Annotation of rpl/lapack/lapack/zlahef_rk.f, revision 1.5
1.1 bertrand 1: *> \brief \b ZLAHEF_RK computes a partial factorization of a complex Hermitian 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 ZLAHEF_RK + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlahef_rk.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlahef_rk.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlahef_rk.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLAHEF_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: *> ZLAHEF_RK computes a partial factorization of a complex Hermitian
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**H U22**H )
44: *>
45: *> A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) 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: *> ZLAHEF_RK is an auxiliary routine called by ZHETRF_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: *> Hermitian 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 Hermitian 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 Hermitian 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 Hermitian 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 Hermitian 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: *> \ingroup complex16HEcomputational
243: *
244: *> \par Contributors:
245: * ==================
246: *>
247: *> \verbatim
248: *>
249: *> December 2016, Igor Kozachenko,
250: *> Computer Science Division,
251: *> University of California, Berkeley
252: *>
253: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
254: *> School of Mathematics,
255: *> University of Manchester
256: *>
257: *> \endverbatim
258: *
259: * =====================================================================
260: SUBROUTINE ZLAHEF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW,
261: $ INFO )
262: *
1.5 ! bertrand 263: * -- LAPACK computational routine --
1.1 bertrand 264: * -- LAPACK is a software package provided by Univ. of Tennessee, --
265: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
266: *
267: * .. Scalar Arguments ..
268: CHARACTER UPLO
269: INTEGER INFO, KB, LDA, LDW, N, NB
270: * ..
271: * .. Array Arguments ..
272: INTEGER IPIV( * )
273: COMPLEX*16 A( LDA, * ), W( LDW, * ), E( * )
274: * ..
275: *
276: * =====================================================================
277: *
278: * .. Parameters ..
279: DOUBLE PRECISION ZERO, ONE
280: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
281: COMPLEX*16 CONE
282: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
283: DOUBLE PRECISION EIGHT, SEVTEN
284: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
285: COMPLEX*16 CZERO
286: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ) )
287: * ..
288: * .. Local Scalars ..
289: LOGICAL DONE
290: INTEGER IMAX, ITEMP, II, J, JB, JJ, JMAX, K, KK, KKW,
291: $ KP, KSTEP, KW, P
292: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, DTEMP, R1, ROWMAX, T,
293: $ SFMIN
294: COMPLEX*16 D11, D21, D22, Z
295: * ..
296: * .. External Functions ..
297: LOGICAL LSAME
298: INTEGER IZAMAX
299: DOUBLE PRECISION DLAMCH
300: EXTERNAL LSAME, IZAMAX, DLAMCH
301: * ..
302: * .. External Subroutines ..
303: EXTERNAL ZCOPY, ZDSCAL, ZGEMM, ZGEMV, ZLACGV, ZSWAP
304: * ..
305: * .. Intrinsic Functions ..
306: INTRINSIC ABS, DBLE, DCONJG, DIMAG, MAX, MIN, SQRT
307: * ..
308: * .. Statement Functions ..
309: DOUBLE PRECISION CABS1
310: * ..
311: * .. Statement Function definitions ..
312: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
313: * ..
314: * .. Executable Statements ..
315: *
316: INFO = 0
317: *
318: * Initialize ALPHA for use in choosing pivot block size.
319: *
320: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
321: *
322: * Compute machine safe minimum
323: *
324: SFMIN = DLAMCH( 'S' )
325: *
326: IF( LSAME( UPLO, 'U' ) ) THEN
327: *
328: * Factorize the trailing columns of A using the upper triangle
329: * of A and working backwards, and compute the matrix W = U12*D
330: * for use in updating A11 (note that conjg(W) is actually stored)
1.4 bertrand 331: * Initialize the first entry of array E, where superdiagonal
1.1 bertrand 332: * elements of D are stored
333: *
334: E( 1 ) = CZERO
335: *
336: * K is the main loop index, decreasing from N in steps of 1 or 2
337: *
338: K = N
339: 10 CONTINUE
340: *
341: * KW is the column of W which corresponds to column K of A
342: *
343: KW = NB + K - N
344: *
345: * Exit from loop
346: *
347: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
348: $ GO TO 30
349: *
350: KSTEP = 1
351: P = K
352: *
353: * Copy column K of A to column KW of W and update it
354: *
355: IF( K.GT.1 )
356: $ CALL ZCOPY( K-1, A( 1, K ), 1, W( 1, KW ), 1 )
357: W( K, KW ) = DBLE( A( K, K ) )
358: IF( K.LT.N ) THEN
359: CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ), LDA,
360: $ W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 )
361: W( K, KW ) = DBLE( W( K, KW ) )
362: END IF
363: *
364: * Determine rows and columns to be interchanged and whether
365: * a 1-by-1 or 2-by-2 pivot block will be used
366: *
367: ABSAKK = ABS( DBLE( W( K, KW ) ) )
368: *
369: * IMAX is the row-index of the largest off-diagonal element in
370: * column K, and COLMAX is its absolute value.
371: * Determine both COLMAX and IMAX.
372: *
373: IF( K.GT.1 ) THEN
374: IMAX = IZAMAX( K-1, W( 1, KW ), 1 )
375: COLMAX = CABS1( W( IMAX, KW ) )
376: ELSE
377: COLMAX = ZERO
378: END IF
379: *
380: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
381: *
382: * Column K is zero or underflow: set INFO and continue
383: *
384: IF( INFO.EQ.0 )
385: $ INFO = K
386: KP = K
387: A( K, K ) = DBLE( W( K, KW ) )
388: IF( K.GT.1 )
389: $ CALL ZCOPY( K-1, W( 1, KW ), 1, A( 1, K ), 1 )
390: *
391: * Set E( K ) to zero
392: *
393: IF( K.GT.1 )
394: $ E( K ) = CZERO
395: *
396: ELSE
397: *
398: * ============================================================
399: *
400: * BEGIN pivot search
401: *
402: * Case(1)
403: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
404: * (used to handle NaN and Inf)
405: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
406: *
407: * no interchange, use 1-by-1 pivot block
408: *
409: KP = K
410: *
411: ELSE
412: *
413: * Lop until pivot found
414: *
415: DONE = .FALSE.
416: *
417: 12 CONTINUE
418: *
419: * BEGIN pivot search loop body
420: *
421: *
422: * Copy column IMAX to column KW-1 of W and update it
423: *
424: IF( IMAX.GT.1 )
425: $ CALL ZCOPY( IMAX-1, A( 1, IMAX ), 1, W( 1, KW-1 ),
426: $ 1 )
427: W( IMAX, KW-1 ) = DBLE( A( IMAX, IMAX ) )
428: *
429: CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
430: $ W( IMAX+1, KW-1 ), 1 )
431: CALL ZLACGV( K-IMAX, W( IMAX+1, KW-1 ), 1 )
432: *
433: IF( K.LT.N ) THEN
434: CALL ZGEMV( 'No transpose', K, N-K, -CONE,
435: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
436: $ CONE, W( 1, KW-1 ), 1 )
437: W( IMAX, KW-1 ) = DBLE( W( IMAX, KW-1 ) )
438: END IF
439: *
440: * JMAX is the column-index of the largest off-diagonal
441: * element in row IMAX, and ROWMAX is its absolute value.
442: * Determine both ROWMAX and JMAX.
443: *
444: IF( IMAX.NE.K ) THEN
445: JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ),
446: $ 1 )
447: ROWMAX = CABS1( W( JMAX, KW-1 ) )
448: ELSE
449: ROWMAX = ZERO
450: END IF
451: *
452: IF( IMAX.GT.1 ) THEN
453: ITEMP = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 )
454: DTEMP = CABS1( W( ITEMP, KW-1 ) )
455: IF( DTEMP.GT.ROWMAX ) THEN
456: ROWMAX = DTEMP
457: JMAX = ITEMP
458: END IF
459: END IF
460: *
461: * Case(2)
462: * Equivalent to testing for
1.5 ! bertrand 463: * ABS( DBLE( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX
1.1 bertrand 464: * (used to handle NaN and Inf)
465: *
466: IF( .NOT.( ABS( DBLE( W( IMAX,KW-1 ) ) )
467: $ .LT.ALPHA*ROWMAX ) ) THEN
468: *
469: * interchange rows and columns K and IMAX,
470: * use 1-by-1 pivot block
471: *
472: KP = IMAX
473: *
474: * copy column KW-1 of W to column KW of W
475: *
476: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
477: *
478: DONE = .TRUE.
479: *
480: * Case(3)
481: * Equivalent to testing for ROWMAX.EQ.COLMAX,
482: * (used to handle NaN and Inf)
483: *
484: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
485: $ THEN
486: *
487: * interchange rows and columns K-1 and IMAX,
488: * use 2-by-2 pivot block
489: *
490: KP = IMAX
491: KSTEP = 2
492: DONE = .TRUE.
493: *
494: * Case(4)
495: ELSE
496: *
497: * Pivot not found: set params and repeat
498: *
499: P = IMAX
500: COLMAX = ROWMAX
501: IMAX = JMAX
502: *
503: * Copy updated JMAXth (next IMAXth) column to Kth of W
504: *
505: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
506: *
507: END IF
508: *
509: *
510: * END pivot search loop body
511: *
512: IF( .NOT.DONE ) GOTO 12
513: *
514: END IF
515: *
516: * END pivot search
517: *
518: * ============================================================
519: *
520: * KK is the column of A where pivoting step stopped
521: *
522: KK = K - KSTEP + 1
523: *
524: * KKW is the column of W which corresponds to column KK of A
525: *
526: KKW = NB + KK - N
527: *
528: * Interchange rows and columns P and K.
529: * Updated column P is already stored in column KW of W.
530: *
531: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
532: *
533: * Copy non-updated column K to column P of submatrix A
534: * at step K. No need to copy element into columns
535: * K and K-1 of A for 2-by-2 pivot, since these columns
536: * will be later overwritten.
537: *
538: A( P, P ) = DBLE( A( K, K ) )
539: CALL ZCOPY( K-1-P, A( P+1, K ), 1, A( P, P+1 ),
540: $ LDA )
541: CALL ZLACGV( K-1-P, A( P, P+1 ), LDA )
542: IF( P.GT.1 )
543: $ CALL ZCOPY( P-1, A( 1, K ), 1, A( 1, P ), 1 )
544: *
545: * Interchange rows K and P in the last K+1 to N columns of A
546: * (columns K and K-1 of A for 2-by-2 pivot will be
547: * later overwritten). Interchange rows K and P
548: * in last KKW to NB columns of W.
549: *
550: IF( K.LT.N )
551: $ CALL ZSWAP( N-K, A( K, K+1 ), LDA, A( P, K+1 ),
552: $ LDA )
553: CALL ZSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ),
554: $ LDW )
555: END IF
556: *
557: * Interchange rows and columns KP and KK.
558: * Updated column KP is already stored in column KKW of W.
559: *
560: IF( KP.NE.KK ) THEN
561: *
562: * Copy non-updated column KK to column KP of submatrix A
563: * at step K. No need to copy element into column K
564: * (or K and K-1 for 2-by-2 pivot) of A, since these columns
565: * will be later overwritten.
566: *
567: A( KP, KP ) = DBLE( A( KK, KK ) )
568: CALL ZCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
569: $ LDA )
570: CALL ZLACGV( KK-1-KP, A( KP, KP+1 ), LDA )
571: IF( KP.GT.1 )
572: $ CALL ZCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
573: *
574: * Interchange rows KK and KP in last K+1 to N columns of A
575: * (columns K (or K and K-1 for 2-by-2 pivot) of A will be
576: * later overwritten). Interchange rows KK and KP
577: * in last KKW to NB columns of W.
578: *
579: IF( K.LT.N )
580: $ CALL ZSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
581: $ LDA )
582: CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
583: $ LDW )
584: END IF
585: *
586: IF( KSTEP.EQ.1 ) THEN
587: *
588: * 1-by-1 pivot block D(k): column kw of W now holds
589: *
590: * W(kw) = U(k)*D(k),
591: *
592: * where U(k) is the k-th column of U
593: *
594: * (1) Store subdiag. elements of column U(k)
595: * and 1-by-1 block D(k) in column k of A.
596: * (NOTE: Diagonal element U(k,k) is a UNIT element
597: * and not stored)
598: * A(k,k) := D(k,k) = W(k,kw)
599: * A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k)
600: *
601: * (NOTE: No need to use for Hermitian matrix
1.5 ! bertrand 602: * A( K, K ) = DBLE( W( K, K) ) to separately copy diagonal
1.1 bertrand 603: * element D(k,k) from W (potentially saves only one load))
604: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
605: IF( K.GT.1 ) THEN
606: *
607: * (NOTE: No need to check if A(k,k) is NOT ZERO,
608: * since that was ensured earlier in pivot search:
609: * case A(k,k) = 0 falls into 2x2 pivot case(3))
610: *
611: * Handle division by a small number
612: *
613: T = DBLE( A( K, K ) )
614: IF( ABS( T ).GE.SFMIN ) THEN
615: R1 = ONE / T
616: CALL ZDSCAL( K-1, R1, A( 1, K ), 1 )
617: ELSE
618: DO 14 II = 1, K-1
619: A( II, K ) = A( II, K ) / T
620: 14 CONTINUE
621: END IF
622: *
623: * (2) Conjugate column W(kw)
624: *
625: CALL ZLACGV( K-1, W( 1, KW ), 1 )
626: *
627: * Store the superdiagonal element of D in array E
628: *
629: E( K ) = CZERO
630: *
631: END IF
632: *
633: ELSE
634: *
635: * 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold
636: *
637: * ( W(kw-1) W(kw) ) = ( U(k-1) U(k) )*D(k)
638: *
639: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
640: * of U
641: *
642: * (1) Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2
643: * block D(k-1:k,k-1:k) in columns k-1 and k of A.
644: * (NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT
645: * block and not stored)
646: * A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw)
647: * A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) =
648: * = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) )
649: *
650: IF( K.GT.2 ) THEN
651: *
652: * Factor out the columns of the inverse of 2-by-2 pivot
653: * block D, so that each column contains 1, to reduce the
654: * number of FLOPS when we multiply panel
655: * ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1).
656: *
657: * D**(-1) = ( d11 cj(d21) )**(-1) =
658: * ( d21 d22 )
659: *
660: * = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) =
661: * ( (-d21) ( d11 ) )
662: *
663: * = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) *
664: *
665: * * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) =
666: * ( ( -1 ) ( d11/conj(d21) ) )
667: *
668: * = 1/(|d21|**2) * 1/(D22*D11-1) *
669: *
670: * * ( d21*( D11 ) conj(d21)*( -1 ) ) =
671: * ( ( -1 ) ( D22 ) )
672: *
673: * = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) =
674: * ( ( -1 ) ( D22 ) )
675: *
676: * = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) =
677: * ( ( -1 ) ( D22 ) )
678: *
679: * Handle division by a small number. (NOTE: order of
680: * operations is important)
681: *
682: * = ( T*(( D11 )/conj(D21)) T*(( -1 )/D21 ) )
683: * ( (( -1 ) ) (( D22 ) ) ),
684: *
685: * where D11 = d22/d21,
686: * D22 = d11/conj(d21),
687: * D21 = d21,
688: * T = 1/(D22*D11-1).
689: *
690: * (NOTE: No need to check for division by ZERO,
691: * since that was ensured earlier in pivot search:
692: * (a) d21 != 0 in 2x2 pivot case(4),
693: * since |d21| should be larger than |d11| and |d22|;
694: * (b) (D22*D11 - 1) != 0, since from (a),
695: * both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.)
696: *
697: D21 = W( K-1, KW )
698: D11 = W( K, KW ) / DCONJG( D21 )
699: D22 = W( K-1, KW-1 ) / D21
700: T = ONE / ( DBLE( D11*D22 )-ONE )
701: *
702: * Update elements in columns A(k-1) and A(k) as
703: * dot products of rows of ( W(kw-1) W(kw) ) and columns
704: * of D**(-1)
705: *
706: DO 20 J = 1, K - 2
707: A( J, K-1 ) = T*( ( D11*W( J, KW-1 )-W( J, KW ) ) /
708: $ D21 )
709: A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
710: $ DCONJG( D21 ) )
711: 20 CONTINUE
712: END IF
713: *
714: * Copy diagonal elements of D(K) to A,
715: * copy superdiagonal element of D(K) to E(K) and
716: * ZERO out superdiagonal entry of A
717: *
718: A( K-1, K-1 ) = W( K-1, KW-1 )
719: A( K-1, K ) = CZERO
720: A( K, K ) = W( K, KW )
721: E( K ) = W( K-1, KW )
722: E( K-1 ) = CZERO
723: *
724: * (2) Conjugate columns W(kw) and W(kw-1)
725: *
726: CALL ZLACGV( K-1, W( 1, KW ), 1 )
727: CALL ZLACGV( K-2, W( 1, KW-1 ), 1 )
728: *
729: END IF
730: *
731: * End column K is nonsingular
732: *
733: END IF
734: *
735: * Store details of the interchanges in IPIV
736: *
737: IF( KSTEP.EQ.1 ) THEN
738: IPIV( K ) = KP
739: ELSE
740: IPIV( K ) = -P
741: IPIV( K-1 ) = -KP
742: END IF
743: *
744: * Decrease K and return to the start of the main loop
745: *
746: K = K - KSTEP
747: GO TO 10
748: *
749: 30 CONTINUE
750: *
751: * Update the upper triangle of A11 (= A(1:k,1:k)) as
752: *
753: * A11 := A11 - U12*D*U12**H = A11 - U12*W**H
754: *
755: * computing blocks of NB columns at a time (note that conjg(W) is
756: * actually stored)
757: *
758: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
759: JB = MIN( NB, K-J+1 )
760: *
761: * Update the upper triangle of the diagonal block
762: *
763: DO 40 JJ = J, J + JB - 1
764: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
765: CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE,
766: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE,
767: $ A( J, JJ ), 1 )
768: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
769: 40 CONTINUE
770: *
771: * Update the rectangular superdiagonal block
772: *
773: IF( J.GE.2 )
774: $ CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB, N-K,
775: $ -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
776: $ CONE, A( 1, J ), LDA )
777: 50 CONTINUE
778: *
779: * Set KB to the number of columns factorized
780: *
781: KB = N - K
782: *
783: ELSE
784: *
785: * Factorize the leading columns of A using the lower triangle
786: * of A and working forwards, and compute the matrix W = L21*D
787: * for use in updating A22 (note that conjg(W) is actually stored)
788: *
1.4 bertrand 789: * Initialize the unused last entry of the subdiagonal array E.
1.1 bertrand 790: *
791: E( N ) = CZERO
792: *
793: * K is the main loop index, increasing from 1 in steps of 1 or 2
794: *
795: K = 1
796: 70 CONTINUE
797: *
798: * Exit from loop
799: *
800: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
801: $ GO TO 90
802: *
803: KSTEP = 1
804: P = K
805: *
806: * Copy column K of A to column K of W and update column K of W
807: *
808: W( K, K ) = DBLE( A( K, K ) )
809: IF( K.LT.N )
810: $ CALL ZCOPY( N-K, A( K+1, K ), 1, W( K+1, K ), 1 )
811: IF( K.GT.1 ) THEN
812: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
813: $ LDA, W( K, 1 ), LDW, CONE, W( K, K ), 1 )
814: W( K, K ) = DBLE( W( K, K ) )
815: END IF
816: *
817: * Determine rows and columns to be interchanged and whether
818: * a 1-by-1 or 2-by-2 pivot block will be used
819: *
820: ABSAKK = ABS( DBLE( W( K, K ) ) )
821: *
822: * IMAX is the row-index of the largest off-diagonal element in
823: * column K, and COLMAX is its absolute value.
824: * Determine both COLMAX and IMAX.
825: *
826: IF( K.LT.N ) THEN
827: IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 )
828: COLMAX = CABS1( W( IMAX, K ) )
829: ELSE
830: COLMAX = ZERO
831: END IF
832: *
833: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
834: *
835: * Column K is zero or underflow: set INFO and continue
836: *
837: IF( INFO.EQ.0 )
838: $ INFO = K
839: KP = K
840: A( K, K ) = DBLE( W( K, K ) )
841: IF( K.LT.N )
842: $ CALL ZCOPY( N-K, W( K+1, K ), 1, A( K+1, K ), 1 )
843: *
844: * Set E( K ) to zero
845: *
846: IF( K.LT.N )
847: $ E( K ) = CZERO
848: *
849: ELSE
850: *
851: * ============================================================
852: *
853: * BEGIN pivot search
854: *
855: * Case(1)
856: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
857: * (used to handle NaN and Inf)
858: *
859: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
860: *
861: * no interchange, use 1-by-1 pivot block
862: *
863: KP = K
864: *
865: ELSE
866: *
867: DONE = .FALSE.
868: *
869: * Loop until pivot found
870: *
871: 72 CONTINUE
872: *
873: * BEGIN pivot search loop body
874: *
875: *
876: * Copy column IMAX to column k+1 of W and update it
877: *
878: CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
879: CALL ZLACGV( IMAX-K, W( K, K+1 ), 1 )
880: W( IMAX, K+1 ) = DBLE( A( IMAX, IMAX ) )
881: *
882: IF( IMAX.LT.N )
883: $ CALL ZCOPY( N-IMAX, A( IMAX+1, IMAX ), 1,
884: $ W( IMAX+1, K+1 ), 1 )
885: *
886: IF( K.GT.1 ) THEN
887: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE,
888: $ A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
889: $ CONE, W( K, K+1 ), 1 )
890: W( IMAX, K+1 ) = DBLE( W( IMAX, K+1 ) )
891: END IF
892: *
893: * JMAX is the column-index of the largest off-diagonal
894: * element in row IMAX, and ROWMAX is its absolute value.
895: * Determine both ROWMAX and JMAX.
896: *
897: IF( IMAX.NE.K ) THEN
898: JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 )
899: ROWMAX = CABS1( W( JMAX, K+1 ) )
900: ELSE
901: ROWMAX = ZERO
902: END IF
903: *
904: IF( IMAX.LT.N ) THEN
905: ITEMP = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
906: DTEMP = CABS1( W( ITEMP, K+1 ) )
907: IF( DTEMP.GT.ROWMAX ) THEN
908: ROWMAX = DTEMP
909: JMAX = ITEMP
910: END IF
911: END IF
912: *
913: * Case(2)
914: * Equivalent to testing for
1.5 ! bertrand 915: * ABS( DBLE( W( IMAX,K+1 ) ) ).GE.ALPHA*ROWMAX
1.1 bertrand 916: * (used to handle NaN and Inf)
917: *
918: IF( .NOT.( ABS( DBLE( W( IMAX,K+1 ) ) )
919: $ .LT.ALPHA*ROWMAX ) ) THEN
920: *
921: * interchange rows and columns K and IMAX,
922: * use 1-by-1 pivot block
923: *
924: KP = IMAX
925: *
926: * copy column K+1 of W to column K of W
927: *
928: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
929: *
930: DONE = .TRUE.
931: *
932: * Case(3)
933: * Equivalent to testing for ROWMAX.EQ.COLMAX,
934: * (used to handle NaN and Inf)
935: *
936: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
937: $ THEN
938: *
939: * interchange rows and columns K+1 and IMAX,
940: * use 2-by-2 pivot block
941: *
942: KP = IMAX
943: KSTEP = 2
944: DONE = .TRUE.
945: *
946: * Case(4)
947: ELSE
948: *
949: * Pivot not found: set params and repeat
950: *
951: P = IMAX
952: COLMAX = ROWMAX
953: IMAX = JMAX
954: *
955: * Copy updated JMAXth (next IMAXth) column to Kth of W
956: *
957: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
958: *
959: END IF
960: *
961: *
962: * End pivot search loop body
963: *
964: IF( .NOT.DONE ) GOTO 72
965: *
966: END IF
967: *
968: * END pivot search
969: *
970: * ============================================================
971: *
972: * KK is the column of A where pivoting step stopped
973: *
974: KK = K + KSTEP - 1
975: *
976: * Interchange rows and columns P and K (only for 2-by-2 pivot).
977: * Updated column P is already stored in column K of W.
978: *
979: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
980: *
981: * Copy non-updated column KK-1 to column P of submatrix A
982: * at step K. No need to copy element into columns
983: * K and K+1 of A for 2-by-2 pivot, since these columns
984: * will be later overwritten.
985: *
986: A( P, P ) = DBLE( A( K, K ) )
987: CALL ZCOPY( P-K-1, A( K+1, K ), 1, A( P, K+1 ), LDA )
988: CALL ZLACGV( P-K-1, A( P, K+1 ), LDA )
989: IF( P.LT.N )
990: $ CALL ZCOPY( N-P, A( P+1, K ), 1, A( P+1, P ), 1 )
991: *
992: * Interchange rows K and P in first K-1 columns of A
993: * (columns K and K+1 of A for 2-by-2 pivot will be
994: * later overwritten). Interchange rows K and P
995: * in first KK columns of W.
996: *
997: IF( K.GT.1 )
998: $ CALL ZSWAP( K-1, A( K, 1 ), LDA, A( P, 1 ), LDA )
999: CALL ZSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
1000: END IF
1001: *
1002: * Interchange rows and columns KP and KK.
1003: * Updated column KP is already stored in column KK of W.
1004: *
1005: IF( KP.NE.KK ) THEN
1006: *
1007: * Copy non-updated column KK to column KP of submatrix A
1008: * at step K. No need to copy element into column K
1009: * (or K and K+1 for 2-by-2 pivot) of A, since these columns
1010: * will be later overwritten.
1011: *
1012: A( KP, KP ) = DBLE( A( KK, KK ) )
1013: CALL ZCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
1014: $ LDA )
1015: CALL ZLACGV( KP-KK-1, A( KP, KK+1 ), LDA )
1016: IF( KP.LT.N )
1017: $ CALL ZCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
1018: *
1019: * Interchange rows KK and KP in first K-1 columns of A
1020: * (column K (or K and K+1 for 2-by-2 pivot) of A will be
1021: * later overwritten). Interchange rows KK and KP
1022: * in first KK columns of W.
1023: *
1024: IF( K.GT.1 )
1025: $ CALL ZSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
1026: CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
1027: END IF
1028: *
1029: IF( KSTEP.EQ.1 ) THEN
1030: *
1031: * 1-by-1 pivot block D(k): column k of W now holds
1032: *
1033: * W(k) = L(k)*D(k),
1034: *
1035: * where L(k) is the k-th column of L
1036: *
1037: * (1) Store subdiag. elements of column L(k)
1038: * and 1-by-1 block D(k) in column k of A.
1039: * (NOTE: Diagonal element L(k,k) is a UNIT element
1040: * and not stored)
1041: * A(k,k) := D(k,k) = W(k,k)
1042: * A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k)
1043: *
1044: * (NOTE: No need to use for Hermitian matrix
1.5 ! bertrand 1045: * A( K, K ) = DBLE( W( K, K) ) to separately copy diagonal
1.1 bertrand 1046: * element D(k,k) from W (potentially saves only one load))
1047: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
1048: IF( K.LT.N ) THEN
1049: *
1050: * (NOTE: No need to check if A(k,k) is NOT ZERO,
1051: * since that was ensured earlier in pivot search:
1052: * case A(k,k) = 0 falls into 2x2 pivot case(3))
1053: *
1054: * Handle division by a small number
1055: *
1056: T = DBLE( A( K, K ) )
1057: IF( ABS( T ).GE.SFMIN ) THEN
1058: R1 = ONE / T
1059: CALL ZDSCAL( N-K, R1, A( K+1, K ), 1 )
1060: ELSE
1061: DO 74 II = K + 1, N
1062: A( II, K ) = A( II, K ) / T
1063: 74 CONTINUE
1064: END IF
1065: *
1066: * (2) Conjugate column W(k)
1067: *
1068: CALL ZLACGV( N-K, W( K+1, K ), 1 )
1069: *
1070: * Store the subdiagonal element of D in array E
1071: *
1072: E( K ) = CZERO
1073: *
1074: END IF
1075: *
1076: ELSE
1077: *
1078: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
1079: *
1080: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
1081: *
1082: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
1083: * of L
1084: *
1085: * (1) Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2
1086: * block D(k:k+1,k:k+1) in columns k and k+1 of A.
1087: * NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT
1088: * block and not stored.
1089: * A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1)
1090: * A(k+2:N,k:k+1) := L(k+2:N,k:k+1) =
1091: * = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) )
1092: *
1093: IF( K.LT.N-1 ) THEN
1094: *
1095: * Factor out the columns of the inverse of 2-by-2 pivot
1096: * block D, so that each column contains 1, to reduce the
1097: * number of FLOPS when we multiply panel
1098: * ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1).
1099: *
1100: * D**(-1) = ( d11 cj(d21) )**(-1) =
1101: * ( d21 d22 )
1102: *
1103: * = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) =
1104: * ( (-d21) ( d11 ) )
1105: *
1106: * = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) *
1107: *
1108: * * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) =
1109: * ( ( -1 ) ( d11/conj(d21) ) )
1110: *
1111: * = 1/(|d21|**2) * 1/(D22*D11-1) *
1112: *
1113: * * ( d21*( D11 ) conj(d21)*( -1 ) ) =
1114: * ( ( -1 ) ( D22 ) )
1115: *
1116: * = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) =
1117: * ( ( -1 ) ( D22 ) )
1118: *
1119: * = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) =
1120: * ( ( -1 ) ( D22 ) )
1121: *
1122: * Handle division by a small number. (NOTE: order of
1123: * operations is important)
1124: *
1125: * = ( T*(( D11 )/conj(D21)) T*(( -1 )/D21 ) )
1126: * ( (( -1 ) ) (( D22 ) ) ),
1127: *
1128: * where D11 = d22/d21,
1129: * D22 = d11/conj(d21),
1130: * D21 = d21,
1131: * T = 1/(D22*D11-1).
1132: *
1133: * (NOTE: No need to check for division by ZERO,
1134: * since that was ensured earlier in pivot search:
1135: * (a) d21 != 0 in 2x2 pivot case(4),
1136: * since |d21| should be larger than |d11| and |d22|;
1137: * (b) (D22*D11 - 1) != 0, since from (a),
1138: * both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.)
1139: *
1140: D21 = W( K+1, K )
1141: D11 = W( K+1, K+1 ) / D21
1142: D22 = W( K, K ) / DCONJG( D21 )
1143: T = ONE / ( DBLE( D11*D22 )-ONE )
1144: *
1145: * Update elements in columns A(k) and A(k+1) as
1146: * dot products of rows of ( W(k) W(k+1) ) and columns
1147: * of D**(-1)
1148: *
1149: DO 80 J = K + 2, N
1150: A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
1151: $ DCONJG( D21 ) )
1152: A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
1153: $ D21 )
1154: 80 CONTINUE
1155: END IF
1156: *
1157: * Copy diagonal elements of D(K) to A,
1158: * copy subdiagonal element of D(K) to E(K) and
1159: * ZERO out subdiagonal entry of A
1160: *
1161: A( K, K ) = W( K, K )
1162: A( K+1, K ) = CZERO
1163: A( K+1, K+1 ) = W( K+1, K+1 )
1164: E( K ) = W( K+1, K )
1165: E( K+1 ) = CZERO
1166: *
1167: * (2) Conjugate columns W(k) and W(k+1)
1168: *
1169: CALL ZLACGV( N-K, W( K+1, K ), 1 )
1170: CALL ZLACGV( N-K-1, W( K+2, K+1 ), 1 )
1171: *
1172: END IF
1173: *
1174: * End column K is nonsingular
1175: *
1176: END IF
1177: *
1178: * Store details of the interchanges in IPIV
1179: *
1180: IF( KSTEP.EQ.1 ) THEN
1181: IPIV( K ) = KP
1182: ELSE
1183: IPIV( K ) = -P
1184: IPIV( K+1 ) = -KP
1185: END IF
1186: *
1187: * Increase K and return to the start of the main loop
1188: *
1189: K = K + KSTEP
1190: GO TO 70
1191: *
1192: 90 CONTINUE
1193: *
1194: * Update the lower triangle of A22 (= A(k:n,k:n)) as
1195: *
1196: * A22 := A22 - L21*D*L21**H = A22 - L21*W**H
1197: *
1198: * computing blocks of NB columns at a time (note that conjg(W) is
1199: * actually stored)
1200: *
1201: DO 110 J = K, N, NB
1202: JB = MIN( NB, N-J+1 )
1203: *
1204: * Update the lower triangle of the diagonal block
1205: *
1206: DO 100 JJ = J, J + JB - 1
1207: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
1208: CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
1209: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
1210: $ A( JJ, JJ ), 1 )
1211: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
1212: 100 CONTINUE
1213: *
1214: * Update the rectangular subdiagonal block
1215: *
1216: IF( J+JB.LE.N )
1217: $ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
1218: $ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ),
1219: $ LDW, CONE, A( J+JB, J ), LDA )
1220: 110 CONTINUE
1221: *
1222: * Set KB to the number of columns factorized
1223: *
1224: KB = K - 1
1225: *
1226: END IF
1227: RETURN
1228: *
1229: * End of ZLAHEF_RK
1230: *
1231: END
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