1: *> \brief \b ZLAHEF_AA
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZLAHEF_AA + 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_aa.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLAHEF_AA( UPLO, J1, M, NB, A, LDA, IPIV,
22: * H, LDH, WORK )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER UPLO
26: * INTEGER J1, M, NB, LDA, LDH
27: * ..
28: * .. Array Arguments ..
29: * INTEGER IPIV( * )
30: * COMPLEX*16 A( LDA, * ), H( LDH, * ), WORK( * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> DLAHEF_AA factorizes a panel of a complex hermitian matrix A using
40: *> the Aasen's algorithm. The panel consists of a set of NB rows of A
41: *> when UPLO is U, or a set of NB columns when UPLO is L.
42: *>
43: *> In order to factorize the panel, the Aasen's algorithm requires the
44: *> last row, or column, of the previous panel. The first row, or column,
45: *> of A is set to be the first row, or column, of an identity matrix,
46: *> which is used to factorize the first panel.
47: *>
48: *> The resulting J-th row of U, or J-th column of L, is stored in the
49: *> (J-1)-th row, or column, of A (without the unit diagonals), while
50: *> the diagonal and subdiagonal of A are overwritten by those of T.
51: *>
52: *> \endverbatim
53: *
54: * Arguments:
55: * ==========
56: *
57: *> \param[in] UPLO
58: *> \verbatim
59: *> UPLO is CHARACTER*1
60: *> = 'U': Upper triangle of A is stored;
61: *> = 'L': Lower triangle of A is stored.
62: *> \endverbatim
63: *>
64: *> \param[in] J1
65: *> \verbatim
66: *> J1 is INTEGER
67: *> The location of the first row, or column, of the panel
68: *> within the submatrix of A, passed to this routine, e.g.,
69: *> when called by ZHETRF_AA, for the first panel, J1 is 1,
70: *> while for the remaining panels, J1 is 2.
71: *> \endverbatim
72: *>
73: *> \param[in] M
74: *> \verbatim
75: *> M is INTEGER
76: *> The dimension of the submatrix. M >= 0.
77: *> \endverbatim
78: *>
79: *> \param[in] NB
80: *> \verbatim
81: *> NB is INTEGER
82: *> The dimension of the panel to be facotorized.
83: *> \endverbatim
84: *>
85: *> \param[in,out] A
86: *> \verbatim
87: *> A is COMPLEX*16 array, dimension (LDA,M) for
88: *> the first panel, while dimension (LDA,M+1) for the
89: *> remaining panels.
90: *>
91: *> On entry, A contains the last row, or column, of
92: *> the previous panel, and the trailing submatrix of A
93: *> to be factorized, except for the first panel, only
94: *> the panel is passed.
95: *>
96: *> On exit, the leading panel is factorized.
97: *> \endverbatim
98: *>
99: *> \param[in] LDA
100: *> \verbatim
101: *> LDA is INTEGER
102: *> The leading dimension of the array A. LDA >= max(1,N).
103: *> \endverbatim
104: *>
105: *> \param[out] IPIV
106: *> \verbatim
107: *> IPIV is INTEGER array, dimension (N)
108: *> Details of the row and column interchanges,
109: *> the row and column k were interchanged with the row and
110: *> column IPIV(k).
111: *> \endverbatim
112: *>
113: *> \param[in,out] H
114: *> \verbatim
115: *> H is COMPLEX*16 workspace, dimension (LDH,NB).
116: *>
117: *> \endverbatim
118: *>
119: *> \param[in] LDH
120: *> \verbatim
121: *> LDH is INTEGER
122: *> The leading dimension of the workspace H. LDH >= max(1,M).
123: *> \endverbatim
124: *>
125: *> \param[out] WORK
126: *> \verbatim
127: *> WORK is COMPLEX*16 workspace, dimension (M).
128: *> \endverbatim
129: *>
130: *
131: * Authors:
132: * ========
133: *
134: *> \author Univ. of Tennessee
135: *> \author Univ. of California Berkeley
136: *> \author Univ. of Colorado Denver
137: *> \author NAG Ltd.
138: *
139: *> \date November 2017
140: *
141: *> \ingroup complex16HEcomputational
142: *
143: * =====================================================================
144: SUBROUTINE ZLAHEF_AA( UPLO, J1, M, NB, A, LDA, IPIV,
145: $ H, LDH, WORK )
146: *
147: * -- LAPACK computational routine (version 3.8.0) --
148: * -- LAPACK is a software package provided by Univ. of Tennessee, --
149: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
150: * November 2017
151: *
152: IMPLICIT NONE
153: *
154: * .. Scalar Arguments ..
155: CHARACTER UPLO
156: INTEGER M, NB, J1, LDA, LDH
157: * ..
158: * .. Array Arguments ..
159: INTEGER IPIV( * )
160: COMPLEX*16 A( LDA, * ), H( LDH, * ), WORK( * )
161: * ..
162: *
163: * =====================================================================
164: * .. Parameters ..
165: COMPLEX*16 ZERO, ONE
166: PARAMETER ( ZERO = (0.0D+0, 0.0D+0), ONE = (1.0D+0, 0.0D+0) )
167: *
168: * .. Local Scalars ..
169: INTEGER J, K, K1, I1, I2, MJ
170: COMPLEX*16 PIV, ALPHA
171: * ..
172: * .. External Functions ..
173: LOGICAL LSAME
174: INTEGER IZAMAX, ILAENV
175: EXTERNAL LSAME, ILAENV, IZAMAX
176: * ..
177: * .. External Subroutines ..
178: EXTERNAL ZGEMM, ZGEMV, ZAXPY, ZLACGV, ZCOPY, ZSCAL, ZSWAP,
179: $ ZLASET, XERBLA
180: * ..
181: * .. Intrinsic Functions ..
182: INTRINSIC DBLE, DCONJG, MAX
183: * ..
184: * .. Executable Statements ..
185: *
186: J = 1
187: *
188: * K1 is the first column of the panel to be factorized
189: * i.e., K1 is 2 for the first block column, and 1 for the rest of the blocks
190: *
191: K1 = (2-J1)+1
192: *
193: IF( LSAME( UPLO, 'U' ) ) THEN
194: *
195: * .....................................................
196: * Factorize A as U**T*D*U using the upper triangle of A
197: * .....................................................
198: *
199: 10 CONTINUE
200: IF ( J.GT.MIN(M, NB) )
201: $ GO TO 20
202: *
203: * K is the column to be factorized
204: * when being called from ZHETRF_AA,
205: * > for the first block column, J1 is 1, hence J1+J-1 is J,
206: * > for the rest of the columns, J1 is 2, and J1+J-1 is J+1,
207: *
208: K = J1+J-1
209: IF( J.EQ.M ) THEN
210: *
211: * Only need to compute T(J, J)
212: *
213: MJ = 1
214: ELSE
215: MJ = M-J+1
216: END IF
217: *
218: * H(J:N, J) := A(J, J:N) - H(J:N, 1:(J-1)) * L(J1:(J-1), J),
219: * where H(J:N, J) has been initialized to be A(J, J:N)
220: *
221: IF( K.GT.2 ) THEN
222: *
223: * K is the column to be factorized
224: * > for the first block column, K is J, skipping the first two
225: * columns
226: * > for the rest of the columns, K is J+1, skipping only the
227: * first column
228: *
229: CALL ZLACGV( J-K1, A( 1, J ), 1 )
230: CALL ZGEMV( 'No transpose', MJ, J-K1,
231: $ -ONE, H( J, K1 ), LDH,
232: $ A( 1, J ), 1,
233: $ ONE, H( J, J ), 1 )
234: CALL ZLACGV( J-K1, A( 1, J ), 1 )
235: END IF
236: *
237: * Copy H(i:n, i) into WORK
238: *
239: CALL ZCOPY( MJ, H( J, J ), 1, WORK( 1 ), 1 )
240: *
241: IF( J.GT.K1 ) THEN
242: *
243: * Compute WORK := WORK - L(J-1, J:N) * T(J-1,J),
244: * where A(J-1, J) stores T(J-1, J) and A(J-2, J:N) stores U(J-1, J:N)
245: *
246: ALPHA = -DCONJG( A( K-1, J ) )
247: CALL ZAXPY( MJ, ALPHA, A( K-2, J ), LDA, WORK( 1 ), 1 )
248: END IF
249: *
250: * Set A(J, J) = T(J, J)
251: *
252: A( K, J ) = DBLE( WORK( 1 ) )
253: *
254: IF( J.LT.M ) THEN
255: *
256: * Compute WORK(2:N) = T(J, J) L(J, (J+1):N)
257: * where A(J, J) stores T(J, J) and A(J-1, (J+1):N) stores U(J, (J+1):N)
258: *
259: IF( K.GT.1 ) THEN
260: ALPHA = -A( K, J )
261: CALL ZAXPY( M-J, ALPHA, A( K-1, J+1 ), LDA,
262: $ WORK( 2 ), 1 )
263: ENDIF
264: *
265: * Find max(|WORK(2:n)|)
266: *
267: I2 = IZAMAX( M-J, WORK( 2 ), 1 ) + 1
268: PIV = WORK( I2 )
269: *
270: * Apply hermitian pivot
271: *
272: IF( (I2.NE.2) .AND. (PIV.NE.0) ) THEN
273: *
274: * Swap WORK(I1) and WORK(I2)
275: *
276: I1 = 2
277: WORK( I2 ) = WORK( I1 )
278: WORK( I1 ) = PIV
279: *
280: * Swap A(I1, I1+1:N) with A(I1+1:N, I2)
281: *
282: I1 = I1+J-1
283: I2 = I2+J-1
284: CALL ZSWAP( I2-I1-1, A( J1+I1-1, I1+1 ), LDA,
285: $ A( J1+I1, I2 ), 1 )
286: CALL ZLACGV( I2-I1, A( J1+I1-1, I1+1 ), LDA )
287: CALL ZLACGV( I2-I1-1, A( J1+I1, I2 ), 1 )
288: *
289: * Swap A(I1, I2+1:N) with A(I2, I2+1:N)
290: *
291: IF( I2.LT.M )
292: $ CALL ZSWAP( M-I2, A( J1+I1-1, I2+1 ), LDA,
293: $ A( J1+I2-1, I2+1 ), LDA )
294: *
295: * Swap A(I1, I1) with A(I2,I2)
296: *
297: PIV = A( I1+J1-1, I1 )
298: A( J1+I1-1, I1 ) = A( J1+I2-1, I2 )
299: A( J1+I2-1, I2 ) = PIV
300: *
301: * Swap H(I1, 1:J1) with H(I2, 1:J1)
302: *
303: CALL ZSWAP( I1-1, H( I1, 1 ), LDH, H( I2, 1 ), LDH )
304: IPIV( I1 ) = I2
305: *
306: IF( I1.GT.(K1-1) ) THEN
307: *
308: * Swap L(1:I1-1, I1) with L(1:I1-1, I2),
309: * skipping the first column
310: *
311: CALL ZSWAP( I1-K1+1, A( 1, I1 ), 1,
312: $ A( 1, I2 ), 1 )
313: END IF
314: ELSE
315: IPIV( J+1 ) = J+1
316: ENDIF
317: *
318: * Set A(J, J+1) = T(J, J+1)
319: *
320: A( K, J+1 ) = WORK( 2 )
321: *
322: IF( J.LT.NB ) THEN
323: *
324: * Copy A(J+1:N, J+1) into H(J:N, J),
325: *
326: CALL ZCOPY( M-J, A( K+1, J+1 ), LDA,
327: $ H( J+1, J+1 ), 1 )
328: END IF
329: *
330: * Compute L(J+2, J+1) = WORK( 3:N ) / T(J, J+1),
331: * where A(J, J+1) = T(J, J+1) and A(J+2:N, J) = L(J+2:N, J+1)
332: *
333: IF( J.LT.(M-1) ) THEN
334: IF( A( K, J+1 ).NE.ZERO ) THEN
335: ALPHA = ONE / A( K, J+1 )
336: CALL ZCOPY( M-J-1, WORK( 3 ), 1, A( K, J+2 ), LDA )
337: CALL ZSCAL( M-J-1, ALPHA, A( K, J+2 ), LDA )
338: ELSE
339: CALL ZLASET( 'Full', 1, M-J-1, ZERO, ZERO,
340: $ A( K, J+2 ), LDA)
341: END IF
342: END IF
343: END IF
344: J = J + 1
345: GO TO 10
346: 20 CONTINUE
347: *
348: ELSE
349: *
350: * .....................................................
351: * Factorize A as L*D*L**T using the lower triangle of A
352: * .....................................................
353: *
354: 30 CONTINUE
355: IF( J.GT.MIN( M, NB ) )
356: $ GO TO 40
357: *
358: * K is the column to be factorized
359: * when being called from ZHETRF_AA,
360: * > for the first block column, J1 is 1, hence J1+J-1 is J,
361: * > for the rest of the columns, J1 is 2, and J1+J-1 is J+1,
362: *
363: K = J1+J-1
364: IF( J.EQ.M ) THEN
365: *
366: * Only need to compute T(J, J)
367: *
368: MJ = 1
369: ELSE
370: MJ = M-J+1
371: END IF
372: *
373: * H(J:N, J) := A(J:N, J) - H(J:N, 1:(J-1)) * L(J, J1:(J-1))^T,
374: * where H(J:N, J) has been initialized to be A(J:N, J)
375: *
376: IF( K.GT.2 ) THEN
377: *
378: * K is the column to be factorized
379: * > for the first block column, K is J, skipping the first two
380: * columns
381: * > for the rest of the columns, K is J+1, skipping only the
382: * first column
383: *
384: CALL ZLACGV( J-K1, A( J, 1 ), LDA )
385: CALL ZGEMV( 'No transpose', MJ, J-K1,
386: $ -ONE, H( J, K1 ), LDH,
387: $ A( J, 1 ), LDA,
388: $ ONE, H( J, J ), 1 )
389: CALL ZLACGV( J-K1, A( J, 1 ), LDA )
390: END IF
391: *
392: * Copy H(J:N, J) into WORK
393: *
394: CALL ZCOPY( MJ, H( J, J ), 1, WORK( 1 ), 1 )
395: *
396: IF( J.GT.K1 ) THEN
397: *
398: * Compute WORK := WORK - L(J:N, J-1) * T(J-1,J),
399: * where A(J-1, J) = T(J-1, J) and A(J, J-2) = L(J, J-1)
400: *
401: ALPHA = -DCONJG( A( J, K-1 ) )
402: CALL ZAXPY( MJ, ALPHA, A( J, K-2 ), 1, WORK( 1 ), 1 )
403: END IF
404: *
405: * Set A(J, J) = T(J, J)
406: *
407: A( J, K ) = DBLE( WORK( 1 ) )
408: *
409: IF( J.LT.M ) THEN
410: *
411: * Compute WORK(2:N) = T(J, J) L((J+1):N, J)
412: * where A(J, J) = T(J, J) and A((J+1):N, J-1) = L((J+1):N, J)
413: *
414: IF( K.GT.1 ) THEN
415: ALPHA = -A( J, K )
416: CALL ZAXPY( M-J, ALPHA, A( J+1, K-1 ), 1,
417: $ WORK( 2 ), 1 )
418: ENDIF
419: *
420: * Find max(|WORK(2:n)|)
421: *
422: I2 = IZAMAX( M-J, WORK( 2 ), 1 ) + 1
423: PIV = WORK( I2 )
424: *
425: * Apply hermitian pivot
426: *
427: IF( (I2.NE.2) .AND. (PIV.NE.0) ) THEN
428: *
429: * Swap WORK(I1) and WORK(I2)
430: *
431: I1 = 2
432: WORK( I2 ) = WORK( I1 )
433: WORK( I1 ) = PIV
434: *
435: * Swap A(I1+1:N, I1) with A(I2, I1+1:N)
436: *
437: I1 = I1+J-1
438: I2 = I2+J-1
439: CALL ZSWAP( I2-I1-1, A( I1+1, J1+I1-1 ), 1,
440: $ A( I2, J1+I1 ), LDA )
441: CALL ZLACGV( I2-I1, A( I1+1, J1+I1-1 ), 1 )
442: CALL ZLACGV( I2-I1-1, A( I2, J1+I1 ), LDA )
443: *
444: * Swap A(I2+1:N, I1) with A(I2+1:N, I2)
445: *
446: IF( I2.LT.M )
447: $ CALL ZSWAP( M-I2, A( I2+1, J1+I1-1 ), 1,
448: $ A( I2+1, J1+I2-1 ), 1 )
449: *
450: * Swap A(I1, I1) with A(I2, I2)
451: *
452: PIV = A( I1, J1+I1-1 )
453: A( I1, J1+I1-1 ) = A( I2, J1+I2-1 )
454: A( I2, J1+I2-1 ) = PIV
455: *
456: * Swap H(I1, I1:J1) with H(I2, I2:J1)
457: *
458: CALL ZSWAP( I1-1, H( I1, 1 ), LDH, H( I2, 1 ), LDH )
459: IPIV( I1 ) = I2
460: *
461: IF( I1.GT.(K1-1) ) THEN
462: *
463: * Swap L(1:I1-1, I1) with L(1:I1-1, I2),
464: * skipping the first column
465: *
466: CALL ZSWAP( I1-K1+1, A( I1, 1 ), LDA,
467: $ A( I2, 1 ), LDA )
468: END IF
469: ELSE
470: IPIV( J+1 ) = J+1
471: ENDIF
472: *
473: * Set A(J+1, J) = T(J+1, J)
474: *
475: A( J+1, K ) = WORK( 2 )
476: *
477: IF( J.LT.NB ) THEN
478: *
479: * Copy A(J+1:N, J+1) into H(J+1:N, J),
480: *
481: CALL ZCOPY( M-J, A( J+1, K+1 ), 1,
482: $ H( J+1, J+1 ), 1 )
483: END IF
484: *
485: * Compute L(J+2, J+1) = WORK( 3:N ) / T(J, J+1),
486: * where A(J, J+1) = T(J, J+1) and A(J+2:N, J) = L(J+2:N, J+1)
487: *
488: IF( J.LT.(M-1) ) THEN
489: IF( A( J+1, K ).NE.ZERO ) THEN
490: ALPHA = ONE / A( J+1, K )
491: CALL ZCOPY( M-J-1, WORK( 3 ), 1, A( J+2, K ), 1 )
492: CALL ZSCAL( M-J-1, ALPHA, A( J+2, K ), 1 )
493: ELSE
494: CALL ZLASET( 'Full', M-J-1, 1, ZERO, ZERO,
495: $ A( J+2, K ), LDA )
496: END IF
497: END IF
498: END IF
499: J = J + 1
500: GO TO 30
501: 40 CONTINUE
502: END IF
503: RETURN
504: *
505: * End of ZLAHEF_AA
506: *
507: END
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