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: *> \date December 2016
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 3.7.0) --
195: * -- LAPACK is a software package provided by Univ. of Tennessee, --
196: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
197: * December 2016
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
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