1: *> \brief \b ZLASYF computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagonal pivoting method.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZLASYF + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlasyf.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, KB, LDA, LDW, N, NB
26: * ..
27: * .. Array Arguments ..
28: * INTEGER IPIV( * )
29: * COMPLEX*16 A( LDA, * ), W( LDW, * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> ZLASYF computes a partial factorization of a complex symmetric matrix
39: *> A using the Bunch-Kaufman diagonal pivoting method. The partial
40: *> factorization has the form:
41: *>
42: *> A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
43: *> ( 0 U22 ) ( 0 D ) ( U12**T U22**T )
44: *>
45: *> A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) 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: *> Note that U**T denotes the transpose of U.
51: *>
52: *> ZLASYF is an auxiliary routine called by ZSYTRF. It uses blocked code
53: *> (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
54: *> A22 (if UPLO = 'L').
55: *> \endverbatim
56: *
57: * Arguments:
58: * ==========
59: *
60: *> \param[in] UPLO
61: *> \verbatim
62: *> UPLO is CHARACTER*1
63: *> Specifies whether the upper or lower triangular part of the
64: *> symmetric matrix A is stored:
65: *> = 'U': Upper triangular
66: *> = 'L': Lower triangular
67: *> \endverbatim
68: *>
69: *> \param[in] N
70: *> \verbatim
71: *> N is INTEGER
72: *> The order of the matrix A. N >= 0.
73: *> \endverbatim
74: *>
75: *> \param[in] NB
76: *> \verbatim
77: *> NB is INTEGER
78: *> The maximum number of columns of the matrix A that should be
79: *> factored. NB should be at least 2 to allow for 2-by-2 pivot
80: *> blocks.
81: *> \endverbatim
82: *>
83: *> \param[out] KB
84: *> \verbatim
85: *> KB is INTEGER
86: *> The number of columns of A that were actually factored.
87: *> KB is either NB-1 or NB, or N if N <= NB.
88: *> \endverbatim
89: *>
90: *> \param[in,out] A
91: *> \verbatim
92: *> A is COMPLEX*16 array, dimension (LDA,N)
93: *> On entry, the symmetric matrix A. If UPLO = 'U', the leading
94: *> n-by-n upper triangular part of A contains the upper
95: *> triangular part of the matrix A, and the strictly lower
96: *> triangular part of A is not referenced. If UPLO = 'L', the
97: *> leading n-by-n lower triangular part of A contains the lower
98: *> triangular part of the matrix A, and the strictly upper
99: *> triangular part of A is not referenced.
100: *> On exit, A contains details of the partial factorization.
101: *> \endverbatim
102: *>
103: *> \param[in] LDA
104: *> \verbatim
105: *> LDA is INTEGER
106: *> The leading dimension of the array A. LDA >= max(1,N).
107: *> \endverbatim
108: *>
109: *> \param[out] IPIV
110: *> \verbatim
111: *> IPIV is INTEGER array, dimension (N)
112: *> Details of the interchanges and the block structure of D.
113: *>
114: *> If UPLO = 'U':
115: *> Only the last KB elements of IPIV are set.
116: *>
117: *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
118: *> interchanged and D(k,k) is a 1-by-1 diagonal block.
119: *>
120: *> If IPIV(k) = IPIV(k-1) < 0, then rows and columns
121: *> k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
122: *> is a 2-by-2 diagonal block.
123: *>
124: *> If UPLO = 'L':
125: *> Only the first KB elements of IPIV are set.
126: *>
127: *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
128: *> interchanged and D(k,k) is a 1-by-1 diagonal block.
129: *>
130: *> If IPIV(k) = IPIV(k+1) < 0, then rows and columns
131: *> k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
132: *> is a 2-by-2 diagonal block.
133: *> \endverbatim
134: *>
135: *> \param[out] W
136: *> \verbatim
137: *> W is COMPLEX*16 array, dimension (LDW,NB)
138: *> \endverbatim
139: *>
140: *> \param[in] LDW
141: *> \verbatim
142: *> LDW is INTEGER
143: *> The leading dimension of the array W. LDW >= max(1,N).
144: *> \endverbatim
145: *>
146: *> \param[out] INFO
147: *> \verbatim
148: *> INFO is INTEGER
149: *> = 0: successful exit
150: *> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
151: *> has been completed, but the block diagonal matrix D is
152: *> exactly singular.
153: *> \endverbatim
154: *
155: * Authors:
156: * ========
157: *
158: *> \author Univ. of Tennessee
159: *> \author Univ. of California Berkeley
160: *> \author Univ. of Colorado Denver
161: *> \author NAG Ltd.
162: *
163: *> \date November 2013
164: *
165: *> \ingroup complex16SYcomputational
166: *
167: *> \par Contributors:
168: * ==================
169: *>
170: *> \verbatim
171: *>
172: *> November 2013, Igor Kozachenko,
173: *> Computer Science Division,
174: *> University of California, Berkeley
175: *> \endverbatim
176: *
177: * =====================================================================
178: SUBROUTINE ZLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
179: *
180: * -- LAPACK computational routine (version 3.5.0) --
181: * -- LAPACK is a software package provided by Univ. of Tennessee, --
182: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
183: * November 2013
184: *
185: * .. Scalar Arguments ..
186: CHARACTER UPLO
187: INTEGER INFO, KB, LDA, LDW, N, NB
188: * ..
189: * .. Array Arguments ..
190: INTEGER IPIV( * )
191: COMPLEX*16 A( LDA, * ), W( LDW, * )
192: * ..
193: *
194: * =====================================================================
195: *
196: * .. Parameters ..
197: DOUBLE PRECISION ZERO, ONE
198: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
199: DOUBLE PRECISION EIGHT, SEVTEN
200: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
201: COMPLEX*16 CONE
202: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
203: * ..
204: * .. Local Scalars ..
205: INTEGER IMAX, J, JB, JJ, JMAX, JP, K, KK, KKW, KP,
206: $ KSTEP, KW
207: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX
208: COMPLEX*16 D11, D21, D22, R1, T, Z
209: * ..
210: * .. External Functions ..
211: LOGICAL LSAME
212: INTEGER IZAMAX
213: EXTERNAL LSAME, IZAMAX
214: * ..
215: * .. External Subroutines ..
216: EXTERNAL ZCOPY, ZGEMM, ZGEMV, ZSCAL, ZSWAP
217: * ..
218: * .. Intrinsic Functions ..
219: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN, SQRT
220: * ..
221: * .. Statement Functions ..
222: DOUBLE PRECISION CABS1
223: * ..
224: * .. Statement Function definitions ..
225: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
226: * ..
227: * .. Executable Statements ..
228: *
229: INFO = 0
230: *
231: * Initialize ALPHA for use in choosing pivot block size.
232: *
233: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
234: *
235: IF( LSAME( UPLO, 'U' ) ) THEN
236: *
237: * Factorize the trailing columns of A using the upper triangle
238: * of A and working backwards, and compute the matrix W = U12*D
239: * for use in updating A11
240: *
241: * K is the main loop index, decreasing from N in steps of 1 or 2
242: *
243: * KW is the column of W which corresponds to column K of A
244: *
245: K = N
246: 10 CONTINUE
247: KW = NB + K - N
248: *
249: * Exit from loop
250: *
251: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
252: $ GO TO 30
253: *
254: * Copy column K of A to column KW of W and update it
255: *
256: CALL ZCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
257: IF( K.LT.N )
258: $ CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ), LDA,
259: $ W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 )
260: *
261: KSTEP = 1
262: *
263: * Determine rows and columns to be interchanged and whether
264: * a 1-by-1 or 2-by-2 pivot block will be used
265: *
266: ABSAKK = CABS1( W( K, KW ) )
267: *
268: * IMAX is the row-index of the largest off-diagonal element in
269:
270: *
271: IF( K.GT.1 ) THEN
272: IMAX = IZAMAX( K-1, W( 1, KW ), 1 )
273: COLMAX = CABS1( W( IMAX, KW ) )
274: ELSE
275: COLMAX = ZERO
276: END IF
277: *
278: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
279: *
280: * Column K is zero or underflow: set INFO and continue
281: *
282: IF( INFO.EQ.0 )
283: $ INFO = K
284: KP = K
285: ELSE
286: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
287: *
288: * no interchange, use 1-by-1 pivot block
289: *
290: KP = K
291: ELSE
292: *
293: * Copy column IMAX to column KW-1 of W and update it
294: *
295: CALL ZCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
296: CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
297: $ W( IMAX+1, KW-1 ), 1 )
298: IF( K.LT.N )
299: $ CALL ZGEMV( 'No transpose', K, N-K, -CONE,
300: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
301: $ CONE, W( 1, KW-1 ), 1 )
302: *
303: * JMAX is the column-index of the largest off-diagonal
304: * element in row IMAX, and ROWMAX is its absolute value
305: *
306: JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 )
307: ROWMAX = CABS1( W( JMAX, KW-1 ) )
308: IF( IMAX.GT.1 ) THEN
309: JMAX = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 )
310: ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, KW-1 ) ) )
311: END IF
312: *
313: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
314: *
315: * no interchange, use 1-by-1 pivot block
316: *
317: KP = K
318: ELSE IF( CABS1( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX ) THEN
319: *
320: * interchange rows and columns K and IMAX, use 1-by-1
321: * pivot block
322: *
323: KP = IMAX
324: *
325: * copy column KW-1 of W to column KW of W
326: *
327: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
328: ELSE
329: *
330: * interchange rows and columns K-1 and IMAX, use 2-by-2
331: * pivot block
332: *
333: KP = IMAX
334: KSTEP = 2
335: END IF
336: END IF
337: *
338: * ============================================================
339: *
340: * KK is the column of A where pivoting step stopped
341: *
342: KK = K - KSTEP + 1
343: *
344: * KKW is the column of W which corresponds to column KK of A
345: *
346: KKW = NB + KK - N
347: *
348: * Interchange rows and columns KP and KK.
349: * Updated column KP is already stored in column KKW of W.
350: *
351: IF( KP.NE.KK ) THEN
352: *
353: * Copy non-updated column KK to column KP of submatrix A
354: * at step K. No need to copy element into column K
355: * (or K and K-1 for 2-by-2 pivot) of A, since these columns
356: * will be later overwritten.
357: *
358: A( KP, KP ) = A( KK, KK )
359: CALL ZCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
360: $ LDA )
361: IF( KP.GT.1 )
362: $ CALL ZCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
363: *
364: * Interchange rows KK and KP in last K+1 to N columns of A
365: * (columns K (or K and K-1 for 2-by-2 pivot) of A will be
366: * later overwritten). Interchange rows KK and KP
367: * in last KKW to NB columns of W.
368: *
369: IF( K.LT.N )
370: $ CALL ZSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
371: $ LDA )
372: CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
373: $ LDW )
374: END IF
375: *
376: IF( KSTEP.EQ.1 ) THEN
377: *
378: * 1-by-1 pivot block D(k): column kw of W now holds
379: *
380: * W(kw) = U(k)*D(k),
381: *
382: * where U(k) is the k-th column of U
383: *
384: * Store subdiag. elements of column U(k)
385: * and 1-by-1 block D(k) in column k of A.
386: * NOTE: Diagonal element U(k,k) is a UNIT element
387: * and not stored.
388: * A(k,k) := D(k,k) = W(k,kw)
389: * A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k)
390: *
391: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
392: R1 = CONE / A( K, K )
393: CALL ZSCAL( K-1, R1, A( 1, K ), 1 )
394: *
395: ELSE
396: *
397: * 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold
398: *
399: * ( W(kw-1) W(kw) ) = ( U(k-1) U(k) )*D(k)
400: *
401: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
402: * of U
403: *
404: * Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2
405: * block D(k-1:k,k-1:k) in columns k-1 and k of A.
406: * NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT
407: * block and not stored.
408: * A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw)
409: * A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) =
410: * = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) )
411: *
412: IF( K.GT.2 ) THEN
413: *
414: * Compose the columns of the inverse of 2-by-2 pivot
415: * block D in the following way to reduce the number
416: * of FLOPS when we myltiply panel ( W(kw-1) W(kw) ) by
417: * this inverse
418: *
419: * D**(-1) = ( d11 d21 )**(-1) =
420: * ( d21 d22 )
421: *
422: * = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) =
423: * ( (-d21 ) ( d11 ) )
424: *
425: * = 1/d21 * 1/((d11/d21)*(d22/d21)-1) *
426: *
427: * * ( ( d22/d21 ) ( -1 ) ) =
428: * ( ( -1 ) ( d11/d21 ) )
429: *
430: * = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) ( -1 ) ) =
431: * ( ( -1 ) ( D22 ) )
432: *
433: * = 1/d21 * T * ( ( D11 ) ( -1 ) )
434: * ( ( -1 ) ( D22 ) )
435: *
436: * = D21 * ( ( D11 ) ( -1 ) )
437: * ( ( -1 ) ( D22 ) )
438: *
439: D21 = W( K-1, KW )
440: D11 = W( K, KW ) / D21
441: D22 = W( K-1, KW-1 ) / D21
442: T = CONE / ( D11*D22-CONE )
443: D21 = T / D21
444: *
445: * Update elements in columns A(k-1) and A(k) as
446: * dot products of rows of ( W(kw-1) W(kw) ) and columns
447: * of D**(-1)
448: *
449: DO 20 J = 1, K - 2
450: A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) )
451: A( J, K ) = D21*( D22*W( J, KW )-W( J, KW-1 ) )
452: 20 CONTINUE
453: END IF
454: *
455: * Copy D(k) to A
456: *
457: A( K-1, K-1 ) = W( K-1, KW-1 )
458: A( K-1, K ) = W( K-1, KW )
459: A( K, K ) = W( K, KW )
460: *
461: END IF
462: *
463: END IF
464: *
465: * Store details of the interchanges in IPIV
466: *
467: IF( KSTEP.EQ.1 ) THEN
468: IPIV( K ) = KP
469: ELSE
470: IPIV( K ) = -KP
471: IPIV( K-1 ) = -KP
472: END IF
473: *
474: * Decrease K and return to the start of the main loop
475: *
476: K = K - KSTEP
477: GO TO 10
478: *
479: 30 CONTINUE
480: *
481: * Update the upper triangle of A11 (= A(1:k,1:k)) as
482: *
483: * A11 := A11 - U12*D*U12**T = A11 - U12*W**T
484: *
485: * computing blocks of NB columns at a time
486: *
487: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
488: JB = MIN( NB, K-J+1 )
489: *
490: * Update the upper triangle of the diagonal block
491: *
492: DO 40 JJ = J, J + JB - 1
493: CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE,
494: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE,
495: $ A( J, JJ ), 1 )
496: 40 CONTINUE
497: *
498: * Update the rectangular superdiagonal block
499: *
500: CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB, N-K,
501: $ -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
502: $ CONE, A( 1, J ), LDA )
503: 50 CONTINUE
504: *
505: * Put U12 in standard form by partially undoing the interchanges
506: * in columns k+1:n looping backwards from k+1 to n
507: *
508: J = K + 1
509: 60 CONTINUE
510: *
511: * Undo the interchanges (if any) of rows JJ and JP at each
512: * step J
513: *
514: * (Here, J is a diagonal index)
515: JJ = J
516: JP = IPIV( J )
517: IF( JP.LT.0 ) THEN
518: JP = -JP
519: * (Here, J is a diagonal index)
520: J = J + 1
521: END IF
522: * (NOTE: Here, J is used to determine row length. Length N-J+1
523: * of the rows to swap back doesn't include diagonal element)
524: J = J + 1
525: IF( JP.NE.JJ .AND. J.LE.N )
526: $ CALL ZSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA )
527: IF( J.LT.N )
528: $ GO TO 60
529: *
530: * Set KB to the number of columns factorized
531: *
532: KB = N - K
533: *
534: ELSE
535: *
536: * Factorize the leading columns of A using the lower triangle
537: * of A and working forwards, and compute the matrix W = L21*D
538: * for use in updating A22
539: *
540: * K is the main loop index, increasing from 1 in steps of 1 or 2
541: *
542: K = 1
543: 70 CONTINUE
544: *
545: * Exit from loop
546: *
547: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
548: $ GO TO 90
549: *
550: * Copy column K of A to column K of W and update it
551: *
552: CALL ZCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
553: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ), LDA,
554: $ W( K, 1 ), LDW, CONE, W( K, K ), 1 )
555: *
556: KSTEP = 1
557: *
558: * Determine rows and columns to be interchanged and whether
559: * a 1-by-1 or 2-by-2 pivot block will be used
560: *
561: ABSAKK = CABS1( W( K, K ) )
562: *
563: * IMAX is the row-index of the largest off-diagonal element in
564:
565: *
566: IF( K.LT.N ) THEN
567: IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 )
568: COLMAX = CABS1( W( IMAX, K ) )
569: ELSE
570: COLMAX = ZERO
571: END IF
572: *
573: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
574: *
575: * Column K is zero or underflow: set INFO and continue
576: *
577: IF( INFO.EQ.0 )
578: $ INFO = K
579: KP = K
580: ELSE
581: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
582: *
583: * no interchange, use 1-by-1 pivot block
584: *
585: KP = K
586: ELSE
587: *
588: * Copy column IMAX to column K+1 of W and update it
589: *
590: CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1 )
591: CALL ZCOPY( N-IMAX+1, A( IMAX, IMAX ), 1, W( IMAX, K+1 ),
592: $ 1 )
593: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
594: $ LDA, W( IMAX, 1 ), LDW, CONE, W( K, K+1 ),
595: $ 1 )
596: *
597: * JMAX is the column-index of the largest off-diagonal
598: * element in row IMAX, and ROWMAX is its absolute value
599: *
600: JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 )
601: ROWMAX = CABS1( W( JMAX, K+1 ) )
602: IF( IMAX.LT.N ) THEN
603: JMAX = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1 )
604: ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, K+1 ) ) )
605: END IF
606: *
607: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
608: *
609: * no interchange, use 1-by-1 pivot block
610: *
611: KP = K
612: ELSE IF( CABS1( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX ) THEN
613: *
614: * interchange rows and columns K and IMAX, use 1-by-1
615: * pivot block
616: *
617: KP = IMAX
618: *
619: * copy column K+1 of W to column K of W
620: *
621: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
622: ELSE
623: *
624: * interchange rows and columns K+1 and IMAX, use 2-by-2
625: * pivot block
626: *
627: KP = IMAX
628: KSTEP = 2
629: END IF
630: END IF
631: *
632: * ============================================================
633: *
634: * KK is the column of A where pivoting step stopped
635: *
636: KK = K + KSTEP - 1
637: *
638: * Interchange rows and columns KP and KK.
639: * Updated column KP is already stored in column KK of W.
640: *
641: IF( KP.NE.KK ) THEN
642: *
643: * Copy non-updated column KK to column KP of submatrix A
644: * at step K. No need to copy element into column K
645: * (or K and K+1 for 2-by-2 pivot) of A, since these columns
646: * will be later overwritten.
647: *
648: A( KP, KP ) = A( KK, KK )
649: CALL ZCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
650: $ LDA )
651: IF( KP.LT.N )
652: $ CALL ZCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
653: *
654: * Interchange rows KK and KP in first K-1 columns of A
655: * (columns K (or K and K+1 for 2-by-2 pivot) of A will be
656: * later overwritten). Interchange rows KK and KP
657: * in first KK columns of W.
658: *
659: IF( K.GT.1 )
660: $ CALL ZSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
661: CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
662: END IF
663: *
664: IF( KSTEP.EQ.1 ) THEN
665: *
666: * 1-by-1 pivot block D(k): column k of W now holds
667: *
668: * W(k) = L(k)*D(k),
669: *
670: * where L(k) is the k-th column of L
671: *
672: * Store subdiag. elements of column L(k)
673: * and 1-by-1 block D(k) in column k of A.
674: * (NOTE: Diagonal element L(k,k) is a UNIT element
675: * and not stored)
676: * A(k,k) := D(k,k) = W(k,k)
677: * A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k)
678: *
679: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
680: IF( K.LT.N ) THEN
681: R1 = CONE / A( K, K )
682: CALL ZSCAL( N-K, R1, A( K+1, K ), 1 )
683: END IF
684: *
685: ELSE
686: *
687: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
688: *
689: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
690: *
691: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
692: * of L
693: *
694: * Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2
695: * block D(k:k+1,k:k+1) in columns k and k+1 of A.
696: * (NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT
697: * block and not stored)
698: * A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1)
699: * A(k+2:N,k:k+1) := L(k+2:N,k:k+1) =
700: * = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) )
701: *
702: IF( K.LT.N-1 ) THEN
703: *
704: * Compose the columns of the inverse of 2-by-2 pivot
705: * block D in the following way to reduce the number
706: * of FLOPS when we myltiply panel ( W(k) W(k+1) ) by
707: * this inverse
708: *
709: * D**(-1) = ( d11 d21 )**(-1) =
710: * ( d21 d22 )
711: *
712: * = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) =
713: * ( (-d21 ) ( d11 ) )
714: *
715: * = 1/d21 * 1/((d11/d21)*(d22/d21)-1) *
716: *
717: * * ( ( d22/d21 ) ( -1 ) ) =
718: * ( ( -1 ) ( d11/d21 ) )
719: *
720: * = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) ( -1 ) ) =
721: * ( ( -1 ) ( D22 ) )
722: *
723: * = 1/d21 * T * ( ( D11 ) ( -1 ) )
724: * ( ( -1 ) ( D22 ) )
725: *
726: * = D21 * ( ( D11 ) ( -1 ) )
727: * ( ( -1 ) ( D22 ) )
728: *
729: D21 = W( K+1, K )
730: D11 = W( K+1, K+1 ) / D21
731: D22 = W( K, K ) / D21
732: T = CONE / ( D11*D22-CONE )
733: D21 = T / D21
734: *
735: * Update elements in columns A(k) and A(k+1) as
736: * dot products of rows of ( W(k) W(k+1) ) and columns
737: * of D**(-1)
738: *
739: DO 80 J = K + 2, N
740: A( J, K ) = D21*( D11*W( J, K )-W( J, K+1 ) )
741: A( J, K+1 ) = D21*( D22*W( J, K+1 )-W( J, K ) )
742: 80 CONTINUE
743: END IF
744: *
745: * Copy D(k) to A
746: *
747: A( K, K ) = W( K, K )
748: A( K+1, K ) = W( K+1, K )
749: A( K+1, K+1 ) = W( K+1, K+1 )
750: *
751: END IF
752: *
753: END IF
754: *
755: * Store details of the interchanges in IPIV
756: *
757: IF( KSTEP.EQ.1 ) THEN
758: IPIV( K ) = KP
759: ELSE
760: IPIV( K ) = -KP
761: IPIV( K+1 ) = -KP
762: END IF
763: *
764: * Increase K and return to the start of the main loop
765: *
766: K = K + KSTEP
767: GO TO 70
768: *
769: 90 CONTINUE
770: *
771: * Update the lower triangle of A22 (= A(k:n,k:n)) as
772: *
773: * A22 := A22 - L21*D*L21**T = A22 - L21*W**T
774: *
775: * computing blocks of NB columns at a time
776: *
777: DO 110 J = K, N, NB
778: JB = MIN( NB, N-J+1 )
779: *
780: * Update the lower triangle of the diagonal block
781: *
782: DO 100 JJ = J, J + JB - 1
783: CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
784: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
785: $ A( JJ, JJ ), 1 )
786: 100 CONTINUE
787: *
788: * Update the rectangular subdiagonal block
789: *
790: IF( J+JB.LE.N )
791: $ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
792: $ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ),
793: $ LDW, CONE, A( J+JB, J ), LDA )
794: 110 CONTINUE
795: *
796: * Put L21 in standard form by partially undoing the interchanges
797: * of rows in columns 1:k-1 looping backwards from k-1 to 1
798: *
799: J = K - 1
800: 120 CONTINUE
801: *
802: * Undo the interchanges (if any) of rows JJ and JP at each
803: * step J
804: *
805: * (Here, J is a diagonal index)
806: JJ = J
807: JP = IPIV( J )
808: IF( JP.LT.0 ) THEN
809: JP = -JP
810: * (Here, J is a diagonal index)
811: J = J - 1
812: END IF
813: * (NOTE: Here, J is used to determine row length. Length J
814: * of the rows to swap back doesn't include diagonal element)
815: J = J - 1
816: IF( JP.NE.JJ .AND. J.GE.1 )
817: $ CALL ZSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA )
818: IF( J.GT.1 )
819: $ GO TO 120
820: *
821: * Set KB to the number of columns factorized
822: *
823: KB = K - 1
824: *
825: END IF
826: RETURN
827: *
828: * End of ZLASYF
829: *
830: END
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