 'L':
103 *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
104 *> interchanged and D(k,k) is a 1-by-1 diagonal block.
105 *>
106 *> If IPIV(k) = IPIV(k+1) < 0, then rows and columns
107 *> k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
108 *> is a 2-by-2 diagonal block.
109 *> \endverbatim
110 *>
111 *> \param[out] INFO
112 *> \verbatim
113 *> INFO is INTEGER
114 *> = 0: successful exit
115 *> < 0: if INFO = -k, the k-th argument had an illegal value
116 *> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
117 *> has been completed, but the block diagonal matrix D is
118 *> exactly singular, and division by zero will occur if it
119 *> is used to solve a system of equations.
120 *> \endverbatim
121 *
122 * Authors:
123 * ========
124 *
125 *> \author Univ. of Tennessee
126 *> \author Univ. of California Berkeley
127 *> \author Univ. of Colorado Denver
128 *> \author NAG Ltd.
129 *
130 *> \date November 2013
131 *
132 *> \ingroup complex16SYcomputational
133 *
134 *> \par Further Details:
135 * =====================
136 *>
137 *> \verbatim
138 *>
139 *> If UPLO = 'U', then A = U*D*U**T, where
140 *> U = P(n)*U(n)* ... *P(k)U(k)* ...,
141 *> i.e., U is a product of terms P(k)*U(k), where k decreases from n to
142 *> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
143 *> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
144 *> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
145 *> that if the diagonal block D(k) is of order s (s = 1 or 2), then
146 *>
147 *> ( I v 0 ) k-s
148 *> U(k) = ( 0 I 0 ) s
149 *> ( 0 0 I ) n-k
150 *> k-s s n-k
151 *>
152 *> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
153 *> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
154 *> and A(k,k), and v overwrites A(1:k-2,k-1:k).
155 *>
156 *> If UPLO = 'L', then A = L*D*L**T, where
157 *> L = P(1)*L(1)* ... *P(k)*L(k)* ...,
158 *> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
159 *> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
160 *> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
161 *> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
162 *> that if the diagonal block D(k) is of order s (s = 1 or 2), then
163 *>
164 *> ( I 0 0 ) k-1
165 *> L(k) = ( 0 I 0 ) s
166 *> ( 0 v I ) n-k-s+1
167 *> k-1 s n-k-s+1
168 *>
169 *> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
170 *> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
171 *> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
172 *> \endverbatim
173 *
174 *> \par Contributors:
175 * ==================
176 *>
177 *> \verbatim
178 *>
179 *> 09-29-06 - patch from
180 *> Bobby Cheng, MathWorks
181 *>
182 *> Replace l.209 and l.377
183 *> IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
184 *> by
185 *> IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
186 *>
187 *> 1-96 - Based on modifications by J. Lewis, Boeing Computer Services
188 *> Company
189 *> \endverbatim
190 *
191 * =====================================================================
192  SUBROUTINE zsytf2( UPLO, N, A, LDA, IPIV, INFO )
193 *
194 * -- LAPACK computational routine (version 3.5.0) --
195 * -- LAPACK is a software package provided by Univ. of Tennessee, --
196 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
197 * November 2013
198 *
199 * .. Scalar Arguments ..
200  CHARACTER uplo
201  INTEGER info, lda, n
202 * ..
203 * .. Array Arguments ..
204  INTEGER ipiv( * )
205  COMPLEX*16 a( lda, * )
206 * ..
207 *
208 * =====================================================================
209 *
210 * .. Parameters ..
211  DOUBLE PRECISION zero, one
212  parameter( zero = 0.0d+0, one = 1.0d+0 )
213  DOUBLE PRECISION eight, sevten
214  parameter( eight = 8.0d+0, sevten = 17.0d+0 )
215  COMPLEX*16 cone
216  parameter( cone = ( 1.0d+0, 0.0d+0 ) )
217 * ..
218 * .. Local Scalars ..
219  LOGICAL upper
220  INTEGER i, imax, j, jmax, k, kk, kp, kstep
221  DOUBLE PRECISION absakk, alpha, colmax, rowmax
222  COMPLEX*16 d11, d12, d21, d22, r1, t, wk, wkm1, wkp1, z
223 * ..
224 * .. External Functions ..
225  LOGICAL disnan, lsame
226  INTEGER izamax
227  EXTERNAL disnan, lsame, izamax
228 * ..
229 * .. External Subroutines ..
230  EXTERNAL xerbla, zscal, zswap, zsyr
231 * ..
232 * .. Intrinsic Functions ..
233  INTRINSIC abs, dble, dimag, max, sqrt
234 * ..
235 * .. Statement Functions ..
236  DOUBLE PRECISION cabs1
237 * ..
238 * .. Statement Function definitions ..
239  cabs1( z ) = abs( dble( z ) ) + abs( dimag( z ) )
240 * ..
241 * .. Executable Statements ..
242 *
243 * Test the input parameters.
244 *
245  info = 0
246  upper = lsame( uplo, 'U' )
247  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
248  info = -1
249  ELSE IF( n.LT.0 ) THEN
250  info = -2
251  ELSE IF( lda.LT.max( 1, n ) ) THEN
252  info = -4
253  END IF
254  IF( info.NE.0 ) THEN
255  CALL xerbla( 'ZSYTF2', -info )
256  RETURN
257  END IF
258 *
259 * Initialize ALPHA for use in choosing pivot block size.
260 *
261  alpha = ( one+sqrt( sevten ) ) / eight
262 *
263  IF( upper ) THEN
264 *
265 * Factorize A as U*D*U**T using the upper triangle of A
266 *
267 * K is the main loop index, decreasing from N to 1 in steps of
268 * 1 or 2
269 *
270  k = n
271  10 CONTINUE
272 *
273 * If K < 1, exit from loop
274 *
275  IF( k.LT.1 )
276  $ go to 70
277  kstep = 1
278 *
279 * Determine rows and columns to be interchanged and whether
280 * a 1-by-1 or 2-by-2 pivot block will be used
281 *
282  absakk = cabs1( a( k, k ) )
283 *
284 * IMAX is the row-index of the largest off-diagonal element in
285 * column K, and COLMAX is its absolute value.
286 * Determine both COLMAX and IMAX.
287 *
288  IF( k.GT.1 ) THEN
289  imax = izamax( k-1, a( 1, k ), 1 )
290  colmax = cabs1( a( imax, k ) )
291  ELSE
292  colmax = zero
293  END IF
294 *
295  IF( max( absakk, colmax ).EQ.zero .OR. disnan(absakk) ) THEN
296 *
297 * Column K is zero or underflow, or contains a NaN:
298 * set INFO and continue
299 *
300  IF( info.EQ.0 )
301  $ info = k
302  kp = k
303  ELSE
304  IF( absakk.GE.alpha*colmax ) THEN
305 *
306 * no interchange, use 1-by-1 pivot block
307 *
308  kp = k
309  ELSE
310 *
311 * JMAX is the column-index of the largest off-diagonal
312 * element in row IMAX, and ROWMAX is its absolute value
313 *
314  jmax = imax + izamax( k-imax, a( imax, imax+1 ), lda )
315  rowmax = cabs1( a( imax, jmax ) )
316  IF( imax.GT.1 ) THEN
317  jmax = izamax( imax-1, a( 1, imax ), 1 )
318  rowmax = max( rowmax, cabs1( a( jmax, imax ) ) )
319  END IF
320 *
321  IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
322 *
323 * no interchange, use 1-by-1 pivot block
324 *
325  kp = k
326  ELSE IF( cabs1( a( imax, imax ) ).GE.alpha*rowmax ) THEN
327 *
328 * interchange rows and columns K and IMAX, use 1-by-1
329 * pivot block
330 *
331  kp = imax
332  ELSE
333 *
334 * interchange rows and columns K-1 and IMAX, use 2-by-2
335 * pivot block
336 *
337  kp = imax
338  kstep = 2
339  END IF
340  END IF
341 *
342  kk = k - kstep + 1
343  IF( kp.NE.kk ) THEN
344 *
345 * Interchange rows and columns KK and KP in the leading
346 * submatrix A(1:k,1:k)
347 *
348  CALL zswap( kp-1, a( 1, kk ), 1, a( 1, kp ), 1 )
349  CALL zswap( kk-kp-1, a( kp+1, kk ), 1, a( kp, kp+1 ),
350  $ lda )
351  t = a( kk, kk )
352  a( kk, kk ) = a( kp, kp )
353  a( kp, kp ) = t
354  IF( kstep.EQ.2 ) THEN
355  t = a( k-1, k )
356  a( k-1, k ) = a( kp, k )
357  a( kp, k ) = t
358  END IF
359  END IF
360 *
361 * Update the leading submatrix
362 *
363  IF( kstep.EQ.1 ) THEN
364 *
365 * 1-by-1 pivot block D(k): column k now holds
366 *
367 * W(k) = U(k)*D(k)
368 *
369 * where U(k) is the k-th column of U
370 *
371 * Perform a rank-1 update of A(1:k-1,1:k-1) as
372 *
373 * A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
374 *
375  r1 = cone / a( k, k )
376  CALL zsyr( uplo, k-1, -r1, a( 1, k ), 1, a, lda )
377 *
378 * Store U(k) in column k
379 *
380  CALL zscal( k-1, r1, a( 1, k ), 1 )
381  ELSE
382 *
383 * 2-by-2 pivot block D(k): columns k and k-1 now hold
384 *
385 * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
386 *
387 * where U(k) and U(k-1) are the k-th and (k-1)-th columns
388 * of U
389 *
390 * Perform a rank-2 update of A(1:k-2,1:k-2) as
391 *
392 * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
393 * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
394 *
395  IF( k.GT.2 ) THEN
396 *
397  d12 = a( k-1, k )
398  d22 = a( k-1, k-1 ) / d12
399  d11 = a( k, k ) / d12
400  t = cone / ( d11*d22-cone )
401  d12 = t / d12
402 *
403  DO 30 j = k - 2, 1, -1
404  wkm1 = d12*( d11*a( j, k-1 )-a( j, k ) )
405  wk = d12*( d22*a( j, k )-a( j, k-1 ) )
406  DO 20 i = j, 1, -1
407  a( i, j ) = a( i, j ) - a( i, k )*wk -
408  $ a( i, k-1 )*wkm1
409  20 CONTINUE
410  a( j, k ) = wk
411  a( j, k-1 ) = wkm1
412  30 CONTINUE
413 *
414  END IF
415 *
416  END IF
417  END IF
418 *
419 * Store details of the interchanges in IPIV
420 *
421  IF( kstep.EQ.1 ) THEN
422  ipiv( k ) = kp
423  ELSE
424  ipiv( k ) = -kp
425  ipiv( k-1 ) = -kp
426  END IF
427 *
428 * Decrease K and return to the start of the main loop
429 *
430  k = k - kstep
431  go to 10
432 *
433  ELSE
434 *
435 * Factorize A as L*D*L**T using the lower triangle of A
436 *
437 * K is the main loop index, increasing from 1 to N in steps of
438 * 1 or 2
439 *
440  k = 1
441  40 CONTINUE
442 *
443 * If K > N, exit from loop
444 *
445  IF( k.GT.n )
446  $ go to 70
447  kstep = 1
448 *
449 * Determine rows and columns to be interchanged and whether
450 * a 1-by-1 or 2-by-2 pivot block will be used
451 *
452  absakk = cabs1( a( k, k ) )
453 *
454 * IMAX is the row-index of the largest off-diagonal element in
455 * column K, and COLMAX is its absolute value.
456 * Determine both COLMAX and IMAX.
457 *
458  IF( k.LT.n ) THEN
459  imax = k + izamax( n-k, a( k+1, k ), 1 )
460  colmax = cabs1( a( imax, k ) )
461  ELSE
462  colmax = zero
463  END IF
464 *
465  IF( max( absakk, colmax ).EQ.zero .OR. disnan(absakk) ) THEN
466 *
467 * Column K is zero or underflow, or contains a NaN:
468 * set INFO and continue
469 *
470  IF( info.EQ.0 )
471  $ info = k
472  kp = k
473  ELSE
474  IF( absakk.GE.alpha*colmax ) THEN
475 *
476 * no interchange, use 1-by-1 pivot block
477 *
478  kp = k
479  ELSE
480 *
481 * JMAX is the column-index of the largest off-diagonal
482 * element in row IMAX, and ROWMAX is its absolute value
483 *
484  jmax = k - 1 + izamax( imax-k, a( imax, k ), lda )
485  rowmax = cabs1( a( imax, jmax ) )
486  IF( imax.LT.n ) THEN
487  jmax = imax + izamax( n-imax, a( imax+1, imax ), 1 )
488  rowmax = max( rowmax, cabs1( a( jmax, imax ) ) )
489  END IF
490 *
491  IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
492 *
493 * no interchange, use 1-by-1 pivot block
494 *
495  kp = k
496  ELSE IF( cabs1( a( imax, imax ) ).GE.alpha*rowmax ) THEN
497 *
498 * interchange rows and columns K and IMAX, use 1-by-1
499 * pivot block
500 *
501  kp = imax
502  ELSE
503 *
504 * interchange rows and columns K+1 and IMAX, use 2-by-2
505 * pivot block
506 *
507  kp = imax
508  kstep = 2
509  END IF
510  END IF
511 *
512  kk = k + kstep - 1
513  IF( kp.NE.kk ) THEN
514 *
515 * Interchange rows and columns KK and KP in the trailing
516 * submatrix A(k:n,k:n)
517 *
518  IF( kp.LT.n )
519  $ CALL zswap( n-kp, a( kp+1, kk ), 1, a( kp+1, kp ), 1 )
520  CALL zswap( kp-kk-1, a( kk+1, kk ), 1, a( kp, kk+1 ),
521  $ lda )
522  t = a( kk, kk )
523  a( kk, kk ) = a( kp, kp )
524  a( kp, kp ) = t
525  IF( kstep.EQ.2 ) THEN
526  t = a( k+1, k )
527  a( k+1, k ) = a( kp, k )
528  a( kp, k ) = t
529  END IF
530  END IF
531 *
532 * Update the trailing submatrix
533 *
534  IF( kstep.EQ.1 ) THEN
535 *
536 * 1-by-1 pivot block D(k): column k now holds
537 *
538 * W(k) = L(k)*D(k)
539 *
540 * where L(k) is the k-th column of L
541 *
542  IF( k.LT.n ) THEN
543 *
544 * Perform a rank-1 update of A(k+1:n,k+1:n) as
545 *
546 * A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
547 *
548  r1 = cone / a( k, k )
549  CALL zsyr( uplo, n-k, -r1, a( k+1, k ), 1,
550  $ a( k+1, k+1 ), lda )
551 *
552 * Store L(k) in column K
553 *
554  CALL zscal( n-k, r1, a( k+1, k ), 1 )
555  END IF
556  ELSE
557 *
558 * 2-by-2 pivot block D(k)
559 *
560  IF( k.LT.n-1 ) THEN
561 *
562 * Perform a rank-2 update of A(k+2:n,k+2:n) as
563 *
564 * A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**T
565 * = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**T
566 *
567 * where L(k) and L(k+1) are the k-th and (k+1)-th
568 * columns of L
569 *
570  d21 = a( k+1, k )
571  d11 = a( k+1, k+1 ) / d21
572  d22 = a( k, k ) / d21
573  t = cone / ( d11*d22-cone )
574  d21 = t / d21
575 *
576  DO 60 j = k + 2, n
577  wk = d21*( d11*a( j, k )-a( j, k+1 ) )
578  wkp1 = d21*( d22*a( j, k+1 )-a( j, k ) )
579  DO 50 i = j, n
580  a( i, j ) = a( i, j ) - a( i, k )*wk -
581  $ a( i, k+1 )*wkp1
582  50 CONTINUE
583  a( j, k ) = wk
584  a( j, k+1 ) = wkp1
585  60 CONTINUE
586  END IF
587  END IF
588  END IF
589 *
590 * Store details of the interchanges in IPIV
591 *
592  IF( kstep.EQ.1 ) THEN
593  ipiv( k ) = kp
594  ELSE
595  ipiv( k ) = -kp
596  ipiv( k+1 ) = -kp
597  END IF
598 *
599 * Increase K and return to the start of the main loop
600 *
601  k = k + kstep
602  go to 40
603 *
604  END IF
605 *
606  70 CONTINUE
607  RETURN
608 *
609 * End of ZSYTF2
610 *
611  END