Annotation of rpl/lapack/lapack/zlasyf.f, revision 1.19
1.14 bertrand 1: *> \brief \b ZLASYF computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagonal pivoting method.
1.9 bertrand 2: *
3: * =========== DOCUMENTATION ===========
4: *
1.14 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.9 bertrand 7: *
8: *> \htmlonly
1.14 bertrand 9: *> Download ZLASYF + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlasyf.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlasyf.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlasyf.f">
1.9 bertrand 15: *> [TXT]</a>
1.14 bertrand 16: *> \endhtmlonly
1.9 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
1.14 bertrand 22: *
1.9 bertrand 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: * ..
1.14 bertrand 31: *
1.9 bertrand 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: *>
1.14 bertrand 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.
1.9 bertrand 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: *
1.14 bertrand 158: *> \author Univ. of Tennessee
159: *> \author Univ. of California Berkeley
160: *> \author Univ. of Colorado Denver
161: *> \author NAG Ltd.
1.9 bertrand 162: *
163: *> \ingroup complex16SYcomputational
164: *
1.14 bertrand 165: *> \par Contributors:
166: * ==================
167: *>
168: *> \verbatim
169: *>
170: *> November 2013, Igor Kozachenko,
171: *> Computer Science Division,
172: *> University of California, Berkeley
173: *> \endverbatim
174: *
1.9 bertrand 175: * =====================================================================
1.1 bertrand 176: SUBROUTINE ZLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
177: *
1.19 ! bertrand 178: * -- LAPACK computational routine --
1.1 bertrand 179: * -- LAPACK is a software package provided by Univ. of Tennessee, --
180: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
181: *
182: * .. Scalar Arguments ..
183: CHARACTER UPLO
184: INTEGER INFO, KB, LDA, LDW, N, NB
185: * ..
186: * .. Array Arguments ..
187: INTEGER IPIV( * )
188: COMPLEX*16 A( LDA, * ), W( LDW, * )
189: * ..
190: *
191: * =====================================================================
192: *
193: * .. Parameters ..
194: DOUBLE PRECISION ZERO, ONE
195: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
196: DOUBLE PRECISION EIGHT, SEVTEN
197: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
198: COMPLEX*16 CONE
199: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
200: * ..
201: * .. Local Scalars ..
202: INTEGER IMAX, J, JB, JJ, JMAX, JP, K, KK, KKW, KP,
203: $ KSTEP, KW
204: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX
205: COMPLEX*16 D11, D21, D22, R1, T, Z
206: * ..
207: * .. External Functions ..
208: LOGICAL LSAME
209: INTEGER IZAMAX
210: EXTERNAL LSAME, IZAMAX
211: * ..
212: * .. External Subroutines ..
213: EXTERNAL ZCOPY, ZGEMM, ZGEMV, ZSCAL, ZSWAP
214: * ..
215: * .. Intrinsic Functions ..
216: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN, SQRT
217: * ..
218: * .. Statement Functions ..
219: DOUBLE PRECISION CABS1
220: * ..
221: * .. Statement Function definitions ..
222: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
223: * ..
224: * .. Executable Statements ..
225: *
226: INFO = 0
227: *
228: * Initialize ALPHA for use in choosing pivot block size.
229: *
230: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
231: *
232: IF( LSAME( UPLO, 'U' ) ) THEN
233: *
234: * Factorize the trailing columns of A using the upper triangle
235: * of A and working backwards, and compute the matrix W = U12*D
236: * for use in updating A11
237: *
238: * K is the main loop index, decreasing from N in steps of 1 or 2
239: *
240: * KW is the column of W which corresponds to column K of A
241: *
242: K = N
243: 10 CONTINUE
244: KW = NB + K - N
245: *
246: * Exit from loop
247: *
248: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
249: $ GO TO 30
250: *
251: * Copy column K of A to column KW of W and update it
252: *
253: CALL ZCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
254: IF( K.LT.N )
255: $ CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ), LDA,
256: $ W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 )
257: *
258: KSTEP = 1
259: *
260: * Determine rows and columns to be interchanged and whether
261: * a 1-by-1 or 2-by-2 pivot block will be used
262: *
263: ABSAKK = CABS1( W( K, KW ) )
264: *
265: * IMAX is the row-index of the largest off-diagonal element in
1.14 bertrand 266:
1.1 bertrand 267: *
268: IF( K.GT.1 ) THEN
269: IMAX = IZAMAX( K-1, W( 1, KW ), 1 )
270: COLMAX = CABS1( W( IMAX, KW ) )
271: ELSE
272: COLMAX = ZERO
273: END IF
274: *
275: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
276: *
1.14 bertrand 277: * Column K is zero or underflow: set INFO and continue
1.1 bertrand 278: *
279: IF( INFO.EQ.0 )
280: $ INFO = K
281: KP = K
282: ELSE
283: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
284: *
285: * no interchange, use 1-by-1 pivot block
286: *
287: KP = K
288: ELSE
289: *
290: * Copy column IMAX to column KW-1 of W and update it
291: *
292: CALL ZCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
293: CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
294: $ W( IMAX+1, KW-1 ), 1 )
295: IF( K.LT.N )
296: $ CALL ZGEMV( 'No transpose', K, N-K, -CONE,
297: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
298: $ CONE, W( 1, KW-1 ), 1 )
299: *
300: * JMAX is the column-index of the largest off-diagonal
301: * element in row IMAX, and ROWMAX is its absolute value
302: *
303: JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 )
304: ROWMAX = CABS1( W( JMAX, KW-1 ) )
305: IF( IMAX.GT.1 ) THEN
306: JMAX = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 )
307: ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, KW-1 ) ) )
308: END IF
309: *
310: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
311: *
312: * no interchange, use 1-by-1 pivot block
313: *
314: KP = K
315: ELSE IF( CABS1( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX ) THEN
316: *
317: * interchange rows and columns K and IMAX, use 1-by-1
318: * pivot block
319: *
320: KP = IMAX
321: *
1.14 bertrand 322: * copy column KW-1 of W to column KW of W
1.1 bertrand 323: *
324: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
325: ELSE
326: *
327: * interchange rows and columns K-1 and IMAX, use 2-by-2
328: * pivot block
329: *
330: KP = IMAX
331: KSTEP = 2
332: END IF
333: END IF
334: *
1.14 bertrand 335: * ============================================================
336: *
337: * KK is the column of A where pivoting step stopped
338: *
1.1 bertrand 339: KK = K - KSTEP + 1
1.14 bertrand 340: *
341: * KKW is the column of W which corresponds to column KK of A
342: *
1.1 bertrand 343: KKW = NB + KK - N
344: *
1.14 bertrand 345: * Interchange rows and columns KP and KK.
346: * Updated column KP is already stored in column KKW of W.
1.1 bertrand 347: *
348: IF( KP.NE.KK ) THEN
349: *
1.14 bertrand 350: * Copy non-updated column KK to column KP of submatrix A
351: * at step K. No need to copy element into column K
352: * (or K and K-1 for 2-by-2 pivot) of A, since these columns
353: * will be later overwritten.
1.1 bertrand 354: *
1.14 bertrand 355: A( KP, KP ) = A( KK, KK )
356: CALL ZCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
1.1 bertrand 357: $ LDA )
1.14 bertrand 358: IF( KP.GT.1 )
359: $ CALL ZCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
1.1 bertrand 360: *
1.14 bertrand 361: * Interchange rows KK and KP in last K+1 to N columns of A
362: * (columns K (or K and K-1 for 2-by-2 pivot) of A will be
363: * later overwritten). Interchange rows KK and KP
364: * in last KKW to NB columns of W.
1.1 bertrand 365: *
1.14 bertrand 366: IF( K.LT.N )
367: $ CALL ZSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
368: $ LDA )
1.1 bertrand 369: CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
370: $ LDW )
371: END IF
372: *
373: IF( KSTEP.EQ.1 ) THEN
374: *
1.14 bertrand 375: * 1-by-1 pivot block D(k): column kw of W now holds
1.1 bertrand 376: *
1.14 bertrand 377: * W(kw) = U(k)*D(k),
1.1 bertrand 378: *
379: * where U(k) is the k-th column of U
380: *
1.14 bertrand 381: * Store subdiag. elements of column U(k)
382: * and 1-by-1 block D(k) in column k of A.
383: * NOTE: Diagonal element U(k,k) is a UNIT element
384: * and not stored.
385: * A(k,k) := D(k,k) = W(k,kw)
386: * A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k)
1.1 bertrand 387: *
388: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
389: R1 = CONE / A( K, K )
390: CALL ZSCAL( K-1, R1, A( 1, K ), 1 )
1.14 bertrand 391: *
1.1 bertrand 392: ELSE
393: *
1.14 bertrand 394: * 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold
1.1 bertrand 395: *
1.14 bertrand 396: * ( W(kw-1) W(kw) ) = ( U(k-1) U(k) )*D(k)
1.1 bertrand 397: *
398: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
399: * of U
400: *
1.14 bertrand 401: * Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2
402: * block D(k-1:k,k-1:k) in columns k-1 and k of A.
403: * NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT
404: * block and not stored.
405: * A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw)
406: * A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) =
407: * = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) )
408: *
1.1 bertrand 409: IF( K.GT.2 ) THEN
410: *
1.14 bertrand 411: * Compose the columns of the inverse of 2-by-2 pivot
412: * block D in the following way to reduce the number
413: * of FLOPS when we myltiply panel ( W(kw-1) W(kw) ) by
414: * this inverse
415: *
416: * D**(-1) = ( d11 d21 )**(-1) =
417: * ( d21 d22 )
418: *
419: * = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) =
420: * ( (-d21 ) ( d11 ) )
421: *
422: * = 1/d21 * 1/((d11/d21)*(d22/d21)-1) *
423: *
424: * * ( ( d22/d21 ) ( -1 ) ) =
425: * ( ( -1 ) ( d11/d21 ) )
426: *
427: * = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) ( -1 ) ) =
428: * ( ( -1 ) ( D22 ) )
429: *
430: * = 1/d21 * T * ( ( D11 ) ( -1 ) )
431: * ( ( -1 ) ( D22 ) )
432: *
433: * = D21 * ( ( D11 ) ( -1 ) )
434: * ( ( -1 ) ( D22 ) )
1.1 bertrand 435: *
436: D21 = W( K-1, KW )
437: D11 = W( K, KW ) / D21
438: D22 = W( K-1, KW-1 ) / D21
439: T = CONE / ( D11*D22-CONE )
440: D21 = T / D21
1.14 bertrand 441: *
442: * Update elements in columns A(k-1) and A(k) as
443: * dot products of rows of ( W(kw-1) W(kw) ) and columns
444: * of D**(-1)
445: *
1.1 bertrand 446: DO 20 J = 1, K - 2
447: A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) )
448: A( J, K ) = D21*( D22*W( J, KW )-W( J, KW-1 ) )
449: 20 CONTINUE
450: END IF
451: *
452: * Copy D(k) to A
453: *
454: A( K-1, K-1 ) = W( K-1, KW-1 )
455: A( K-1, K ) = W( K-1, KW )
456: A( K, K ) = W( K, KW )
1.14 bertrand 457: *
1.1 bertrand 458: END IF
1.14 bertrand 459: *
1.1 bertrand 460: END IF
461: *
462: * Store details of the interchanges in IPIV
463: *
464: IF( KSTEP.EQ.1 ) THEN
465: IPIV( K ) = KP
466: ELSE
467: IPIV( K ) = -KP
468: IPIV( K-1 ) = -KP
469: END IF
470: *
471: * Decrease K and return to the start of the main loop
472: *
473: K = K - KSTEP
474: GO TO 10
475: *
476: 30 CONTINUE
477: *
478: * Update the upper triangle of A11 (= A(1:k,1:k)) as
479: *
1.8 bertrand 480: * A11 := A11 - U12*D*U12**T = A11 - U12*W**T
1.1 bertrand 481: *
482: * computing blocks of NB columns at a time
483: *
484: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
485: JB = MIN( NB, K-J+1 )
486: *
487: * Update the upper triangle of the diagonal block
488: *
489: DO 40 JJ = J, J + JB - 1
490: CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE,
491: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE,
492: $ A( J, JJ ), 1 )
493: 40 CONTINUE
494: *
495: * Update the rectangular superdiagonal block
496: *
497: CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB, N-K,
498: $ -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
499: $ CONE, A( 1, J ), LDA )
500: 50 CONTINUE
501: *
502: * Put U12 in standard form by partially undoing the interchanges
1.14 bertrand 503: * in columns k+1:n looping backwards from k+1 to n
1.1 bertrand 504: *
505: J = K + 1
506: 60 CONTINUE
1.14 bertrand 507: *
508: * Undo the interchanges (if any) of rows JJ and JP at each
509: * step J
510: *
511: * (Here, J is a diagonal index)
512: JJ = J
513: JP = IPIV( J )
514: IF( JP.LT.0 ) THEN
515: JP = -JP
516: * (Here, J is a diagonal index)
517: J = J + 1
518: END IF
519: * (NOTE: Here, J is used to determine row length. Length N-J+1
520: * of the rows to swap back doesn't include diagonal element)
1.1 bertrand 521: J = J + 1
1.14 bertrand 522: IF( JP.NE.JJ .AND. J.LE.N )
523: $ CALL ZSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA )
524: IF( J.LT.N )
1.1 bertrand 525: $ GO TO 60
526: *
527: * Set KB to the number of columns factorized
528: *
529: KB = N - K
530: *
531: ELSE
532: *
533: * Factorize the leading columns of A using the lower triangle
534: * of A and working forwards, and compute the matrix W = L21*D
535: * for use in updating A22
536: *
537: * K is the main loop index, increasing from 1 in steps of 1 or 2
538: *
539: K = 1
540: 70 CONTINUE
541: *
542: * Exit from loop
543: *
544: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
545: $ GO TO 90
546: *
547: * Copy column K of A to column K of W and update it
548: *
549: CALL ZCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
550: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ), LDA,
551: $ W( K, 1 ), LDW, CONE, W( K, K ), 1 )
552: *
553: KSTEP = 1
554: *
555: * Determine rows and columns to be interchanged and whether
556: * a 1-by-1 or 2-by-2 pivot block will be used
557: *
558: ABSAKK = CABS1( W( K, K ) )
559: *
560: * IMAX is the row-index of the largest off-diagonal element in
1.14 bertrand 561:
1.1 bertrand 562: *
563: IF( K.LT.N ) THEN
564: IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 )
565: COLMAX = CABS1( W( IMAX, K ) )
566: ELSE
567: COLMAX = ZERO
568: END IF
569: *
570: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
571: *
1.14 bertrand 572: * Column K is zero or underflow: set INFO and continue
1.1 bertrand 573: *
574: IF( INFO.EQ.0 )
575: $ INFO = K
576: KP = K
577: ELSE
578: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
579: *
580: * no interchange, use 1-by-1 pivot block
581: *
582: KP = K
583: ELSE
584: *
585: * Copy column IMAX to column K+1 of W and update it
586: *
587: CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1 )
588: CALL ZCOPY( N-IMAX+1, A( IMAX, IMAX ), 1, W( IMAX, K+1 ),
589: $ 1 )
590: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
591: $ LDA, W( IMAX, 1 ), LDW, CONE, W( K, K+1 ),
592: $ 1 )
593: *
594: * JMAX is the column-index of the largest off-diagonal
595: * element in row IMAX, and ROWMAX is its absolute value
596: *
597: JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 )
598: ROWMAX = CABS1( W( JMAX, K+1 ) )
599: IF( IMAX.LT.N ) THEN
600: JMAX = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1 )
601: ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, K+1 ) ) )
602: END IF
603: *
604: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
605: *
606: * no interchange, use 1-by-1 pivot block
607: *
608: KP = K
609: ELSE IF( CABS1( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX ) THEN
610: *
611: * interchange rows and columns K and IMAX, use 1-by-1
612: * pivot block
613: *
614: KP = IMAX
615: *
1.14 bertrand 616: * copy column K+1 of W to column K of W
1.1 bertrand 617: *
618: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
619: ELSE
620: *
621: * interchange rows and columns K+1 and IMAX, use 2-by-2
622: * pivot block
623: *
624: KP = IMAX
625: KSTEP = 2
626: END IF
627: END IF
628: *
1.14 bertrand 629: * ============================================================
630: *
631: * KK is the column of A where pivoting step stopped
632: *
1.1 bertrand 633: KK = K + KSTEP - 1
634: *
1.14 bertrand 635: * Interchange rows and columns KP and KK.
636: * Updated column KP is already stored in column KK of W.
1.1 bertrand 637: *
638: IF( KP.NE.KK ) THEN
639: *
1.14 bertrand 640: * Copy non-updated column KK to column KP of submatrix A
641: * at step K. No need to copy element into column K
642: * (or K and K+1 for 2-by-2 pivot) of A, since these columns
643: * will be later overwritten.
1.1 bertrand 644: *
1.14 bertrand 645: A( KP, KP ) = A( KK, KK )
646: CALL ZCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
647: $ LDA )
648: IF( KP.LT.N )
649: $ CALL ZCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
1.1 bertrand 650: *
1.14 bertrand 651: * Interchange rows KK and KP in first K-1 columns of A
652: * (columns K (or K and K+1 for 2-by-2 pivot) of A will be
653: * later overwritten). Interchange rows KK and KP
654: * in first KK columns of W.
1.1 bertrand 655: *
1.14 bertrand 656: IF( K.GT.1 )
657: $ CALL ZSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
1.1 bertrand 658: CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
659: END IF
660: *
661: IF( KSTEP.EQ.1 ) THEN
662: *
663: * 1-by-1 pivot block D(k): column k of W now holds
664: *
1.14 bertrand 665: * W(k) = L(k)*D(k),
1.1 bertrand 666: *
667: * where L(k) is the k-th column of L
668: *
1.14 bertrand 669: * Store subdiag. elements of column L(k)
670: * and 1-by-1 block D(k) in column k of A.
671: * (NOTE: Diagonal element L(k,k) is a UNIT element
672: * and not stored)
673: * A(k,k) := D(k,k) = W(k,k)
674: * A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k)
1.1 bertrand 675: *
676: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
677: IF( K.LT.N ) THEN
678: R1 = CONE / A( K, K )
679: CALL ZSCAL( N-K, R1, A( K+1, K ), 1 )
680: END IF
1.14 bertrand 681: *
1.1 bertrand 682: ELSE
683: *
684: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
685: *
686: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
687: *
688: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
689: * of L
690: *
1.14 bertrand 691: * Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2
692: * block D(k:k+1,k:k+1) in columns k and k+1 of A.
693: * (NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT
694: * block and not stored)
695: * A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1)
696: * A(k+2:N,k:k+1) := L(k+2:N,k:k+1) =
697: * = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) )
698: *
1.1 bertrand 699: IF( K.LT.N-1 ) THEN
700: *
1.14 bertrand 701: * Compose the columns of the inverse of 2-by-2 pivot
702: * block D in the following way to reduce the number
703: * of FLOPS when we myltiply panel ( W(k) W(k+1) ) by
704: * this inverse
705: *
706: * D**(-1) = ( d11 d21 )**(-1) =
707: * ( d21 d22 )
708: *
709: * = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) =
710: * ( (-d21 ) ( d11 ) )
711: *
712: * = 1/d21 * 1/((d11/d21)*(d22/d21)-1) *
713: *
714: * * ( ( d22/d21 ) ( -1 ) ) =
715: * ( ( -1 ) ( d11/d21 ) )
716: *
717: * = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) ( -1 ) ) =
718: * ( ( -1 ) ( D22 ) )
719: *
720: * = 1/d21 * T * ( ( D11 ) ( -1 ) )
721: * ( ( -1 ) ( D22 ) )
722: *
723: * = D21 * ( ( D11 ) ( -1 ) )
724: * ( ( -1 ) ( D22 ) )
1.1 bertrand 725: *
726: D21 = W( K+1, K )
727: D11 = W( K+1, K+1 ) / D21
728: D22 = W( K, K ) / D21
729: T = CONE / ( D11*D22-CONE )
730: D21 = T / D21
1.14 bertrand 731: *
732: * Update elements in columns A(k) and A(k+1) as
733: * dot products of rows of ( W(k) W(k+1) ) and columns
734: * of D**(-1)
735: *
1.1 bertrand 736: DO 80 J = K + 2, N
737: A( J, K ) = D21*( D11*W( J, K )-W( J, K+1 ) )
738: A( J, K+1 ) = D21*( D22*W( J, K+1 )-W( J, K ) )
739: 80 CONTINUE
740: END IF
741: *
742: * Copy D(k) to A
743: *
744: A( K, K ) = W( K, K )
745: A( K+1, K ) = W( K+1, K )
746: A( K+1, K+1 ) = W( K+1, K+1 )
1.14 bertrand 747: *
1.1 bertrand 748: END IF
1.14 bertrand 749: *
1.1 bertrand 750: END IF
751: *
752: * Store details of the interchanges in IPIV
753: *
754: IF( KSTEP.EQ.1 ) THEN
755: IPIV( K ) = KP
756: ELSE
757: IPIV( K ) = -KP
758: IPIV( K+1 ) = -KP
759: END IF
760: *
761: * Increase K and return to the start of the main loop
762: *
763: K = K + KSTEP
764: GO TO 70
765: *
766: 90 CONTINUE
767: *
768: * Update the lower triangle of A22 (= A(k:n,k:n)) as
769: *
1.8 bertrand 770: * A22 := A22 - L21*D*L21**T = A22 - L21*W**T
1.1 bertrand 771: *
772: * computing blocks of NB columns at a time
773: *
774: DO 110 J = K, N, NB
775: JB = MIN( NB, N-J+1 )
776: *
777: * Update the lower triangle of the diagonal block
778: *
779: DO 100 JJ = J, J + JB - 1
780: CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
781: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
782: $ A( JJ, JJ ), 1 )
783: 100 CONTINUE
784: *
785: * Update the rectangular subdiagonal block
786: *
787: IF( J+JB.LE.N )
788: $ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
789: $ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ),
790: $ LDW, CONE, A( J+JB, J ), LDA )
791: 110 CONTINUE
792: *
793: * Put L21 in standard form by partially undoing the interchanges
1.14 bertrand 794: * of rows in columns 1:k-1 looping backwards from k-1 to 1
1.1 bertrand 795: *
796: J = K - 1
797: 120 CONTINUE
1.14 bertrand 798: *
799: * Undo the interchanges (if any) of rows JJ and JP at each
800: * step J
801: *
802: * (Here, J is a diagonal index)
803: JJ = J
804: JP = IPIV( J )
805: IF( JP.LT.0 ) THEN
806: JP = -JP
807: * (Here, J is a diagonal index)
808: J = J - 1
809: END IF
810: * (NOTE: Here, J is used to determine row length. Length J
811: * of the rows to swap back doesn't include diagonal element)
1.1 bertrand 812: J = J - 1
1.14 bertrand 813: IF( JP.NE.JJ .AND. J.GE.1 )
814: $ CALL ZSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA )
815: IF( J.GT.1 )
1.1 bertrand 816: $ GO TO 120
817: *
818: * Set KB to the number of columns factorized
819: *
820: KB = K - 1
821: *
822: END IF
823: RETURN
824: *
825: * End of ZLASYF
826: *
827: END
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