1: SUBROUTINE ZLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
2: *
3: * -- LAPACK routine (version 3.2) --
4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6: * November 2006
7: *
8: * .. Scalar Arguments ..
9: CHARACTER UPLO
10: INTEGER INFO, KB, LDA, LDW, N, NB
11: * ..
12: * .. Array Arguments ..
13: INTEGER IPIV( * )
14: COMPLEX*16 A( LDA, * ), W( LDW, * )
15: * ..
16: *
17: * Purpose
18: * =======
19: *
20: * ZLASYF computes a partial factorization of a complex symmetric matrix
21: * A using the Bunch-Kaufman diagonal pivoting method. The partial
22: * factorization has the form:
23: *
24: * A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
25: * ( 0 U22 ) ( 0 D ) ( U12' U22' )
26: *
27: * A = ( L11 0 ) ( D 0 ) ( L11' L21' ) if UPLO = 'L'
28: * ( L21 I ) ( 0 A22 ) ( 0 I )
29: *
30: * where the order of D is at most NB. The actual order is returned in
31: * the argument KB, and is either NB or NB-1, or N if N <= NB.
32: * Note that U' denotes the transpose of U.
33: *
34: * ZLASYF is an auxiliary routine called by ZSYTRF. It uses blocked code
35: * (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
36: * A22 (if UPLO = 'L').
37: *
38: * Arguments
39: * =========
40: *
41: * UPLO (input) CHARACTER*1
42: * Specifies whether the upper or lower triangular part of the
43: * symmetric matrix A is stored:
44: * = 'U': Upper triangular
45: * = 'L': Lower triangular
46: *
47: * N (input) INTEGER
48: * The order of the matrix A. N >= 0.
49: *
50: * NB (input) INTEGER
51: * The maximum number of columns of the matrix A that should be
52: * factored. NB should be at least 2 to allow for 2-by-2 pivot
53: * blocks.
54: *
55: * KB (output) INTEGER
56: * The number of columns of A that were actually factored.
57: * KB is either NB-1 or NB, or N if N <= NB.
58: *
59: * A (input/output) COMPLEX*16 array, dimension (LDA,N)
60: * On entry, the symmetric matrix A. If UPLO = 'U', the leading
61: * n-by-n upper triangular part of A contains the upper
62: * triangular part of the matrix A, and the strictly lower
63: * triangular part of A is not referenced. If UPLO = 'L', the
64: * leading n-by-n lower triangular part of A contains the lower
65: * triangular part of the matrix A, and the strictly upper
66: * triangular part of A is not referenced.
67: * On exit, A contains details of the partial factorization.
68: *
69: * LDA (input) INTEGER
70: * The leading dimension of the array A. LDA >= max(1,N).
71: *
72: * IPIV (output) INTEGER array, dimension (N)
73: * Details of the interchanges and the block structure of D.
74: * If UPLO = 'U', only the last KB elements of IPIV are set;
75: * if UPLO = 'L', only the first KB elements are set.
76: *
77: * If IPIV(k) > 0, then rows and columns k and IPIV(k) were
78: * interchanged and D(k,k) is a 1-by-1 diagonal block.
79: * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
80: * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
81: * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
82: * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
83: * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
84: *
85: * W (workspace) COMPLEX*16 array, dimension (LDW,NB)
86: *
87: * LDW (input) INTEGER
88: * The leading dimension of the array W. LDW >= max(1,N).
89: *
90: * INFO (output) INTEGER
91: * = 0: successful exit
92: * > 0: if INFO = k, D(k,k) is exactly zero. The factorization
93: * has been completed, but the block diagonal matrix D is
94: * exactly singular.
95: *
96: * =====================================================================
97: *
98: * .. Parameters ..
99: DOUBLE PRECISION ZERO, ONE
100: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
101: DOUBLE PRECISION EIGHT, SEVTEN
102: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
103: COMPLEX*16 CONE
104: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
105: * ..
106: * .. Local Scalars ..
107: INTEGER IMAX, J, JB, JJ, JMAX, JP, K, KK, KKW, KP,
108: $ KSTEP, KW
109: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX
110: COMPLEX*16 D11, D21, D22, R1, T, Z
111: * ..
112: * .. External Functions ..
113: LOGICAL LSAME
114: INTEGER IZAMAX
115: EXTERNAL LSAME, IZAMAX
116: * ..
117: * .. External Subroutines ..
118: EXTERNAL ZCOPY, ZGEMM, ZGEMV, ZSCAL, ZSWAP
119: * ..
120: * .. Intrinsic Functions ..
121: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN, SQRT
122: * ..
123: * .. Statement Functions ..
124: DOUBLE PRECISION CABS1
125: * ..
126: * .. Statement Function definitions ..
127: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
128: * ..
129: * .. Executable Statements ..
130: *
131: INFO = 0
132: *
133: * Initialize ALPHA for use in choosing pivot block size.
134: *
135: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
136: *
137: IF( LSAME( UPLO, 'U' ) ) THEN
138: *
139: * Factorize the trailing columns of A using the upper triangle
140: * of A and working backwards, and compute the matrix W = U12*D
141: * for use in updating A11
142: *
143: * K is the main loop index, decreasing from N in steps of 1 or 2
144: *
145: * KW is the column of W which corresponds to column K of A
146: *
147: K = N
148: 10 CONTINUE
149: KW = NB + K - N
150: *
151: * Exit from loop
152: *
153: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
154: $ GO TO 30
155: *
156: * Copy column K of A to column KW of W and update it
157: *
158: CALL ZCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
159: IF( K.LT.N )
160: $ CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ), LDA,
161: $ W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 )
162: *
163: KSTEP = 1
164: *
165: * Determine rows and columns to be interchanged and whether
166: * a 1-by-1 or 2-by-2 pivot block will be used
167: *
168: ABSAKK = CABS1( W( K, KW ) )
169: *
170: * IMAX is the row-index of the largest off-diagonal element in
171: * column K, and COLMAX is its absolute value
172: *
173: IF( K.GT.1 ) THEN
174: IMAX = IZAMAX( K-1, W( 1, KW ), 1 )
175: COLMAX = CABS1( W( IMAX, KW ) )
176: ELSE
177: COLMAX = ZERO
178: END IF
179: *
180: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
181: *
182: * Column K is zero: set INFO and continue
183: *
184: IF( INFO.EQ.0 )
185: $ INFO = K
186: KP = K
187: ELSE
188: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
189: *
190: * no interchange, use 1-by-1 pivot block
191: *
192: KP = K
193: ELSE
194: *
195: * Copy column IMAX to column KW-1 of W and update it
196: *
197: CALL ZCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
198: CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
199: $ W( IMAX+1, KW-1 ), 1 )
200: IF( K.LT.N )
201: $ CALL ZGEMV( 'No transpose', K, N-K, -CONE,
202: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
203: $ CONE, W( 1, KW-1 ), 1 )
204: *
205: * JMAX is the column-index of the largest off-diagonal
206: * element in row IMAX, and ROWMAX is its absolute value
207: *
208: JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 )
209: ROWMAX = CABS1( W( JMAX, KW-1 ) )
210: IF( IMAX.GT.1 ) THEN
211: JMAX = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 )
212: ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, KW-1 ) ) )
213: END IF
214: *
215: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
216: *
217: * no interchange, use 1-by-1 pivot block
218: *
219: KP = K
220: ELSE IF( CABS1( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX ) THEN
221: *
222: * interchange rows and columns K and IMAX, use 1-by-1
223: * pivot block
224: *
225: KP = IMAX
226: *
227: * copy column KW-1 of W to column KW
228: *
229: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
230: ELSE
231: *
232: * interchange rows and columns K-1 and IMAX, use 2-by-2
233: * pivot block
234: *
235: KP = IMAX
236: KSTEP = 2
237: END IF
238: END IF
239: *
240: KK = K - KSTEP + 1
241: KKW = NB + KK - N
242: *
243: * Updated column KP is already stored in column KKW of W
244: *
245: IF( KP.NE.KK ) THEN
246: *
247: * Copy non-updated column KK to column KP
248: *
249: A( KP, K ) = A( KK, K )
250: CALL ZCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
251: $ LDA )
252: CALL ZCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 )
253: *
254: * Interchange rows KK and KP in last KK columns of A and W
255: *
256: CALL ZSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA )
257: CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
258: $ LDW )
259: END IF
260: *
261: IF( KSTEP.EQ.1 ) THEN
262: *
263: * 1-by-1 pivot block D(k): column KW of W now holds
264: *
265: * W(k) = U(k)*D(k)
266: *
267: * where U(k) is the k-th column of U
268: *
269: * Store U(k) in column k of A
270: *
271: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
272: R1 = CONE / A( K, K )
273: CALL ZSCAL( K-1, R1, A( 1, K ), 1 )
274: ELSE
275: *
276: * 2-by-2 pivot block D(k): columns KW and KW-1 of W now
277: * hold
278: *
279: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
280: *
281: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
282: * of U
283: *
284: IF( K.GT.2 ) THEN
285: *
286: * Store U(k) and U(k-1) in columns k and k-1 of A
287: *
288: D21 = W( K-1, KW )
289: D11 = W( K, KW ) / D21
290: D22 = W( K-1, KW-1 ) / D21
291: T = CONE / ( D11*D22-CONE )
292: D21 = T / D21
293: DO 20 J = 1, K - 2
294: A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) )
295: A( J, K ) = D21*( D22*W( J, KW )-W( J, KW-1 ) )
296: 20 CONTINUE
297: END IF
298: *
299: * Copy D(k) to A
300: *
301: A( K-1, K-1 ) = W( K-1, KW-1 )
302: A( K-1, K ) = W( K-1, KW )
303: A( K, K ) = W( K, KW )
304: END IF
305: END IF
306: *
307: * Store details of the interchanges in IPIV
308: *
309: IF( KSTEP.EQ.1 ) THEN
310: IPIV( K ) = KP
311: ELSE
312: IPIV( K ) = -KP
313: IPIV( K-1 ) = -KP
314: END IF
315: *
316: * Decrease K and return to the start of the main loop
317: *
318: K = K - KSTEP
319: GO TO 10
320: *
321: 30 CONTINUE
322: *
323: * Update the upper triangle of A11 (= A(1:k,1:k)) as
324: *
325: * A11 := A11 - U12*D*U12' = A11 - U12*W'
326: *
327: * computing blocks of NB columns at a time
328: *
329: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
330: JB = MIN( NB, K-J+1 )
331: *
332: * Update the upper triangle of the diagonal block
333: *
334: DO 40 JJ = J, J + JB - 1
335: CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE,
336: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE,
337: $ A( J, JJ ), 1 )
338: 40 CONTINUE
339: *
340: * Update the rectangular superdiagonal block
341: *
342: CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB, N-K,
343: $ -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
344: $ CONE, A( 1, J ), LDA )
345: 50 CONTINUE
346: *
347: * Put U12 in standard form by partially undoing the interchanges
348: * in columns k+1:n
349: *
350: J = K + 1
351: 60 CONTINUE
352: JJ = J
353: JP = IPIV( J )
354: IF( JP.LT.0 ) THEN
355: JP = -JP
356: J = J + 1
357: END IF
358: J = J + 1
359: IF( JP.NE.JJ .AND. J.LE.N )
360: $ CALL ZSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA )
361: IF( J.LE.N )
362: $ GO TO 60
363: *
364: * Set KB to the number of columns factorized
365: *
366: KB = N - K
367: *
368: ELSE
369: *
370: * Factorize the leading columns of A using the lower triangle
371: * of A and working forwards, and compute the matrix W = L21*D
372: * for use in updating A22
373: *
374: * K is the main loop index, increasing from 1 in steps of 1 or 2
375: *
376: K = 1
377: 70 CONTINUE
378: *
379: * Exit from loop
380: *
381: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
382: $ GO TO 90
383: *
384: * Copy column K of A to column K of W and update it
385: *
386: CALL ZCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
387: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ), LDA,
388: $ W( K, 1 ), LDW, CONE, W( K, K ), 1 )
389: *
390: KSTEP = 1
391: *
392: * Determine rows and columns to be interchanged and whether
393: * a 1-by-1 or 2-by-2 pivot block will be used
394: *
395: ABSAKK = CABS1( W( K, K ) )
396: *
397: * IMAX is the row-index of the largest off-diagonal element in
398: * column K, and COLMAX is its absolute value
399: *
400: IF( K.LT.N ) THEN
401: IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 )
402: COLMAX = CABS1( W( IMAX, K ) )
403: ELSE
404: COLMAX = ZERO
405: END IF
406: *
407: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
408: *
409: * Column K is zero: set INFO and continue
410: *
411: IF( INFO.EQ.0 )
412: $ INFO = K
413: KP = K
414: ELSE
415: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
416: *
417: * no interchange, use 1-by-1 pivot block
418: *
419: KP = K
420: ELSE
421: *
422: * Copy column IMAX to column K+1 of W and update it
423: *
424: CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1 )
425: CALL ZCOPY( N-IMAX+1, A( IMAX, IMAX ), 1, W( IMAX, K+1 ),
426: $ 1 )
427: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
428: $ LDA, W( IMAX, 1 ), LDW, CONE, W( K, K+1 ),
429: $ 1 )
430: *
431: * JMAX is the column-index of the largest off-diagonal
432: * element in row IMAX, and ROWMAX is its absolute value
433: *
434: JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 )
435: ROWMAX = CABS1( W( JMAX, K+1 ) )
436: IF( IMAX.LT.N ) THEN
437: JMAX = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1 )
438: ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, K+1 ) ) )
439: END IF
440: *
441: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
442: *
443: * no interchange, use 1-by-1 pivot block
444: *
445: KP = K
446: ELSE IF( CABS1( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX ) THEN
447: *
448: * interchange rows and columns K and IMAX, use 1-by-1
449: * pivot block
450: *
451: KP = IMAX
452: *
453: * copy column K+1 of W to column K
454: *
455: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
456: ELSE
457: *
458: * interchange rows and columns K+1 and IMAX, use 2-by-2
459: * pivot block
460: *
461: KP = IMAX
462: KSTEP = 2
463: END IF
464: END IF
465: *
466: KK = K + KSTEP - 1
467: *
468: * Updated column KP is already stored in column KK of W
469: *
470: IF( KP.NE.KK ) THEN
471: *
472: * Copy non-updated column KK to column KP
473: *
474: A( KP, K ) = A( KK, K )
475: CALL ZCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA )
476: CALL ZCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 )
477: *
478: * Interchange rows KK and KP in first KK columns of A and W
479: *
480: CALL ZSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
481: CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
482: END IF
483: *
484: IF( KSTEP.EQ.1 ) THEN
485: *
486: * 1-by-1 pivot block D(k): column k of W now holds
487: *
488: * W(k) = L(k)*D(k)
489: *
490: * where L(k) is the k-th column of L
491: *
492: * Store L(k) in column k of A
493: *
494: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
495: IF( K.LT.N ) THEN
496: R1 = CONE / A( K, K )
497: CALL ZSCAL( N-K, R1, A( K+1, K ), 1 )
498: END IF
499: ELSE
500: *
501: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
502: *
503: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
504: *
505: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
506: * of L
507: *
508: IF( K.LT.N-1 ) THEN
509: *
510: * Store L(k) and L(k+1) in columns k and k+1 of A
511: *
512: D21 = W( K+1, K )
513: D11 = W( K+1, K+1 ) / D21
514: D22 = W( K, K ) / D21
515: T = CONE / ( D11*D22-CONE )
516: D21 = T / D21
517: DO 80 J = K + 2, N
518: A( J, K ) = D21*( D11*W( J, K )-W( J, K+1 ) )
519: A( J, K+1 ) = D21*( D22*W( J, K+1 )-W( J, K ) )
520: 80 CONTINUE
521: END IF
522: *
523: * Copy D(k) to A
524: *
525: A( K, K ) = W( K, K )
526: A( K+1, K ) = W( K+1, K )
527: A( K+1, K+1 ) = W( K+1, K+1 )
528: END IF
529: END IF
530: *
531: * Store details of the interchanges in IPIV
532: *
533: IF( KSTEP.EQ.1 ) THEN
534: IPIV( K ) = KP
535: ELSE
536: IPIV( K ) = -KP
537: IPIV( K+1 ) = -KP
538: END IF
539: *
540: * Increase K and return to the start of the main loop
541: *
542: K = K + KSTEP
543: GO TO 70
544: *
545: 90 CONTINUE
546: *
547: * Update the lower triangle of A22 (= A(k:n,k:n)) as
548: *
549: * A22 := A22 - L21*D*L21' = A22 - L21*W'
550: *
551: * computing blocks of NB columns at a time
552: *
553: DO 110 J = K, N, NB
554: JB = MIN( NB, N-J+1 )
555: *
556: * Update the lower triangle of the diagonal block
557: *
558: DO 100 JJ = J, J + JB - 1
559: CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
560: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
561: $ A( JJ, JJ ), 1 )
562: 100 CONTINUE
563: *
564: * Update the rectangular subdiagonal block
565: *
566: IF( J+JB.LE.N )
567: $ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
568: $ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ),
569: $ LDW, CONE, A( J+JB, J ), LDA )
570: 110 CONTINUE
571: *
572: * Put L21 in standard form by partially undoing the interchanges
573: * in columns 1:k-1
574: *
575: J = K - 1
576: 120 CONTINUE
577: JJ = J
578: JP = IPIV( J )
579: IF( JP.LT.0 ) THEN
580: JP = -JP
581: J = J - 1
582: END IF
583: J = J - 1
584: IF( JP.NE.JJ .AND. J.GE.1 )
585: $ CALL ZSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA )
586: IF( J.GE.1 )
587: $ GO TO 120
588: *
589: * Set KB to the number of columns factorized
590: *
591: KB = K - 1
592: *
593: END IF
594: RETURN
595: *
596: * End of ZLASYF
597: *
598: END
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