1: *> \brief \b ZLAHEF computes a partial factorization of a complex Hermitian indefinite matrix using the Bunch-Kaufman diagonal pivoting method (blocked algorithm, calling Level 3 BLAS).
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZLAHEF + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlahef.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLAHEF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, KB, LDA, LDW, N, NB
26: * ..
27: * .. Array Arguments ..
28: * INTEGER IPIV( * )
29: * COMPLEX*16 A( LDA, * ), W( LDW, * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> ZLAHEF computes a partial factorization of a complex Hermitian
39: *> matrix A using the Bunch-Kaufman diagonal pivoting method. The
40: *> 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: *> Note that U**H denotes the conjugate transpose of U.
51: *>
52: *> ZLAHEF is an auxiliary routine called by ZHETRF. 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: *> Hermitian 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 Hermitian matrix A. If UPLO = 'U', the leading
94: *> n-by-n upper triangular part of A contains the upper
95: *> triangular part of the matrix A, and the strictly lower
96: *> triangular part of A is not referenced. If UPLO = 'L', the
97: *> leading n-by-n lower triangular part of A contains the lower
98: *> triangular part of the matrix A, and the strictly upper
99: *> triangular part of A is not referenced.
100: *> On exit, A contains details of the partial factorization.
101: *> \endverbatim
102: *>
103: *> \param[in] LDA
104: *> \verbatim
105: *> LDA is INTEGER
106: *> The leading dimension of the array A. LDA >= max(1,N).
107: *> \endverbatim
108: *>
109: *> \param[out] IPIV
110: *> \verbatim
111: *> IPIV is INTEGER array, dimension (N)
112: *> Details of the interchanges and the block structure of D.
113: *>
114: *> If UPLO = 'U':
115: *> Only the last KB elements of IPIV are set.
116: *>
117: *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
118: *> interchanged and D(k,k) is a 1-by-1 diagonal block.
119: *>
120: *> If IPIV(k) = IPIV(k-1) < 0, then rows and columns
121: *> k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
122: *> is a 2-by-2 diagonal block.
123: *>
124: *> If UPLO = 'L':
125: *> Only the first KB elements of IPIV are set.
126: *>
127: *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
128: *> interchanged and D(k,k) is a 1-by-1 diagonal block.
129: *>
130: *> If IPIV(k) = IPIV(k+1) < 0, then rows and columns
131: *> k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
132: *> is a 2-by-2 diagonal block.
133: *> \endverbatim
134: *>
135: *> \param[out] W
136: *> \verbatim
137: *> W is COMPLEX*16 array, dimension (LDW,NB)
138: *> \endverbatim
139: *>
140: *> \param[in] LDW
141: *> \verbatim
142: *> LDW is INTEGER
143: *> The leading dimension of the array W. LDW >= max(1,N).
144: *> \endverbatim
145: *>
146: *> \param[out] INFO
147: *> \verbatim
148: *> INFO is INTEGER
149: *> = 0: successful exit
150: *> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
151: *> has been completed, but the block diagonal matrix D is
152: *> exactly singular.
153: *> \endverbatim
154: *
155: * Authors:
156: * ========
157: *
158: *> \author Univ. of Tennessee
159: *> \author Univ. of California Berkeley
160: *> \author Univ. of Colorado Denver
161: *> \author NAG Ltd.
162: *
163: *> \date December 2016
164: *
165: *> \ingroup complex16HEcomputational
166: *
167: *> \par Contributors:
168: * ==================
169: *>
170: *> \verbatim
171: *>
172: *> December 2016, Igor Kozachenko,
173: *> Computer Science Division,
174: *> University of California, Berkeley
175: *> \endverbatim
176: *
177: * =====================================================================
178: SUBROUTINE ZLAHEF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
179: *
180: * -- LAPACK computational routine (version 3.7.0) --
181: * -- LAPACK is a software package provided by Univ. of Tennessee, --
182: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
183: * December 2016
184: *
185: * .. Scalar Arguments ..
186: CHARACTER UPLO
187: INTEGER INFO, KB, LDA, LDW, N, NB
188: * ..
189: * .. Array Arguments ..
190: INTEGER IPIV( * )
191: COMPLEX*16 A( LDA, * ), W( LDW, * )
192: * ..
193: *
194: * =====================================================================
195: *
196: * .. Parameters ..
197: DOUBLE PRECISION ZERO, ONE
198: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
199: COMPLEX*16 CONE
200: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
201: DOUBLE PRECISION EIGHT, SEVTEN
202: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
203: * ..
204: * .. Local Scalars ..
205: INTEGER IMAX, J, JB, JJ, JMAX, JP, K, KK, KKW, KP,
206: $ KSTEP, KW
207: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, R1, ROWMAX, T
208: COMPLEX*16 D11, D21, D22, Z
209: * ..
210: * .. External Functions ..
211: LOGICAL LSAME
212: INTEGER IZAMAX
213: EXTERNAL LSAME, IZAMAX
214: * ..
215: * .. External Subroutines ..
216: EXTERNAL ZCOPY, ZDSCAL, ZGEMM, ZGEMV, ZLACGV, ZSWAP
217: * ..
218: * .. Intrinsic Functions ..
219: INTRINSIC ABS, DBLE, DCONJG, DIMAG, MAX, MIN, SQRT
220: * ..
221: * .. Statement Functions ..
222: DOUBLE PRECISION CABS1
223: * ..
224: * .. Statement Function definitions ..
225: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
226: * ..
227: * .. Executable Statements ..
228: *
229: INFO = 0
230: *
231: * Initialize ALPHA for use in choosing pivot block size.
232: *
233: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
234: *
235: IF( LSAME( UPLO, 'U' ) ) THEN
236: *
237: * Factorize the trailing columns of A using the upper triangle
238: * of A and working backwards, and compute the matrix W = U12*D
239: * for use in updating A11 (note that conjg(W) is actually stored)
240: *
241: * K is the main loop index, decreasing from N in steps of 1 or 2
242: *
243: * KW is the column of W which corresponds to column K of A
244: *
245: K = N
246: 10 CONTINUE
247: KW = NB + K - N
248: *
249: * Exit from loop
250: *
251: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
252: $ GO TO 30
253: *
254: KSTEP = 1
255: *
256: * Copy column K of A to column KW of W and update it
257: *
258: CALL ZCOPY( K-1, A( 1, K ), 1, W( 1, KW ), 1 )
259: W( K, KW ) = DBLE( A( K, K ) )
260: IF( K.LT.N ) THEN
261: CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ), LDA,
262: $ W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 )
263: W( K, KW ) = DBLE( W( K, KW ) )
264: END IF
265: *
266: * Determine rows and columns to be interchanged and whether
267: * a 1-by-1 or 2-by-2 pivot block will be used
268: *
269: ABSAKK = ABS( DBLE( W( K, KW ) ) )
270: *
271: * IMAX is the row-index of the largest off-diagonal element in
272: * column K, and COLMAX is its absolute value.
273: * Determine both COLMAX and IMAX.
274: *
275: IF( K.GT.1 ) THEN
276: IMAX = IZAMAX( K-1, W( 1, KW ), 1 )
277: COLMAX = CABS1( W( IMAX, KW ) )
278: ELSE
279: COLMAX = ZERO
280: END IF
281: *
282: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
283: *
284: * Column K is zero or underflow: set INFO and continue
285: *
286: IF( INFO.EQ.0 )
287: $ INFO = K
288: KP = K
289: A( K, K ) = DBLE( A( K, K ) )
290: ELSE
291: *
292: * ============================================================
293: *
294: * BEGIN pivot search
295: *
296: * Case(1)
297: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
298: *
299: * no interchange, use 1-by-1 pivot block
300: *
301: KP = K
302: ELSE
303: *
304: * BEGIN pivot search along IMAX row
305: *
306: *
307: * Copy column IMAX to column KW-1 of W and update it
308: *
309: CALL ZCOPY( IMAX-1, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
310: W( IMAX, KW-1 ) = DBLE( A( IMAX, IMAX ) )
311: CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
312: $ W( IMAX+1, KW-1 ), 1 )
313: CALL ZLACGV( K-IMAX, W( IMAX+1, KW-1 ), 1 )
314: IF( K.LT.N ) THEN
315: CALL ZGEMV( 'No transpose', K, N-K, -CONE,
316: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
317: $ CONE, W( 1, KW-1 ), 1 )
318: W( IMAX, KW-1 ) = DBLE( W( IMAX, KW-1 ) )
319: END IF
320: *
321: * JMAX is the column-index of the largest off-diagonal
322: * element in row IMAX, and ROWMAX is its absolute value.
323: * Determine only ROWMAX.
324: *
325: JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 )
326: ROWMAX = CABS1( W( JMAX, KW-1 ) )
327: IF( IMAX.GT.1 ) THEN
328: JMAX = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 )
329: ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, KW-1 ) ) )
330: END IF
331: *
332: * Case(2)
333: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
334: *
335: * no interchange, use 1-by-1 pivot block
336: *
337: KP = K
338: *
339: * Case(3)
340: ELSE IF( ABS( DBLE( W( IMAX, KW-1 ) ) ).GE.ALPHA*ROWMAX )
341: $ THEN
342: *
343: * interchange rows and columns K and IMAX, use 1-by-1
344: * pivot block
345: *
346: KP = IMAX
347: *
348: * copy column KW-1 of W to column KW of W
349: *
350: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
351: *
352: * Case(4)
353: ELSE
354: *
355: * interchange rows and columns K-1 and IMAX, use 2-by-2
356: * pivot block
357: *
358: KP = IMAX
359: KSTEP = 2
360: END IF
361: *
362: *
363: * END pivot search along IMAX row
364: *
365: END IF
366: *
367: * END pivot search
368: *
369: * ============================================================
370: *
371: * KK is the column of A where pivoting step stopped
372: *
373: KK = K - KSTEP + 1
374: *
375: * KKW is the column of W which corresponds to column KK of A
376: *
377: KKW = NB + KK - N
378: *
379: * Interchange rows and columns KP and KK.
380: * Updated column KP is already stored in column KKW of W.
381: *
382: IF( KP.NE.KK ) THEN
383: *
384: * Copy non-updated column KK to column KP of submatrix A
385: * at step K. No need to copy element into column K
386: * (or K and K-1 for 2-by-2 pivot) of A, since these columns
387: * will be later overwritten.
388: *
389: A( KP, KP ) = DBLE( A( KK, KK ) )
390: CALL ZCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
391: $ LDA )
392: CALL ZLACGV( KK-1-KP, A( KP, KP+1 ), LDA )
393: IF( KP.GT.1 )
394: $ CALL ZCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
395: *
396: * Interchange rows KK and KP in last K+1 to N columns of A
397: * (columns K (or K and K-1 for 2-by-2 pivot) of A will be
398: * later overwritten). Interchange rows KK and KP
399: * in last KKW to NB columns of W.
400: *
401: IF( K.LT.N )
402: $ CALL ZSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
403: $ LDA )
404: CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
405: $ LDW )
406: END IF
407: *
408: IF( KSTEP.EQ.1 ) THEN
409: *
410: * 1-by-1 pivot block D(k): column kw of W now holds
411: *
412: * W(kw) = U(k)*D(k),
413: *
414: * where U(k) is the k-th column of U
415: *
416: * (1) Store subdiag. elements of column U(k)
417: * and 1-by-1 block D(k) in column k of A.
418: * (NOTE: Diagonal element U(k,k) is a UNIT element
419: * and not stored)
420: * A(k,k) := D(k,k) = W(k,kw)
421: * A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k)
422: *
423: * (NOTE: No need to use for Hermitian matrix
424: * A( K, K ) = DBLE( W( K, K) ) to separately copy diagonal
425: * element D(k,k) from W (potentially saves only one load))
426: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
427: IF( K.GT.1 ) THEN
428: *
429: * (NOTE: No need to check if A(k,k) is NOT ZERO,
430: * since that was ensured earlier in pivot search:
431: * case A(k,k) = 0 falls into 2x2 pivot case(4))
432: *
433: R1 = ONE / DBLE( A( K, K ) )
434: CALL ZDSCAL( K-1, R1, A( 1, K ), 1 )
435: *
436: * (2) Conjugate column W(kw)
437: *
438: CALL ZLACGV( K-1, W( 1, KW ), 1 )
439: END IF
440: *
441: ELSE
442: *
443: * 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold
444: *
445: * ( W(kw-1) W(kw) ) = ( U(k-1) U(k) )*D(k)
446: *
447: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
448: * of U
449: *
450: * (1) Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2
451: * block D(k-1:k,k-1:k) in columns k-1 and k of A.
452: * (NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT
453: * block and not stored)
454: * A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw)
455: * A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) =
456: * = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) )
457: *
458: IF( K.GT.2 ) THEN
459: *
460: * Factor out the columns of the inverse of 2-by-2 pivot
461: * block D, so that each column contains 1, to reduce the
462: * number of FLOPS when we multiply panel
463: * ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1).
464: *
465: * D**(-1) = ( d11 cj(d21) )**(-1) =
466: * ( d21 d22 )
467: *
468: * = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) =
469: * ( (-d21) ( d11 ) )
470: *
471: * = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) *
472: *
473: * * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) =
474: * ( ( -1 ) ( d11/conj(d21) ) )
475: *
476: * = 1/(|d21|**2) * 1/(D22*D11-1) *
477: *
478: * * ( d21*( D11 ) conj(d21)*( -1 ) ) =
479: * ( ( -1 ) ( D22 ) )
480: *
481: * = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) =
482: * ( ( -1 ) ( D22 ) )
483: *
484: * = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) =
485: * ( ( -1 ) ( D22 ) )
486: *
487: * = ( conj(D21)*( D11 ) D21*( -1 ) )
488: * ( ( -1 ) ( D22 ) ),
489: *
490: * where D11 = d22/d21,
491: * D22 = d11/conj(d21),
492: * D21 = T/d21,
493: * T = 1/(D22*D11-1).
494: *
495: * (NOTE: No need to check for division by ZERO,
496: * since that was ensured earlier in pivot search:
497: * (a) d21 != 0, since in 2x2 pivot case(4)
498: * |d21| should be larger than |d11| and |d22|;
499: * (b) (D22*D11 - 1) != 0, since from (a),
500: * both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.)
501: *
502: D21 = W( K-1, KW )
503: D11 = W( K, KW ) / DCONJG( D21 )
504: D22 = W( K-1, KW-1 ) / D21
505: T = ONE / ( DBLE( D11*D22 )-ONE )
506: D21 = T / D21
507: *
508: * Update elements in columns A(k-1) and A(k) as
509: * dot products of rows of ( W(kw-1) W(kw) ) and columns
510: * of D**(-1)
511: *
512: DO 20 J = 1, K - 2
513: A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) )
514: A( J, K ) = DCONJG( D21 )*
515: $ ( D22*W( J, KW )-W( J, KW-1 ) )
516: 20 CONTINUE
517: END IF
518: *
519: * Copy D(k) to A
520: *
521: A( K-1, K-1 ) = W( K-1, KW-1 )
522: A( K-1, K ) = W( K-1, KW )
523: A( K, K ) = W( K, KW )
524: *
525: * (2) Conjugate columns W(kw) and W(kw-1)
526: *
527: CALL ZLACGV( K-1, W( 1, KW ), 1 )
528: CALL ZLACGV( K-2, W( 1, KW-1 ), 1 )
529: *
530: END IF
531: *
532: END IF
533: *
534: * Store details of the interchanges in IPIV
535: *
536: IF( KSTEP.EQ.1 ) THEN
537: IPIV( K ) = KP
538: ELSE
539: IPIV( K ) = -KP
540: IPIV( K-1 ) = -KP
541: END IF
542: *
543: * Decrease K and return to the start of the main loop
544: *
545: K = K - KSTEP
546: GO TO 10
547: *
548: 30 CONTINUE
549: *
550: * Update the upper triangle of A11 (= A(1:k,1:k)) as
551: *
552: * A11 := A11 - U12*D*U12**H = A11 - U12*W**H
553: *
554: * computing blocks of NB columns at a time (note that conjg(W) is
555: * actually stored)
556: *
557: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
558: JB = MIN( NB, K-J+1 )
559: *
560: * Update the upper triangle of the diagonal block
561: *
562: DO 40 JJ = J, J + JB - 1
563: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
564: CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE,
565: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE,
566: $ A( J, JJ ), 1 )
567: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
568: 40 CONTINUE
569: *
570: * Update the rectangular superdiagonal block
571: *
572: CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB, N-K,
573: $ -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
574: $ CONE, A( 1, J ), LDA )
575: 50 CONTINUE
576: *
577: * Put U12 in standard form by partially undoing the interchanges
578: * in columns k+1:n looping backwards from k+1 to n
579: *
580: J = K + 1
581: 60 CONTINUE
582: *
583: * Undo the interchanges (if any) of rows JJ and JP at each
584: * step J
585: *
586: * (Here, J is a diagonal index)
587: JJ = J
588: JP = IPIV( J )
589: IF( JP.LT.0 ) THEN
590: JP = -JP
591: * (Here, J is a diagonal index)
592: J = J + 1
593: END IF
594: * (NOTE: Here, J is used to determine row length. Length N-J+1
595: * of the rows to swap back doesn't include diagonal element)
596: J = J + 1
597: IF( JP.NE.JJ .AND. J.LE.N )
598: $ CALL ZSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA )
599: IF( J.LT.N )
600: $ GO TO 60
601: *
602: * Set KB to the number of columns factorized
603: *
604: KB = N - K
605: *
606: ELSE
607: *
608: * Factorize the leading columns of A using the lower triangle
609: * of A and working forwards, and compute the matrix W = L21*D
610: * for use in updating A22 (note that conjg(W) is actually stored)
611: *
612: * K is the main loop index, increasing from 1 in steps of 1 or 2
613: *
614: K = 1
615: 70 CONTINUE
616: *
617: * Exit from loop
618: *
619: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
620: $ GO TO 90
621: *
622: KSTEP = 1
623: *
624: * Copy column K of A to column K of W and update it
625: *
626: W( K, K ) = DBLE( A( K, K ) )
627: IF( K.LT.N )
628: $ CALL ZCOPY( N-K, A( K+1, K ), 1, W( K+1, K ), 1 )
629: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ), LDA,
630: $ W( K, 1 ), LDW, CONE, W( K, K ), 1 )
631: W( K, K ) = DBLE( W( K, K ) )
632: *
633: * Determine rows and columns to be interchanged and whether
634: * a 1-by-1 or 2-by-2 pivot block will be used
635: *
636: ABSAKK = ABS( DBLE( W( K, K ) ) )
637: *
638: * IMAX is the row-index of the largest off-diagonal element in
639: * column K, and COLMAX is its absolute value.
640: * Determine both COLMAX and IMAX.
641: *
642: IF( K.LT.N ) THEN
643: IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 )
644: COLMAX = CABS1( W( IMAX, K ) )
645: ELSE
646: COLMAX = ZERO
647: END IF
648: *
649: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
650: *
651: * Column K is zero or underflow: set INFO and continue
652: *
653: IF( INFO.EQ.0 )
654: $ INFO = K
655: KP = K
656: A( K, K ) = DBLE( A( K, K ) )
657: ELSE
658: *
659: * ============================================================
660: *
661: * BEGIN pivot search
662: *
663: * Case(1)
664: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
665: *
666: * no interchange, use 1-by-1 pivot block
667: *
668: KP = K
669: ELSE
670: *
671: * BEGIN pivot search along IMAX row
672: *
673: *
674: * Copy column IMAX to column K+1 of W and update it
675: *
676: CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1 )
677: CALL ZLACGV( IMAX-K, W( K, K+1 ), 1 )
678: W( IMAX, K+1 ) = DBLE( A( IMAX, IMAX ) )
679: IF( IMAX.LT.N )
680: $ CALL ZCOPY( N-IMAX, A( IMAX+1, IMAX ), 1,
681: $ W( IMAX+1, K+1 ), 1 )
682: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
683: $ LDA, W( IMAX, 1 ), LDW, CONE, W( K, K+1 ),
684: $ 1 )
685: W( IMAX, K+1 ) = DBLE( W( IMAX, K+1 ) )
686: *
687: * JMAX is the column-index of the largest off-diagonal
688: * element in row IMAX, and ROWMAX is its absolute value.
689: * Determine only ROWMAX.
690: *
691: JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 )
692: ROWMAX = CABS1( W( JMAX, K+1 ) )
693: IF( IMAX.LT.N ) THEN
694: JMAX = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1 )
695: ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, K+1 ) ) )
696: END IF
697: *
698: * Case(2)
699: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
700: *
701: * no interchange, use 1-by-1 pivot block
702: *
703: KP = K
704: *
705: * Case(3)
706: ELSE IF( ABS( DBLE( W( IMAX, K+1 ) ) ).GE.ALPHA*ROWMAX )
707: $ THEN
708: *
709: * interchange rows and columns K and IMAX, use 1-by-1
710: * pivot block
711: *
712: KP = IMAX
713: *
714: * copy column K+1 of W to column K of W
715: *
716: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
717: *
718: * Case(4)
719: ELSE
720: *
721: * interchange rows and columns K+1 and IMAX, use 2-by-2
722: * pivot block
723: *
724: KP = IMAX
725: KSTEP = 2
726: END IF
727: *
728: *
729: * END pivot search along IMAX row
730: *
731: END IF
732: *
733: * END pivot search
734: *
735: * ============================================================
736: *
737: * KK is the column of A where pivoting step stopped
738: *
739: KK = K + KSTEP - 1
740: *
741: * Interchange rows and columns KP and KK.
742: * Updated column KP is already stored in column KK of W.
743: *
744: IF( KP.NE.KK ) THEN
745: *
746: * Copy non-updated column KK to column KP of submatrix A
747: * at step K. No need to copy element into column K
748: * (or K and K+1 for 2-by-2 pivot) of A, since these columns
749: * will be later overwritten.
750: *
751: A( KP, KP ) = DBLE( A( KK, KK ) )
752: CALL ZCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
753: $ LDA )
754: CALL ZLACGV( KP-KK-1, A( KP, KK+1 ), LDA )
755: IF( KP.LT.N )
756: $ CALL ZCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
757: *
758: * Interchange rows KK and KP in first K-1 columns of A
759: * (columns K (or K and K+1 for 2-by-2 pivot) of A will be
760: * later overwritten). Interchange rows KK and KP
761: * in first KK columns of W.
762: *
763: IF( K.GT.1 )
764: $ CALL ZSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
765: CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
766: END IF
767: *
768: IF( KSTEP.EQ.1 ) THEN
769: *
770: * 1-by-1 pivot block D(k): column k of W now holds
771: *
772: * W(k) = L(k)*D(k),
773: *
774: * where L(k) is the k-th column of L
775: *
776: * (1) Store subdiag. elements of column L(k)
777: * and 1-by-1 block D(k) in column k of A.
778: * (NOTE: Diagonal element L(k,k) is a UNIT element
779: * and not stored)
780: * A(k,k) := D(k,k) = W(k,k)
781: * A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k)
782: *
783: * (NOTE: No need to use for Hermitian matrix
784: * A( K, K ) = DBLE( W( K, K) ) to separately copy diagonal
785: * element D(k,k) from W (potentially saves only one load))
786: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
787: IF( K.LT.N ) THEN
788: *
789: * (NOTE: No need to check if A(k,k) is NOT ZERO,
790: * since that was ensured earlier in pivot search:
791: * case A(k,k) = 0 falls into 2x2 pivot case(4))
792: *
793: R1 = ONE / DBLE( A( K, K ) )
794: CALL ZDSCAL( N-K, R1, A( K+1, K ), 1 )
795: *
796: * (2) Conjugate column W(k)
797: *
798: CALL ZLACGV( N-K, W( K+1, K ), 1 )
799: END IF
800: *
801: ELSE
802: *
803: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
804: *
805: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
806: *
807: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
808: * of L
809: *
810: * (1) Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2
811: * block D(k:k+1,k:k+1) in columns k and k+1 of A.
812: * (NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT
813: * block and not stored)
814: * A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1)
815: * A(k+2:N,k:k+1) := L(k+2:N,k:k+1) =
816: * = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) )
817: *
818: IF( K.LT.N-1 ) THEN
819: *
820: * Factor out the columns of the inverse of 2-by-2 pivot
821: * block D, so that each column contains 1, to reduce the
822: * number of FLOPS when we multiply panel
823: * ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1).
824: *
825: * D**(-1) = ( d11 cj(d21) )**(-1) =
826: * ( d21 d22 )
827: *
828: * = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) =
829: * ( (-d21) ( d11 ) )
830: *
831: * = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) *
832: *
833: * * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) =
834: * ( ( -1 ) ( d11/conj(d21) ) )
835: *
836: * = 1/(|d21|**2) * 1/(D22*D11-1) *
837: *
838: * * ( d21*( D11 ) conj(d21)*( -1 ) ) =
839: * ( ( -1 ) ( D22 ) )
840: *
841: * = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) =
842: * ( ( -1 ) ( D22 ) )
843: *
844: * = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) =
845: * ( ( -1 ) ( D22 ) )
846: *
847: * = ( conj(D21)*( D11 ) D21*( -1 ) )
848: * ( ( -1 ) ( D22 ) ),
849: *
850: * where D11 = d22/d21,
851: * D22 = d11/conj(d21),
852: * D21 = T/d21,
853: * T = 1/(D22*D11-1).
854: *
855: * (NOTE: No need to check for division by ZERO,
856: * since that was ensured earlier in pivot search:
857: * (a) d21 != 0, since in 2x2 pivot case(4)
858: * |d21| should be larger than |d11| and |d22|;
859: * (b) (D22*D11 - 1) != 0, since from (a),
860: * both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.)
861: *
862: D21 = W( K+1, K )
863: D11 = W( K+1, K+1 ) / D21
864: D22 = W( K, K ) / DCONJG( D21 )
865: T = ONE / ( DBLE( D11*D22 )-ONE )
866: D21 = T / D21
867: *
868: * Update elements in columns A(k) and A(k+1) as
869: * dot products of rows of ( W(k) W(k+1) ) and columns
870: * of D**(-1)
871: *
872: DO 80 J = K + 2, N
873: A( J, K ) = DCONJG( D21 )*
874: $ ( D11*W( J, K )-W( J, K+1 ) )
875: A( J, K+1 ) = D21*( D22*W( J, K+1 )-W( J, K ) )
876: 80 CONTINUE
877: END IF
878: *
879: * Copy D(k) to A
880: *
881: A( K, K ) = W( K, K )
882: A( K+1, K ) = W( K+1, K )
883: A( K+1, K+1 ) = W( K+1, K+1 )
884: *
885: * (2) Conjugate columns W(k) and W(k+1)
886: *
887: CALL ZLACGV( N-K, W( K+1, K ), 1 )
888: CALL ZLACGV( N-K-1, W( K+2, K+1 ), 1 )
889: *
890: END IF
891: *
892: END IF
893: *
894: * Store details of the interchanges in IPIV
895: *
896: IF( KSTEP.EQ.1 ) THEN
897: IPIV( K ) = KP
898: ELSE
899: IPIV( K ) = -KP
900: IPIV( K+1 ) = -KP
901: END IF
902: *
903: * Increase K and return to the start of the main loop
904: *
905: K = K + KSTEP
906: GO TO 70
907: *
908: 90 CONTINUE
909: *
910: * Update the lower triangle of A22 (= A(k:n,k:n)) as
911: *
912: * A22 := A22 - L21*D*L21**H = A22 - L21*W**H
913: *
914: * computing blocks of NB columns at a time (note that conjg(W) is
915: * actually stored)
916: *
917: DO 110 J = K, N, NB
918: JB = MIN( NB, N-J+1 )
919: *
920: * Update the lower triangle of the diagonal block
921: *
922: DO 100 JJ = J, J + JB - 1
923: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
924: CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
925: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
926: $ A( JJ, JJ ), 1 )
927: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
928: 100 CONTINUE
929: *
930: * Update the rectangular subdiagonal block
931: *
932: IF( J+JB.LE.N )
933: $ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
934: $ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ),
935: $ LDW, CONE, A( J+JB, J ), LDA )
936: 110 CONTINUE
937: *
938: * Put L21 in standard form by partially undoing the interchanges
939: * of rows in columns 1:k-1 looping backwards from k-1 to 1
940: *
941: J = K - 1
942: 120 CONTINUE
943: *
944: * Undo the interchanges (if any) of rows JJ and JP at each
945: * step J
946: *
947: * (Here, J is a diagonal index)
948: JJ = J
949: JP = IPIV( J )
950: IF( JP.LT.0 ) THEN
951: JP = -JP
952: * (Here, J is a diagonal index)
953: J = J - 1
954: END IF
955: * (NOTE: Here, J is used to determine row length. Length J
956: * of the rows to swap back doesn't include diagonal element)
957: J = J - 1
958: IF( JP.NE.JJ .AND. J.GE.1 )
959: $ CALL ZSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA )
960: IF( J.GT.1 )
961: $ GO TO 120
962: *
963: * Set KB to the number of columns factorized
964: *
965: KB = K - 1
966: *
967: END IF
968: RETURN
969: *
970: * End of ZLAHEF
971: *
972: END
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