LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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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
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zsytf2.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zsytf2.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zsytf2.f">
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*> \ingroup complex16SYcomputational
131*
132*> \par Further Details:
133* =====================
134*>
135*> \verbatim
136*>
137*> If UPLO = 'U', then A = U*D*U**T, where
138*> U = P(n)*U(n)* ... *P(k)U(k)* ...,
139*> i.e., U is a product of terms P(k)*U(k), where k decreases from n to
140*> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
141*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
142*> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
143*> that if the diagonal block D(k) is of order s (s = 1 or 2), then
144*>
145*> ( I v 0 ) k-s
146*> U(k) = ( 0 I 0 ) s
147*> ( 0 0 I ) n-k
148*> k-s s n-k
149*>
150*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
151*> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
152*> and A(k,k), and v overwrites A(1:k-2,k-1:k).
153*>
154*> If UPLO = 'L', then A = L*D*L**T, where
155*> L = P(1)*L(1)* ... *P(k)*L(k)* ...,
156*> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
157*> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
158*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
159*> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
160*> that if the diagonal block D(k) is of order s (s = 1 or 2), then
161*>
162*> ( I 0 0 ) k-1
163*> L(k) = ( 0 I 0 ) s
164*> ( 0 v I ) n-k-s+1
165*> k-1 s n-k-s+1
166*>
167*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
168*> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
169*> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
170*> \endverbatim
171*
172*> \par Contributors:
173* ==================
174*>
175*> \verbatim
176*>
177*> 09-29-06 - patch from
178*> Bobby Cheng, MathWorks
179*>
180*> Replace l.209 and l.377
181*> IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
182*> by
183*> IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
184*>
185*> 1-96 - Based on modifications by J. Lewis, Boeing Computer Services
186*> Company
187*> \endverbatim
188*
189* =====================================================================
190 SUBROUTINE zsytf2( UPLO, N, A, LDA, IPIV, INFO )
191*
192* -- LAPACK computational routine --
193* -- LAPACK is a software package provided by Univ. of Tennessee, --
194* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
195*
196* .. Scalar Arguments ..
197 CHARACTER UPLO
198 INTEGER INFO, LDA, N
199* ..
200* .. Array Arguments ..
201 INTEGER IPIV( * )
202 COMPLEX*16 A( LDA, * )
203* ..
204*
205* =====================================================================
206*
207* .. Parameters ..
208 DOUBLE PRECISION ZERO, ONE
209 parameter( zero = 0.0d+0, one = 1.0d+0 )
210 DOUBLE PRECISION EIGHT, SEVTEN
211 parameter( eight = 8.0d+0, sevten = 17.0d+0 )
212 COMPLEX*16 CONE
213 parameter( cone = ( 1.0d+0, 0.0d+0 ) )
214* ..
215* .. Local Scalars ..
216 LOGICAL UPPER
217 INTEGER I, IMAX, J, JMAX, K, KK, KP, KSTEP
218 DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX
219 COMPLEX*16 D11, D12, D21, D22, R1, T, WK, WKM1, WKP1, Z
220* ..
221* .. External Functions ..
222 LOGICAL DISNAN, LSAME
223 INTEGER IZAMAX
224 EXTERNAL disnan, lsame, izamax
225* ..
226* .. External Subroutines ..
227 EXTERNAL xerbla, zscal, zswap, zsyr
228* ..
229* .. Intrinsic Functions ..
230 INTRINSIC abs, dble, dimag, max, sqrt
231* ..
232* .. Statement Functions ..
233 DOUBLE PRECISION CABS1
234* ..
235* .. Statement Function definitions ..
236 cabs1( z ) = abs( dble( z ) ) + abs( dimag( z ) )
237* ..
238* .. Executable Statements ..
239*
240* Test the input parameters.
241*
242 info = 0
243 upper = lsame( uplo, 'U' )
244 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
245 info = -1
246 ELSE IF( n.LT.0 ) THEN
247 info = -2
248 ELSE IF( lda.LT.max( 1, n ) ) THEN
249 info = -4
250 END IF
251 IF( info.NE.0 ) THEN
252 CALL xerbla( 'ZSYTF2', -info )
253 RETURN
254 END IF
255*
256* Initialize ALPHA for use in choosing pivot block size.
257*
258 alpha = ( one+sqrt( sevten ) ) / eight
259*
260 IF( upper ) THEN
261*
262* Factorize A as U*D*U**T using the upper triangle of A
263*
264* K is the main loop index, decreasing from N to 1 in steps of
265* 1 or 2
266*
267 k = n
268 10 CONTINUE
269*
270* If K < 1, exit from loop
271*
272 IF( k.LT.1 )
273 $ GO TO 70
274 kstep = 1
275*
276* Determine rows and columns to be interchanged and whether
277* a 1-by-1 or 2-by-2 pivot block will be used
278*
279 absakk = cabs1( a( k, k ) )
280*
281* IMAX is the row-index of the largest off-diagonal element in
282* column K, and COLMAX is its absolute value.
283* Determine both COLMAX and IMAX.
284*
285 IF( k.GT.1 ) THEN
286 imax = izamax( k-1, a( 1, k ), 1 )
287 colmax = cabs1( a( imax, k ) )
288 ELSE
289 colmax = zero
290 END IF
291*
292 IF( max( absakk, colmax ).EQ.zero .OR. disnan(absakk) ) THEN
293*
294* Column K is zero or underflow, or contains a NaN:
295* set INFO and continue
296*
297 IF( info.EQ.0 )
298 $ info = k
299 kp = k
300 ELSE
301 IF( absakk.GE.alpha*colmax ) THEN
302*
303* no interchange, use 1-by-1 pivot block
304*
305 kp = k
306 ELSE
307*
308* JMAX is the column-index of the largest off-diagonal
309* element in row IMAX, and ROWMAX is its absolute value
310*
311 jmax = imax + izamax( k-imax, a( imax, imax+1 ), lda )
312 rowmax = cabs1( a( imax, jmax ) )
313 IF( imax.GT.1 ) THEN
314 jmax = izamax( imax-1, a( 1, imax ), 1 )
315 rowmax = max( rowmax, cabs1( a( jmax, imax ) ) )
316 END IF
317*
318 IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
319*
320* no interchange, use 1-by-1 pivot block
321*
322 kp = k
323 ELSE IF( cabs1( a( imax, imax ) ).GE.alpha*rowmax ) THEN
324*
325* interchange rows and columns K and IMAX, use 1-by-1
326* pivot block
327*
328 kp = imax
329 ELSE
330*
331* interchange rows and columns K-1 and IMAX, use 2-by-2
332* pivot block
333*
334 kp = imax
335 kstep = 2
336 END IF
337 END IF
338*
339 kk = k - kstep + 1
340 IF( kp.NE.kk ) THEN
341*
342* Interchange rows and columns KK and KP in the leading
343* submatrix A(1:k,1:k)
344*
345 CALL zswap( kp-1, a( 1, kk ), 1, a( 1, kp ), 1 )
346 CALL zswap( kk-kp-1, a( kp+1, kk ), 1, a( kp, kp+1 ),
347 $ lda )
348 t = a( kk, kk )
349 a( kk, kk ) = a( kp, kp )
350 a( kp, kp ) = t
351 IF( kstep.EQ.2 ) THEN
352 t = a( k-1, k )
353 a( k-1, k ) = a( kp, k )
354 a( kp, k ) = t
355 END IF
356 END IF
357*
358* Update the leading submatrix
359*
360 IF( kstep.EQ.1 ) THEN
361*
362* 1-by-1 pivot block D(k): column k now holds
363*
364* W(k) = U(k)*D(k)
365*
366* where U(k) is the k-th column of U
367*
368* Perform a rank-1 update of A(1:k-1,1:k-1) as
369*
370* A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
371*
372 r1 = cone / a( k, k )
373 CALL zsyr( uplo, k-1, -r1, a( 1, k ), 1, a, lda )
374*
375* Store U(k) in column k
376*
377 CALL zscal( k-1, r1, a( 1, k ), 1 )
378 ELSE
379*
380* 2-by-2 pivot block D(k): columns k and k-1 now hold
381*
382* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
383*
384* where U(k) and U(k-1) are the k-th and (k-1)-th columns
385* of U
386*
387* Perform a rank-2 update of A(1:k-2,1:k-2) as
388*
389* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
390* = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
391*
392 IF( k.GT.2 ) THEN
393*
394 d12 = a( k-1, k )
395 d22 = a( k-1, k-1 ) / d12
396 d11 = a( k, k ) / d12
397 t = cone / ( d11*d22-cone )
398 d12 = t / d12
399*
400 DO 30 j = k - 2, 1, -1
401 wkm1 = d12*( d11*a( j, k-1 )-a( j, k ) )
402 wk = d12*( d22*a( j, k )-a( j, k-1 ) )
403 DO 20 i = j, 1, -1
404 a( i, j ) = a( i, j ) - a( i, k )*wk -
405 $ a( i, k-1 )*wkm1
406 20 CONTINUE
407 a( j, k ) = wk
408 a( j, k-1 ) = wkm1
409 30 CONTINUE
410*
411 END IF
412*
413 END IF
414 END IF
415*
416* Store details of the interchanges in IPIV
417*
418 IF( kstep.EQ.1 ) THEN
419 ipiv( k ) = kp
420 ELSE
421 ipiv( k ) = -kp
422 ipiv( k-1 ) = -kp
423 END IF
424*
425* Decrease K and return to the start of the main loop
426*
427 k = k - kstep
428 GO TO 10
429*
430 ELSE
431*
432* Factorize A as L*D*L**T using the lower triangle of A
433*
434* K is the main loop index, increasing from 1 to N in steps of
435* 1 or 2
436*
437 k = 1
438 40 CONTINUE
439*
440* If K > N, exit from loop
441*
442 IF( k.GT.n )
443 $ GO TO 70
444 kstep = 1
445*
446* Determine rows and columns to be interchanged and whether
447* a 1-by-1 or 2-by-2 pivot block will be used
448*
449 absakk = cabs1( a( k, k ) )
450*
451* IMAX is the row-index of the largest off-diagonal element in
452* column K, and COLMAX is its absolute value.
453* Determine both COLMAX and IMAX.
454*
455 IF( k.LT.n ) THEN
456 imax = k + izamax( n-k, a( k+1, k ), 1 )
457 colmax = cabs1( a( imax, k ) )
458 ELSE
459 colmax = zero
460 END IF
461*
462 IF( max( absakk, colmax ).EQ.zero .OR. disnan(absakk) ) THEN
463*
464* Column K is zero or underflow, or contains a NaN:
465* set INFO and continue
466*
467 IF( info.EQ.0 )
468 $ info = k
469 kp = k
470 ELSE
471 IF( absakk.GE.alpha*colmax ) THEN
472*
473* no interchange, use 1-by-1 pivot block
474*
475 kp = k
476 ELSE
477*
478* JMAX is the column-index of the largest off-diagonal
479* element in row IMAX, and ROWMAX is its absolute value
480*
481 jmax = k - 1 + izamax( imax-k, a( imax, k ), lda )
482 rowmax = cabs1( a( imax, jmax ) )
483 IF( imax.LT.n ) THEN
484 jmax = imax + izamax( n-imax, a( imax+1, imax ), 1 )
485 rowmax = max( rowmax, cabs1( a( jmax, imax ) ) )
486 END IF
487*
488 IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
489*
490* no interchange, use 1-by-1 pivot block
491*
492 kp = k
493 ELSE IF( cabs1( a( imax, imax ) ).GE.alpha*rowmax ) THEN
494*
495* interchange rows and columns K and IMAX, use 1-by-1
496* pivot block
497*
498 kp = imax
499 ELSE
500*
501* interchange rows and columns K+1 and IMAX, use 2-by-2
502* pivot block
503*
504 kp = imax
505 kstep = 2
506 END IF
507 END IF
508*
509 kk = k + kstep - 1
510 IF( kp.NE.kk ) THEN
511*
512* Interchange rows and columns KK and KP in the trailing
513* submatrix A(k:n,k:n)
514*
515 IF( kp.LT.n )
516 $ CALL zswap( n-kp, a( kp+1, kk ), 1, a( kp+1, kp ), 1 )
517 CALL zswap( kp-kk-1, a( kk+1, kk ), 1, a( kp, kk+1 ),
518 $ lda )
519 t = a( kk, kk )
520 a( kk, kk ) = a( kp, kp )
521 a( kp, kp ) = t
522 IF( kstep.EQ.2 ) THEN
523 t = a( k+1, k )
524 a( k+1, k ) = a( kp, k )
525 a( kp, k ) = t
526 END IF
527 END IF
528*
529* Update the trailing submatrix
530*
531 IF( kstep.EQ.1 ) THEN
532*
533* 1-by-1 pivot block D(k): column k now holds
534*
535* W(k) = L(k)*D(k)
536*
537* where L(k) is the k-th column of L
538*
539 IF( k.LT.n ) THEN
540*
541* Perform a rank-1 update of A(k+1:n,k+1:n) as
542*
543* A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
544*
545 r1 = cone / a( k, k )
546 CALL zsyr( uplo, n-k, -r1, a( k+1, k ), 1,
547 $ a( k+1, k+1 ), lda )
548*
549* Store L(k) in column K
550*
551 CALL zscal( n-k, r1, a( k+1, k ), 1 )
552 END IF
553 ELSE
554*
555* 2-by-2 pivot block D(k)
556*
557 IF( k.LT.n-1 ) THEN
558*
559* Perform a rank-2 update of A(k+2:n,k+2:n) as
560*
561* A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**T
562* = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**T
563*
564* where L(k) and L(k+1) are the k-th and (k+1)-th
565* columns of L
566*
567 d21 = a( k+1, k )
568 d11 = a( k+1, k+1 ) / d21
569 d22 = a( k, k ) / d21
570 t = cone / ( d11*d22-cone )
571 d21 = t / d21
572*
573 DO 60 j = k + 2, n
574 wk = d21*( d11*a( j, k )-a( j, k+1 ) )
575 wkp1 = d21*( d22*a( j, k+1 )-a( j, k ) )
576 DO 50 i = j, n
577 a( i, j ) = a( i, j ) - a( i, k )*wk -
578 $ a( i, k+1 )*wkp1
579 50 CONTINUE
580 a( j, k ) = wk
581 a( j, k+1 ) = wkp1
582 60 CONTINUE
583 END IF
584 END IF
585 END IF
586*
587* Store details of the interchanges in IPIV
588*
589 IF( kstep.EQ.1 ) THEN
590 ipiv( k ) = kp
591 ELSE
592 ipiv( k ) = -kp
593 ipiv( k+1 ) = -kp
594 END IF
595*
596* Increase K and return to the start of the main loop
597*
598 k = k + kstep
599 GO TO 40
600*
601 END IF
602*
603 70 CONTINUE
604 RETURN
605*
606* End of ZSYTF2
607*
608 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zswap(N, ZX, INCX, ZY, INCY)
ZSWAP
Definition: zswap.f:81
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:78
subroutine zsyr(UPLO, N, ALPHA, X, INCX, A, LDA)
ZSYR performs the symmetric rank-1 update of a complex symmetric matrix.
Definition: zsyr.f:135
subroutine zsytf2(UPLO, N, A, LDA, IPIV, INFO)
ZSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting ...
Definition: zsytf2.f:191