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: *> \ingroup complex16HEcomputational
140: *
141: * =====================================================================
142: SUBROUTINE ZLAHEF_AA( UPLO, J1, M, NB, A, LDA, IPIV,
143: $ H, LDH, WORK )
144: *
145: * -- LAPACK computational routine --
146: * -- LAPACK is a software package provided by Univ. of Tennessee, --
147: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
148: *
149: IMPLICIT NONE
150: *
151: * .. Scalar Arguments ..
152: CHARACTER UPLO
153: INTEGER M, NB, J1, LDA, LDH
154: * ..
155: * .. Array Arguments ..
156: INTEGER IPIV( * )
157: COMPLEX*16 A( LDA, * ), H( LDH, * ), WORK( * )
158: * ..
159: *
160: * =====================================================================
161: * .. Parameters ..
162: COMPLEX*16 ZERO, ONE
163: PARAMETER ( ZERO = (0.0D+0, 0.0D+0), ONE = (1.0D+0, 0.0D+0) )
164: *
165: * .. Local Scalars ..
166: INTEGER J, K, K1, I1, I2, MJ
167: COMPLEX*16 PIV, ALPHA
168: * ..
169: * .. External Functions ..
170: LOGICAL LSAME
171: INTEGER IZAMAX, ILAENV
172: EXTERNAL LSAME, ILAENV, IZAMAX
173: * ..
174: * .. External Subroutines ..
175: EXTERNAL ZGEMM, ZGEMV, ZAXPY, ZLACGV, ZCOPY, ZSCAL, ZSWAP,
176: $ ZLASET, XERBLA
177: * ..
178: * .. Intrinsic Functions ..
179: INTRINSIC DBLE, DCONJG, MAX
180: * ..
181: * .. Executable Statements ..
182: *
183: J = 1
184: *
185: * K1 is the first column of the panel to be factorized
186: * i.e., K1 is 2 for the first block column, and 1 for the rest of the blocks
187: *
188: K1 = (2-J1)+1
189: *
190: IF( LSAME( UPLO, 'U' ) ) THEN
191: *
192: * .....................................................
193: * Factorize A as U**T*D*U using the upper triangle of A
194: * .....................................................
195: *
196: 10 CONTINUE
197: IF ( J.GT.MIN(M, NB) )
198: $ GO TO 20
199: *
200: * K is the column to be factorized
201: * when being called from ZHETRF_AA,
202: * > for the first block column, J1 is 1, hence J1+J-1 is J,
203: * > for the rest of the columns, J1 is 2, and J1+J-1 is J+1,
204: *
205: K = J1+J-1
206: IF( J.EQ.M ) THEN
207: *
208: * Only need to compute T(J, J)
209: *
210: MJ = 1
211: ELSE
212: MJ = M-J+1
213: END IF
214: *
215: * H(J:N, J) := A(J, J:N) - H(J:N, 1:(J-1)) * L(J1:(J-1), J),
216: * where H(J:N, J) has been initialized to be A(J, J:N)
217: *
218: IF( K.GT.2 ) THEN
219: *
220: * K is the column to be factorized
221: * > for the first block column, K is J, skipping the first two
222: * columns
223: * > for the rest of the columns, K is J+1, skipping only the
224: * first column
225: *
226: CALL ZLACGV( J-K1, A( 1, J ), 1 )
227: CALL ZGEMV( 'No transpose', MJ, J-K1,
228: $ -ONE, H( J, K1 ), LDH,
229: $ A( 1, J ), 1,
230: $ ONE, H( J, J ), 1 )
231: CALL ZLACGV( J-K1, A( 1, J ), 1 )
232: END IF
233: *
234: * Copy H(i:n, i) into WORK
235: *
236: CALL ZCOPY( MJ, H( J, J ), 1, WORK( 1 ), 1 )
237: *
238: IF( J.GT.K1 ) THEN
239: *
240: * Compute WORK := WORK - L(J-1, J:N) * T(J-1,J),
241: * where A(J-1, J) stores T(J-1, J) and A(J-2, J:N) stores U(J-1, J:N)
242: *
243: ALPHA = -DCONJG( A( K-1, J ) )
244: CALL ZAXPY( MJ, ALPHA, A( K-2, J ), LDA, WORK( 1 ), 1 )
245: END IF
246: *
247: * Set A(J, J) = T(J, J)
248: *
249: A( K, J ) = DBLE( WORK( 1 ) )
250: *
251: IF( J.LT.M ) THEN
252: *
253: * Compute WORK(2:N) = T(J, J) L(J, (J+1):N)
254: * where A(J, J) stores T(J, J) and A(J-1, (J+1):N) stores U(J, (J+1):N)
255: *
256: IF( K.GT.1 ) THEN
257: ALPHA = -A( K, J )
258: CALL ZAXPY( M-J, ALPHA, A( K-1, J+1 ), LDA,
259: $ WORK( 2 ), 1 )
260: ENDIF
261: *
262: * Find max(|WORK(2:n)|)
263: *
264: I2 = IZAMAX( M-J, WORK( 2 ), 1 ) + 1
265: PIV = WORK( I2 )
266: *
267: * Apply hermitian pivot
268: *
269: IF( (I2.NE.2) .AND. (PIV.NE.0) ) THEN
270: *
271: * Swap WORK(I1) and WORK(I2)
272: *
273: I1 = 2
274: WORK( I2 ) = WORK( I1 )
275: WORK( I1 ) = PIV
276: *
277: * Swap A(I1, I1+1:N) with A(I1+1:N, I2)
278: *
279: I1 = I1+J-1
280: I2 = I2+J-1
281: CALL ZSWAP( I2-I1-1, A( J1+I1-1, I1+1 ), LDA,
282: $ A( J1+I1, I2 ), 1 )
283: CALL ZLACGV( I2-I1, A( J1+I1-1, I1+1 ), LDA )
284: CALL ZLACGV( I2-I1-1, A( J1+I1, I2 ), 1 )
285: *
286: * Swap A(I1, I2+1:N) with A(I2, I2+1:N)
287: *
288: IF( I2.LT.M )
289: $ CALL ZSWAP( M-I2, A( J1+I1-1, I2+1 ), LDA,
290: $ A( J1+I2-1, I2+1 ), LDA )
291: *
292: * Swap A(I1, I1) with A(I2,I2)
293: *
294: PIV = A( I1+J1-1, I1 )
295: A( J1+I1-1, I1 ) = A( J1+I2-1, I2 )
296: A( J1+I2-1, I2 ) = PIV
297: *
298: * Swap H(I1, 1:J1) with H(I2, 1:J1)
299: *
300: CALL ZSWAP( I1-1, H( I1, 1 ), LDH, H( I2, 1 ), LDH )
301: IPIV( I1 ) = I2
302: *
303: IF( I1.GT.(K1-1) ) THEN
304: *
305: * Swap L(1:I1-1, I1) with L(1:I1-1, I2),
306: * skipping the first column
307: *
308: CALL ZSWAP( I1-K1+1, A( 1, I1 ), 1,
309: $ A( 1, I2 ), 1 )
310: END IF
311: ELSE
312: IPIV( J+1 ) = J+1
313: ENDIF
314: *
315: * Set A(J, J+1) = T(J, J+1)
316: *
317: A( K, J+1 ) = WORK( 2 )
318: *
319: IF( J.LT.NB ) THEN
320: *
321: * Copy A(J+1:N, J+1) into H(J:N, J),
322: *
323: CALL ZCOPY( M-J, A( K+1, J+1 ), LDA,
324: $ H( J+1, J+1 ), 1 )
325: END IF
326: *
327: * Compute L(J+2, J+1) = WORK( 3:N ) / T(J, J+1),
328: * where A(J, J+1) = T(J, J+1) and A(J+2:N, J) = L(J+2:N, J+1)
329: *
330: IF( J.LT.(M-1) ) THEN
331: IF( A( K, J+1 ).NE.ZERO ) THEN
332: ALPHA = ONE / A( K, J+1 )
333: CALL ZCOPY( M-J-1, WORK( 3 ), 1, A( K, J+2 ), LDA )
334: CALL ZSCAL( M-J-1, ALPHA, A( K, J+2 ), LDA )
335: ELSE
336: CALL ZLASET( 'Full', 1, M-J-1, ZERO, ZERO,
337: $ A( K, J+2 ), LDA)
338: END IF
339: END IF
340: END IF
341: J = J + 1
342: GO TO 10
343: 20 CONTINUE
344: *
345: ELSE
346: *
347: * .....................................................
348: * Factorize A as L*D*L**T using the lower triangle of A
349: * .....................................................
350: *
351: 30 CONTINUE
352: IF( J.GT.MIN( M, NB ) )
353: $ GO TO 40
354: *
355: * K is the column to be factorized
356: * when being called from ZHETRF_AA,
357: * > for the first block column, J1 is 1, hence J1+J-1 is J,
358: * > for the rest of the columns, J1 is 2, and J1+J-1 is J+1,
359: *
360: K = J1+J-1
361: IF( J.EQ.M ) THEN
362: *
363: * Only need to compute T(J, J)
364: *
365: MJ = 1
366: ELSE
367: MJ = M-J+1
368: END IF
369: *
370: * H(J:N, J) := A(J:N, J) - H(J:N, 1:(J-1)) * L(J, J1:(J-1))^T,
371: * where H(J:N, J) has been initialized to be A(J:N, J)
372: *
373: IF( K.GT.2 ) THEN
374: *
375: * K is the column to be factorized
376: * > for the first block column, K is J, skipping the first two
377: * columns
378: * > for the rest of the columns, K is J+1, skipping only the
379: * first column
380: *
381: CALL ZLACGV( J-K1, A( J, 1 ), LDA )
382: CALL ZGEMV( 'No transpose', MJ, J-K1,
383: $ -ONE, H( J, K1 ), LDH,
384: $ A( J, 1 ), LDA,
385: $ ONE, H( J, J ), 1 )
386: CALL ZLACGV( J-K1, A( J, 1 ), LDA )
387: END IF
388: *
389: * Copy H(J:N, J) into WORK
390: *
391: CALL ZCOPY( MJ, H( J, J ), 1, WORK( 1 ), 1 )
392: *
393: IF( J.GT.K1 ) THEN
394: *
395: * Compute WORK := WORK - L(J:N, J-1) * T(J-1,J),
396: * where A(J-1, J) = T(J-1, J) and A(J, J-2) = L(J, J-1)
397: *
398: ALPHA = -DCONJG( A( J, K-1 ) )
399: CALL ZAXPY( MJ, ALPHA, A( J, K-2 ), 1, WORK( 1 ), 1 )
400: END IF
401: *
402: * Set A(J, J) = T(J, J)
403: *
404: A( J, K ) = DBLE( WORK( 1 ) )
405: *
406: IF( J.LT.M ) THEN
407: *
408: * Compute WORK(2:N) = T(J, J) L((J+1):N, J)
409: * where A(J, J) = T(J, J) and A((J+1):N, J-1) = L((J+1):N, J)
410: *
411: IF( K.GT.1 ) THEN
412: ALPHA = -A( J, K )
413: CALL ZAXPY( M-J, ALPHA, A( J+1, K-1 ), 1,
414: $ WORK( 2 ), 1 )
415: ENDIF
416: *
417: * Find max(|WORK(2:n)|)
418: *
419: I2 = IZAMAX( M-J, WORK( 2 ), 1 ) + 1
420: PIV = WORK( I2 )
421: *
422: * Apply hermitian pivot
423: *
424: IF( (I2.NE.2) .AND. (PIV.NE.0) ) THEN
425: *
426: * Swap WORK(I1) and WORK(I2)
427: *
428: I1 = 2
429: WORK( I2 ) = WORK( I1 )
430: WORK( I1 ) = PIV
431: *
432: * Swap A(I1+1:N, I1) with A(I2, I1+1:N)
433: *
434: I1 = I1+J-1
435: I2 = I2+J-1
436: CALL ZSWAP( I2-I1-1, A( I1+1, J1+I1-1 ), 1,
437: $ A( I2, J1+I1 ), LDA )
438: CALL ZLACGV( I2-I1, A( I1+1, J1+I1-1 ), 1 )
439: CALL ZLACGV( I2-I1-1, A( I2, J1+I1 ), LDA )
440: *
441: * Swap A(I2+1:N, I1) with A(I2+1:N, I2)
442: *
443: IF( I2.LT.M )
444: $ CALL ZSWAP( M-I2, A( I2+1, J1+I1-1 ), 1,
445: $ A( I2+1, J1+I2-1 ), 1 )
446: *
447: * Swap A(I1, I1) with A(I2, I2)
448: *
449: PIV = A( I1, J1+I1-1 )
450: A( I1, J1+I1-1 ) = A( I2, J1+I2-1 )
451: A( I2, J1+I2-1 ) = PIV
452: *
453: * Swap H(I1, I1:J1) with H(I2, I2:J1)
454: *
455: CALL ZSWAP( I1-1, H( I1, 1 ), LDH, H( I2, 1 ), LDH )
456: IPIV( I1 ) = I2
457: *
458: IF( I1.GT.(K1-1) ) THEN
459: *
460: * Swap L(1:I1-1, I1) with L(1:I1-1, I2),
461: * skipping the first column
462: *
463: CALL ZSWAP( I1-K1+1, A( I1, 1 ), LDA,
464: $ A( I2, 1 ), LDA )
465: END IF
466: ELSE
467: IPIV( J+1 ) = J+1
468: ENDIF
469: *
470: * Set A(J+1, J) = T(J+1, J)
471: *
472: A( J+1, K ) = WORK( 2 )
473: *
474: IF( J.LT.NB ) THEN
475: *
476: * Copy A(J+1:N, J+1) into H(J+1:N, J),
477: *
478: CALL ZCOPY( M-J, A( J+1, K+1 ), 1,
479: $ H( J+1, J+1 ), 1 )
480: END IF
481: *
482: * Compute L(J+2, J+1) = WORK( 3:N ) / T(J, J+1),
483: * where A(J, J+1) = T(J, J+1) and A(J+2:N, J) = L(J+2:N, J+1)
484: *
485: IF( J.LT.(M-1) ) THEN
486: IF( A( J+1, K ).NE.ZERO ) THEN
487: ALPHA = ONE / A( J+1, K )
488: CALL ZCOPY( M-J-1, WORK( 3 ), 1, A( J+2, K ), 1 )
489: CALL ZSCAL( M-J-1, ALPHA, A( J+2, K ), 1 )
490: ELSE
491: CALL ZLASET( 'Full', M-J-1, 1, ZERO, ZERO,
492: $ A( J+2, K ), LDA )
493: END IF
494: END IF
495: END IF
496: J = J + 1
497: GO TO 30
498: 40 CONTINUE
499: END IF
500: RETURN
501: *
502: * End of ZLAHEF_AA
503: *
504: END
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