1: *> \brief \b ZLASYF_RK computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) 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_RK + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlasyf_rk.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlasyf_rk.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLASYF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW,
22: * INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER UPLO
26: * INTEGER INFO, KB, LDA, LDW, N, NB
27: * ..
28: * .. Array Arguments ..
29: * INTEGER IPIV( * )
30: * COMPLEX*16 A( LDA, * ), E( * ), W( LDW, * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *> ZLASYF_RK computes a partial factorization of a complex symmetric
39: *> matrix A using the bounded Bunch-Kaufman (rook) diagonal
40: *> pivoting method. The partial 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: *>
51: *> ZLASYF_RK is an auxiliary routine called by ZSYTRF_RK. It uses
52: *> blocked code (calling Level 3 BLAS) to update the submatrix
53: *> A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
54: *> \endverbatim
55: *
56: * Arguments:
57: * ==========
58: *
59: *> \param[in] UPLO
60: *> \verbatim
61: *> UPLO is CHARACTER*1
62: *> Specifies whether the upper or lower triangular part of the
63: *> symmetric matrix A is stored:
64: *> = 'U': Upper triangular
65: *> = 'L': Lower triangular
66: *> \endverbatim
67: *>
68: *> \param[in] N
69: *> \verbatim
70: *> N is INTEGER
71: *> The order of the matrix A. N >= 0.
72: *> \endverbatim
73: *>
74: *> \param[in] NB
75: *> \verbatim
76: *> NB is INTEGER
77: *> The maximum number of columns of the matrix A that should be
78: *> factored. NB should be at least 2 to allow for 2-by-2 pivot
79: *> blocks.
80: *> \endverbatim
81: *>
82: *> \param[out] KB
83: *> \verbatim
84: *> KB is INTEGER
85: *> The number of columns of A that were actually factored.
86: *> KB is either NB-1 or NB, or N if N <= NB.
87: *> \endverbatim
88: *>
89: *> \param[in,out] A
90: *> \verbatim
91: *> A is COMPLEX*16 array, dimension (LDA,N)
92: *> On entry, the symmetric matrix A.
93: *> If UPLO = 'U': the leading N-by-N upper triangular part
94: *> of A contains the upper triangular part of the matrix A,
95: *> and the strictly lower triangular part of A is not
96: *> referenced.
97: *>
98: *> If UPLO = 'L': the leading N-by-N lower triangular part
99: *> of A contains the lower triangular part of the matrix A,
100: *> and the strictly upper triangular part of A is not
101: *> referenced.
102: *>
103: *> On exit, contains:
104: *> a) ONLY diagonal elements of the symmetric block diagonal
105: *> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
106: *> (superdiagonal (or subdiagonal) elements of D
107: *> are stored on exit in array E), and
108: *> b) If UPLO = 'U': factor U in the superdiagonal part of A.
109: *> If UPLO = 'L': factor L in the subdiagonal part of A.
110: *> \endverbatim
111: *>
112: *> \param[in] LDA
113: *> \verbatim
114: *> LDA is INTEGER
115: *> The leading dimension of the array A. LDA >= max(1,N).
116: *> \endverbatim
117: *>
118: *> \param[out] E
119: *> \verbatim
120: *> E is COMPLEX*16 array, dimension (N)
121: *> On exit, contains the superdiagonal (or subdiagonal)
122: *> elements of the symmetric block diagonal matrix D
123: *> with 1-by-1 or 2-by-2 diagonal blocks, where
124: *> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
125: *> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
126: *>
127: *> NOTE: For 1-by-1 diagonal block D(k), where
128: *> 1 <= k <= N, the element E(k) is set to 0 in both
129: *> UPLO = 'U' or UPLO = 'L' cases.
130: *> \endverbatim
131: *>
132: *> \param[out] IPIV
133: *> \verbatim
134: *> IPIV is INTEGER array, dimension (N)
135: *> IPIV describes the permutation matrix P in the factorization
136: *> of matrix A as follows. The absolute value of IPIV(k)
137: *> represents the index of row and column that were
138: *> interchanged with the k-th row and column. The value of UPLO
139: *> describes the order in which the interchanges were applied.
140: *> Also, the sign of IPIV represents the block structure of
141: *> the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
142: *> diagonal blocks which correspond to 1 or 2 interchanges
143: *> at each factorization step.
144: *>
145: *> If UPLO = 'U',
146: *> ( in factorization order, k decreases from N to 1 ):
147: *> a) A single positive entry IPIV(k) > 0 means:
148: *> D(k,k) is a 1-by-1 diagonal block.
149: *> If IPIV(k) != k, rows and columns k and IPIV(k) were
150: *> interchanged in the submatrix A(1:N,N-KB+1:N);
151: *> If IPIV(k) = k, no interchange occurred.
152: *>
153: *>
154: *> b) A pair of consecutive negative entries
155: *> IPIV(k) < 0 and IPIV(k-1) < 0 means:
156: *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
157: *> (NOTE: negative entries in IPIV appear ONLY in pairs).
158: *> 1) If -IPIV(k) != k, rows and columns
159: *> k and -IPIV(k) were interchanged
160: *> in the matrix A(1:N,N-KB+1:N).
161: *> If -IPIV(k) = k, no interchange occurred.
162: *> 2) If -IPIV(k-1) != k-1, rows and columns
163: *> k-1 and -IPIV(k-1) were interchanged
164: *> in the submatrix A(1:N,N-KB+1:N).
165: *> If -IPIV(k-1) = k-1, no interchange occurred.
166: *>
167: *> c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.
168: *>
169: *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
170: *>
171: *> If UPLO = 'L',
172: *> ( in factorization order, k increases from 1 to N ):
173: *> a) A single positive entry IPIV(k) > 0 means:
174: *> D(k,k) is a 1-by-1 diagonal block.
175: *> If IPIV(k) != k, rows and columns k and IPIV(k) were
176: *> interchanged in the submatrix A(1:N,1:KB).
177: *> If IPIV(k) = k, no interchange occurred.
178: *>
179: *> b) A pair of consecutive negative entries
180: *> IPIV(k) < 0 and IPIV(k+1) < 0 means:
181: *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
182: *> (NOTE: negative entries in IPIV appear ONLY in pairs).
183: *> 1) If -IPIV(k) != k, rows and columns
184: *> k and -IPIV(k) were interchanged
185: *> in the submatrix A(1:N,1:KB).
186: *> If -IPIV(k) = k, no interchange occurred.
187: *> 2) If -IPIV(k+1) != k+1, rows and columns
188: *> k-1 and -IPIV(k-1) were interchanged
189: *> in the submatrix A(1:N,1:KB).
190: *> If -IPIV(k+1) = k+1, no interchange occurred.
191: *>
192: *> c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.
193: *>
194: *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
195: *> \endverbatim
196: *>
197: *> \param[out] W
198: *> \verbatim
199: *> W is COMPLEX*16 array, dimension (LDW,NB)
200: *> \endverbatim
201: *>
202: *> \param[in] LDW
203: *> \verbatim
204: *> LDW is INTEGER
205: *> The leading dimension of the array W. LDW >= max(1,N).
206: *> \endverbatim
207: *>
208: *> \param[out] INFO
209: *> \verbatim
210: *> INFO is INTEGER
211: *> = 0: successful exit
212: *>
213: *> < 0: If INFO = -k, the k-th argument had an illegal value
214: *>
215: *> > 0: If INFO = k, the matrix A is singular, because:
216: *> If UPLO = 'U': column k in the upper
217: *> triangular part of A contains all zeros.
218: *> If UPLO = 'L': column k in the lower
219: *> triangular part of A contains all zeros.
220: *>
221: *> Therefore D(k,k) is exactly zero, and superdiagonal
222: *> elements of column k of U (or subdiagonal elements of
223: *> column k of L ) are all zeros. The factorization has
224: *> been completed, but the block diagonal matrix D is
225: *> exactly singular, and division by zero will occur if
226: *> it is used to solve a system of equations.
227: *>
228: *> NOTE: INFO only stores the first occurrence of
229: *> a singularity, any subsequent occurrence of singularity
230: *> is not stored in INFO even though the factorization
231: *> always completes.
232: *> \endverbatim
233: *
234: * Authors:
235: * ========
236: *
237: *> \author Univ. of Tennessee
238: *> \author Univ. of California Berkeley
239: *> \author Univ. of Colorado Denver
240: *> \author NAG Ltd.
241: *
242: *> \date December 2016
243: *
244: *> \ingroup complex16SYcomputational
245: *
246: *> \par Contributors:
247: * ==================
248: *>
249: *> \verbatim
250: *>
251: *> December 2016, Igor Kozachenko,
252: *> Computer Science Division,
253: *> University of California, Berkeley
254: *>
255: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
256: *> School of Mathematics,
257: *> University of Manchester
258: *>
259: *> \endverbatim
260: *
261: * =====================================================================
262: SUBROUTINE ZLASYF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW,
263: $ INFO )
264: *
265: * -- LAPACK computational routine (version 3.7.0) --
266: * -- LAPACK is a software package provided by Univ. of Tennessee, --
267: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
268: * December 2016
269: *
270: * .. Scalar Arguments ..
271: CHARACTER UPLO
272: INTEGER INFO, KB, LDA, LDW, N, NB
273: * ..
274: * .. Array Arguments ..
275: INTEGER IPIV( * )
276: COMPLEX*16 A( LDA, * ), E( * ), W( LDW, * )
277: * ..
278: *
279: * =====================================================================
280: *
281: * .. Parameters ..
282: DOUBLE PRECISION ZERO, ONE
283: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
284: DOUBLE PRECISION EIGHT, SEVTEN
285: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
286: COMPLEX*16 CONE, CZERO
287: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ),
288: $ CZERO = ( 0.0D+0, 0.0D+0 ) )
289: * ..
290: * .. Local Scalars ..
291: LOGICAL DONE
292: INTEGER IMAX, ITEMP, J, JB, JJ, JMAX, K, KK, KW, KKW,
293: $ KP, KSTEP, P, II
294: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX, SFMIN, DTEMP
295: COMPLEX*16 D11, D12, D21, D22, R1, T, Z
296: * ..
297: * .. External Functions ..
298: LOGICAL LSAME
299: INTEGER IZAMAX
300: DOUBLE PRECISION DLAMCH
301: EXTERNAL LSAME, IZAMAX, DLAMCH
302: * ..
303: * .. External Subroutines ..
304: EXTERNAL ZCOPY, ZGEMM, ZGEMV, ZSCAL, ZSWAP
305: * ..
306: * .. Intrinsic Functions ..
307: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN, SQRT
308: * ..
309: * .. Statement Functions ..
310: DOUBLE PRECISION CABS1
311: * ..
312: * .. Statement Function definitions ..
313: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
314: * ..
315: * .. Executable Statements ..
316: *
317: INFO = 0
318: *
319: * Initialize ALPHA for use in choosing pivot block size.
320: *
321: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
322: *
323: * Compute machine safe minimum
324: *
325: SFMIN = DLAMCH( 'S' )
326: *
327: IF( LSAME( UPLO, 'U' ) ) THEN
328: *
329: * Factorize the trailing columns of A using the upper triangle
330: * of A and working backwards, and compute the matrix W = U12*D
331: * for use in updating A11
332: *
333: * Initialize the first entry of array E, where superdiagonal
334: * elements of D are stored
335: *
336: E( 1 ) = CZERO
337: *
338: * K is the main loop index, decreasing from N in steps of 1 or 2
339: *
340: K = N
341: 10 CONTINUE
342: *
343: * KW is the column of W which corresponds to column K of A
344: *
345: KW = NB + K - N
346: *
347: * Exit from loop
348: *
349: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
350: $ GO TO 30
351: *
352: KSTEP = 1
353: P = K
354: *
355: * Copy column K of A to column KW of W and update it
356: *
357: CALL ZCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
358: IF( K.LT.N )
359: $ CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ),
360: $ LDA, W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 )
361: *
362: * Determine rows and columns to be interchanged and whether
363: * a 1-by-1 or 2-by-2 pivot block will be used
364: *
365: ABSAKK = CABS1( W( K, KW ) )
366: *
367: * IMAX is the row-index of the largest off-diagonal element in
368: * column K, and COLMAX is its absolute value.
369: * Determine both COLMAX and IMAX.
370: *
371: IF( K.GT.1 ) THEN
372: IMAX = IZAMAX( K-1, W( 1, KW ), 1 )
373: COLMAX = CABS1( W( IMAX, KW ) )
374: ELSE
375: COLMAX = ZERO
376: END IF
377: *
378: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
379: *
380: * Column K is zero or underflow: set INFO and continue
381: *
382: IF( INFO.EQ.0 )
383: $ INFO = K
384: KP = K
385: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
386: *
387: * Set E( K ) to zero
388: *
389: IF( K.GT.1 )
390: $ E( K ) = CZERO
391: *
392: ELSE
393: *
394: * ============================================================
395: *
396: * Test for interchange
397: *
398: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
399: * (used to handle NaN and Inf)
400: *
401: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
402: *
403: * no interchange, use 1-by-1 pivot block
404: *
405: KP = K
406: *
407: ELSE
408: *
409: DONE = .FALSE.
410: *
411: * Loop until pivot found
412: *
413: 12 CONTINUE
414: *
415: * Begin pivot search loop body
416: *
417: *
418: * Copy column IMAX to column KW-1 of W and update it
419: *
420: CALL ZCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
421: CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
422: $ W( IMAX+1, KW-1 ), 1 )
423: *
424: IF( K.LT.N )
425: $ CALL ZGEMV( 'No transpose', K, N-K, -CONE,
426: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
427: $ CONE, W( 1, KW-1 ), 1 )
428: *
429: * JMAX is the column-index of the largest off-diagonal
430: * element in row IMAX, and ROWMAX is its absolute value.
431: * Determine both ROWMAX and JMAX.
432: *
433: IF( IMAX.NE.K ) THEN
434: JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ),
435: $ 1 )
436: ROWMAX = CABS1( W( JMAX, KW-1 ) )
437: ELSE
438: ROWMAX = ZERO
439: END IF
440: *
441: IF( IMAX.GT.1 ) THEN
442: ITEMP = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 )
443: DTEMP = CABS1( W( ITEMP, KW-1 ) )
444: IF( DTEMP.GT.ROWMAX ) THEN
445: ROWMAX = DTEMP
446: JMAX = ITEMP
447: END IF
448: END IF
449: *
450: * Equivalent to testing for
451: * CABS1( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX
452: * (used to handle NaN and Inf)
453: *
454: IF( .NOT.(CABS1( W( IMAX, KW-1 ) ).LT.ALPHA*ROWMAX ) )
455: $ THEN
456: *
457: * interchange rows and columns K and IMAX,
458: * use 1-by-1 pivot block
459: *
460: KP = IMAX
461: *
462: * copy column KW-1 of W to column KW of W
463: *
464: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
465: *
466: DONE = .TRUE.
467: *
468: * Equivalent to testing for ROWMAX.EQ.COLMAX,
469: * (used to handle NaN and Inf)
470: *
471: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
472: $ THEN
473: *
474: * interchange rows and columns K-1 and IMAX,
475: * use 2-by-2 pivot block
476: *
477: KP = IMAX
478: KSTEP = 2
479: DONE = .TRUE.
480: ELSE
481: *
482: * Pivot not found: set params and repeat
483: *
484: P = IMAX
485: COLMAX = ROWMAX
486: IMAX = JMAX
487: *
488: * Copy updated JMAXth (next IMAXth) column to Kth of W
489: *
490: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
491: *
492: END IF
493: *
494: * End pivot search loop body
495: *
496: IF( .NOT. DONE ) GOTO 12
497: *
498: END IF
499: *
500: * ============================================================
501: *
502: KK = K - KSTEP + 1
503: *
504: * KKW is the column of W which corresponds to column KK of A
505: *
506: KKW = NB + KK - N
507: *
508: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
509: *
510: * Copy non-updated column K to column P
511: *
512: CALL ZCOPY( K-P, A( P+1, K ), 1, A( P, P+1 ), LDA )
513: CALL ZCOPY( P, A( 1, K ), 1, A( 1, P ), 1 )
514: *
515: * Interchange rows K and P in last N-K+1 columns of A
516: * and last N-K+2 columns of W
517: *
518: CALL ZSWAP( N-K+1, A( K, K ), LDA, A( P, K ), LDA )
519: CALL ZSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ), LDW )
520: END IF
521: *
522: * Updated column KP is already stored in column KKW of W
523: *
524: IF( KP.NE.KK ) THEN
525: *
526: * Copy non-updated column KK to column KP
527: *
528: A( KP, K ) = A( KK, K )
529: CALL ZCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
530: $ LDA )
531: CALL ZCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 )
532: *
533: * Interchange rows KK and KP in last N-KK+1 columns
534: * of A and W
535: *
536: CALL ZSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA )
537: CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
538: $ LDW )
539: END IF
540: *
541: IF( KSTEP.EQ.1 ) THEN
542: *
543: * 1-by-1 pivot block D(k): column KW of W now holds
544: *
545: * W(k) = U(k)*D(k)
546: *
547: * where U(k) is the k-th column of U
548: *
549: * Store U(k) in column k of A
550: *
551: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
552: IF( K.GT.1 ) THEN
553: IF( CABS1( A( K, K ) ).GE.SFMIN ) THEN
554: R1 = CONE / A( K, K )
555: CALL ZSCAL( K-1, R1, A( 1, K ), 1 )
556: ELSE IF( A( K, K ).NE.CZERO ) THEN
557: DO 14 II = 1, K - 1
558: A( II, K ) = A( II, K ) / A( K, K )
559: 14 CONTINUE
560: END IF
561: *
562: * Store the superdiagonal element of D in array E
563: *
564: E( K ) = CZERO
565: *
566: END IF
567: *
568: ELSE
569: *
570: * 2-by-2 pivot block D(k): columns KW and KW-1 of W now
571: * hold
572: *
573: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
574: *
575: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
576: * of U
577: *
578: IF( K.GT.2 ) THEN
579: *
580: * Store U(k) and U(k-1) in columns k and k-1 of A
581: *
582: D12 = W( K-1, KW )
583: D11 = W( K, KW ) / D12
584: D22 = W( K-1, KW-1 ) / D12
585: T = CONE / ( D11*D22-CONE )
586: DO 20 J = 1, K - 2
587: A( J, K-1 ) = T*( (D11*W( J, KW-1 )-W( J, KW ) ) /
588: $ D12 )
589: A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
590: $ D12 )
591: 20 CONTINUE
592: END IF
593: *
594: * Copy diagonal elements of D(K) to A,
595: * copy superdiagonal element of D(K) to E(K) and
596: * ZERO out superdiagonal entry of A
597: *
598: A( K-1, K-1 ) = W( K-1, KW-1 )
599: A( K-1, K ) = CZERO
600: A( K, K ) = W( K, KW )
601: E( K ) = W( K-1, KW )
602: E( K-1 ) = CZERO
603: *
604: END IF
605: *
606: * End column K is nonsingular
607: *
608: END IF
609: *
610: * Store details of the interchanges in IPIV
611: *
612: IF( KSTEP.EQ.1 ) THEN
613: IPIV( K ) = KP
614: ELSE
615: IPIV( K ) = -P
616: IPIV( K-1 ) = -KP
617: END IF
618: *
619: * Decrease K and return to the start of the main loop
620: *
621: K = K - KSTEP
622: GO TO 10
623: *
624: 30 CONTINUE
625: *
626: * Update the upper triangle of A11 (= A(1:k,1:k)) as
627: *
628: * A11 := A11 - U12*D*U12**T = A11 - U12*W**T
629: *
630: * computing blocks of NB columns at a time
631: *
632: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
633: JB = MIN( NB, K-J+1 )
634: *
635: * Update the upper triangle of the diagonal block
636: *
637: DO 40 JJ = J, J + JB - 1
638: CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE,
639: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE,
640: $ A( J, JJ ), 1 )
641: 40 CONTINUE
642: *
643: * Update the rectangular superdiagonal block
644: *
645: IF( J.GE.2 )
646: $ CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB,
647: $ N-K, -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ),
648: $ LDW, CONE, A( 1, J ), LDA )
649: 50 CONTINUE
650: *
651: * Set KB to the number of columns factorized
652: *
653: KB = N - K
654: *
655: ELSE
656: *
657: * Factorize the leading columns of A using the lower triangle
658: * of A and working forwards, and compute the matrix W = L21*D
659: * for use in updating A22
660: *
661: * Initialize the unused last entry of the subdiagonal array E.
662: *
663: E( N ) = CZERO
664: *
665: * K is the main loop index, increasing from 1 in steps of 1 or 2
666: *
667: K = 1
668: 70 CONTINUE
669: *
670: * Exit from loop
671: *
672: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
673: $ GO TO 90
674: *
675: KSTEP = 1
676: P = K
677: *
678: * Copy column K of A to column K of W and update it
679: *
680: CALL ZCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
681: IF( K.GT.1 )
682: $ CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
683: $ LDA, W( K, 1 ), LDW, CONE, W( K, K ), 1 )
684: *
685: * Determine rows and columns to be interchanged and whether
686: * a 1-by-1 or 2-by-2 pivot block will be used
687: *
688: ABSAKK = CABS1( W( K, K ) )
689: *
690: * IMAX is the row-index of the largest off-diagonal element in
691: * column K, and COLMAX is its absolute value.
692: * Determine both COLMAX and IMAX.
693: *
694: IF( K.LT.N ) THEN
695: IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 )
696: COLMAX = CABS1( W( IMAX, K ) )
697: ELSE
698: COLMAX = ZERO
699: END IF
700: *
701: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
702: *
703: * Column K is zero or underflow: set INFO and continue
704: *
705: IF( INFO.EQ.0 )
706: $ INFO = K
707: KP = K
708: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
709: *
710: * Set E( K ) to zero
711: *
712: IF( K.LT.N )
713: $ E( K ) = CZERO
714: *
715: ELSE
716: *
717: * ============================================================
718: *
719: * Test for interchange
720: *
721: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
722: * (used to handle NaN and Inf)
723: *
724: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
725: *
726: * no interchange, use 1-by-1 pivot block
727: *
728: KP = K
729: *
730: ELSE
731: *
732: DONE = .FALSE.
733: *
734: * Loop until pivot found
735: *
736: 72 CONTINUE
737: *
738: * Begin pivot search loop body
739: *
740: *
741: * Copy column IMAX to column K+1 of W and update it
742: *
743: CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
744: CALL ZCOPY( N-IMAX+1, A( IMAX, IMAX ), 1,
745: $ W( IMAX, K+1 ), 1 )
746: IF( K.GT.1 )
747: $ CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE,
748: $ A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
749: $ CONE, W( K, K+1 ), 1 )
750: *
751: * JMAX is the column-index of the largest off-diagonal
752: * element in row IMAX, and ROWMAX is its absolute value.
753: * Determine both ROWMAX and JMAX.
754: *
755: IF( IMAX.NE.K ) THEN
756: JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 )
757: ROWMAX = CABS1( W( JMAX, K+1 ) )
758: ELSE
759: ROWMAX = ZERO
760: END IF
761: *
762: IF( IMAX.LT.N ) THEN
763: ITEMP = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
764: DTEMP = CABS1( W( ITEMP, K+1 ) )
765: IF( DTEMP.GT.ROWMAX ) THEN
766: ROWMAX = DTEMP
767: JMAX = ITEMP
768: END IF
769: END IF
770: *
771: * Equivalent to testing for
772: * CABS1( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX
773: * (used to handle NaN and Inf)
774: *
775: IF( .NOT.( CABS1( W( IMAX, K+1 ) ).LT.ALPHA*ROWMAX ) )
776: $ THEN
777: *
778: * interchange rows and columns K and IMAX,
779: * use 1-by-1 pivot block
780: *
781: KP = IMAX
782: *
783: * copy column K+1 of W to column K of W
784: *
785: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
786: *
787: DONE = .TRUE.
788: *
789: * Equivalent to testing for ROWMAX.EQ.COLMAX,
790: * (used to handle NaN and Inf)
791: *
792: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
793: $ THEN
794: *
795: * interchange rows and columns K+1 and IMAX,
796: * use 2-by-2 pivot block
797: *
798: KP = IMAX
799: KSTEP = 2
800: DONE = .TRUE.
801: ELSE
802: *
803: * Pivot not found: set params and repeat
804: *
805: P = IMAX
806: COLMAX = ROWMAX
807: IMAX = JMAX
808: *
809: * Copy updated JMAXth (next IMAXth) column to Kth of W
810: *
811: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
812: *
813: END IF
814: *
815: * End pivot search loop body
816: *
817: IF( .NOT. DONE ) GOTO 72
818: *
819: END IF
820: *
821: * ============================================================
822: *
823: KK = K + KSTEP - 1
824: *
825: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
826: *
827: * Copy non-updated column K to column P
828: *
829: CALL ZCOPY( P-K, A( K, K ), 1, A( P, K ), LDA )
830: CALL ZCOPY( N-P+1, A( P, K ), 1, A( P, P ), 1 )
831: *
832: * Interchange rows K and P in first K columns of A
833: * and first K+1 columns of W
834: *
835: CALL ZSWAP( K, A( K, 1 ), LDA, A( P, 1 ), LDA )
836: CALL ZSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
837: END IF
838: *
839: * Updated column KP is already stored in column KK of W
840: *
841: IF( KP.NE.KK ) THEN
842: *
843: * Copy non-updated column KK to column KP
844: *
845: A( KP, K ) = A( KK, K )
846: CALL ZCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA )
847: CALL ZCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 )
848: *
849: * Interchange rows KK and KP in first KK columns of A and W
850: *
851: CALL ZSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
852: CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
853: END IF
854: *
855: IF( KSTEP.EQ.1 ) THEN
856: *
857: * 1-by-1 pivot block D(k): column k of W now holds
858: *
859: * W(k) = L(k)*D(k)
860: *
861: * where L(k) is the k-th column of L
862: *
863: * Store L(k) in column k of A
864: *
865: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
866: IF( K.LT.N ) THEN
867: IF( CABS1( A( K, K ) ).GE.SFMIN ) THEN
868: R1 = CONE / A( K, K )
869: CALL ZSCAL( N-K, R1, A( K+1, K ), 1 )
870: ELSE IF( A( K, K ).NE.CZERO ) THEN
871: DO 74 II = K + 1, N
872: A( II, K ) = A( II, K ) / A( K, K )
873: 74 CONTINUE
874: END IF
875: *
876: * Store the subdiagonal element of D in array E
877: *
878: E( K ) = CZERO
879: *
880: END IF
881: *
882: ELSE
883: *
884: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
885: *
886: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
887: *
888: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
889: * of L
890: *
891: IF( K.LT.N-1 ) THEN
892: *
893: * Store L(k) and L(k+1) in columns k and k+1 of A
894: *
895: D21 = W( K+1, K )
896: D11 = W( K+1, K+1 ) / D21
897: D22 = W( K, K ) / D21
898: T = CONE / ( D11*D22-CONE )
899: DO 80 J = K + 2, N
900: A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
901: $ D21 )
902: A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
903: $ D21 )
904: 80 CONTINUE
905: END IF
906: *
907: * Copy diagonal elements of D(K) to A,
908: * copy subdiagonal element of D(K) to E(K) and
909: * ZERO out subdiagonal entry of A
910: *
911: A( K, K ) = W( K, K )
912: A( K+1, K ) = CZERO
913: A( K+1, K+1 ) = W( K+1, K+1 )
914: E( K ) = W( K+1, K )
915: E( K+1 ) = CZERO
916: *
917: END IF
918: *
919: * End column K is nonsingular
920: *
921: END IF
922: *
923: * Store details of the interchanges in IPIV
924: *
925: IF( KSTEP.EQ.1 ) THEN
926: IPIV( K ) = KP
927: ELSE
928: IPIV( K ) = -P
929: IPIV( K+1 ) = -KP
930: END IF
931: *
932: * Increase K and return to the start of the main loop
933: *
934: K = K + KSTEP
935: GO TO 70
936: *
937: 90 CONTINUE
938: *
939: * Update the lower triangle of A22 (= A(k:n,k:n)) as
940: *
941: * A22 := A22 - L21*D*L21**T = A22 - L21*W**T
942: *
943: * computing blocks of NB columns at a time
944: *
945: DO 110 J = K, N, NB
946: JB = MIN( NB, N-J+1 )
947: *
948: * Update the lower triangle of the diagonal block
949: *
950: DO 100 JJ = J, J + JB - 1
951: CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
952: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
953: $ A( JJ, JJ ), 1 )
954: 100 CONTINUE
955: *
956: * Update the rectangular subdiagonal block
957: *
958: IF( J+JB.LE.N )
959: $ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
960: $ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ),
961: $ LDW, CONE, A( J+JB, J ), LDA )
962: 110 CONTINUE
963: *
964: * Set KB to the number of columns factorized
965: *
966: KB = K - 1
967: *
968: END IF
969: *
970: RETURN
971: *
972: * End of ZLASYF_RK
973: *
974: END
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