1: *> \brief \b ZSYTRF_AA
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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9: *> Download ZSYTRF_AA + dependencies
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14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zsytrf_aa.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZSYTRF_AA( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER N, LDA, LWORK, INFO
26: * ..
27: * .. Array Arguments ..
28: * INTEGER IPIV( * )
29: * COMPLEX*16 A( LDA, * ), WORK( * )
30: * ..
31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> ZSYTRF_AA computes the factorization of a complex symmetric matrix A
38: *> using the Aasen's algorithm. The form of the factorization is
39: *>
40: *> A = U**T*T*U or A = L*T*L**T
41: *>
42: *> where U (or L) is a product of permutation and unit upper (lower)
43: *> triangular matrices, and T is a complex symmetric tridiagonal matrix.
44: *>
45: *> This is the blocked version of the algorithm, calling Level 3 BLAS.
46: *> \endverbatim
47: *
48: * Arguments:
49: * ==========
50: *
51: *> \param[in] UPLO
52: *> \verbatim
53: *> UPLO is CHARACTER*1
54: *> = 'U': Upper triangle of A is stored;
55: *> = 'L': Lower triangle of A is stored.
56: *> \endverbatim
57: *>
58: *> \param[in] N
59: *> \verbatim
60: *> N is INTEGER
61: *> The order of the matrix A. N >= 0.
62: *> \endverbatim
63: *>
64: *> \param[in,out] A
65: *> \verbatim
66: *> A is COMPLEX*16 array, dimension (LDA,N)
67: *> On entry, the symmetric matrix A. If UPLO = 'U', the leading
68: *> N-by-N upper triangular part of A contains the upper
69: *> triangular part of the matrix A, and the strictly lower
70: *> triangular part of A is not referenced. If UPLO = 'L', the
71: *> leading N-by-N lower triangular part of A contains the lower
72: *> triangular part of the matrix A, and the strictly upper
73: *> triangular part of A is not referenced.
74: *>
75: *> On exit, the tridiagonal matrix is stored in the diagonals
76: *> and the subdiagonals of A just below (or above) the diagonals,
77: *> and L is stored below (or above) the subdiaonals, when UPLO
78: *> is 'L' (or 'U').
79: *> \endverbatim
80: *>
81: *> \param[in] LDA
82: *> \verbatim
83: *> LDA is INTEGER
84: *> The leading dimension of the array A. LDA >= max(1,N).
85: *> \endverbatim
86: *>
87: *> \param[out] IPIV
88: *> \verbatim
89: *> IPIV is INTEGER array, dimension (N)
90: *> On exit, it contains the details of the interchanges, i.e.,
91: *> the row and column k of A were interchanged with the
92: *> row and column IPIV(k).
93: *> \endverbatim
94: *>
95: *> \param[out] WORK
96: *> \verbatim
97: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
98: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
99: *> \endverbatim
100: *>
101: *> \param[in] LWORK
102: *> \verbatim
103: *> LWORK is INTEGER
104: *> The length of WORK. LWORK >=MAX(1,2*N). For optimum performance
105: *> LWORK >= N*(1+NB), where NB is the optimal blocksize.
106: *>
107: *> If LWORK = -1, then a workspace query is assumed; the routine
108: *> only calculates the optimal size of the WORK array, returns
109: *> this value as the first entry of the WORK array, and no error
110: *> message related to LWORK is issued by XERBLA.
111: *> \endverbatim
112: *>
113: *> \param[out] INFO
114: *> \verbatim
115: *> INFO is INTEGER
116: *> = 0: successful exit
117: *> < 0: if INFO = -i, the i-th argument had an illegal value.
118: *> \endverbatim
119: *
120: * Authors:
121: * ========
122: *
123: *> \author Univ. of Tennessee
124: *> \author Univ. of California Berkeley
125: *> \author Univ. of Colorado Denver
126: *> \author NAG Ltd.
127: *
128: *> \ingroup complex16SYcomputational
129: *
130: * =====================================================================
131: SUBROUTINE ZSYTRF_AA( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)
132: *
133: * -- LAPACK computational routine --
134: * -- LAPACK is a software package provided by Univ. of Tennessee, --
135: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
136: *
137: IMPLICIT NONE
138: *
139: * .. Scalar Arguments ..
140: CHARACTER UPLO
141: INTEGER N, LDA, LWORK, INFO
142: * ..
143: * .. Array Arguments ..
144: INTEGER IPIV( * )
145: COMPLEX*16 A( LDA, * ), WORK( * )
146: * ..
147: *
148: * =====================================================================
149: * .. Parameters ..
150: COMPLEX*16 ZERO, ONE
151: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
152: *
153: * .. Local Scalars ..
154: LOGICAL LQUERY, UPPER
155: INTEGER J, LWKOPT
156: INTEGER NB, MJ, NJ, K1, K2, J1, J2, J3, JB
157: COMPLEX*16 ALPHA
158: * ..
159: * .. External Functions ..
160: LOGICAL LSAME
161: INTEGER ILAENV
162: EXTERNAL LSAME, ILAENV
163: * ..
164: * .. External Subroutines ..
165: EXTERNAL ZLASYF_AA, ZGEMM, ZGEMV, ZSCAL, ZCOPY,
166: $ ZSWAP, XERBLA
167: * ..
168: * .. Intrinsic Functions ..
169: INTRINSIC MAX
170: * ..
171: * .. Executable Statements ..
172: *
173: * Determine the block size
174: *
175: NB = ILAENV( 1, 'ZSYTRF_AA', UPLO, N, -1, -1, -1 )
176: *
177: * Test the input parameters.
178: *
179: INFO = 0
180: UPPER = LSAME( UPLO, 'U' )
181: LQUERY = ( LWORK.EQ.-1 )
182: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
183: INFO = -1
184: ELSE IF( N.LT.0 ) THEN
185: INFO = -2
186: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
187: INFO = -4
188: ELSE IF( LWORK.LT.MAX( 1, 2*N ) .AND. .NOT.LQUERY ) THEN
189: INFO = -7
190: END IF
191: *
192: IF( INFO.EQ.0 ) THEN
193: LWKOPT = (NB+1)*N
194: WORK( 1 ) = LWKOPT
195: END IF
196: *
197: IF( INFO.NE.0 ) THEN
198: CALL XERBLA( 'ZSYTRF_AA', -INFO )
199: RETURN
200: ELSE IF( LQUERY ) THEN
201: RETURN
202: END IF
203: *
204: * Quick return
205: *
206: IF ( N.EQ.0 ) THEN
207: RETURN
208: ENDIF
209: IPIV( 1 ) = 1
210: IF ( N.EQ.1 ) THEN
211: RETURN
212: END IF
213: *
214: * Adjust block size based on the workspace size
215: *
216: IF( LWORK.LT.((1+NB)*N) ) THEN
217: NB = ( LWORK-N ) / N
218: END IF
219: *
220: IF( UPPER ) THEN
221: *
222: * .....................................................
223: * Factorize A as U**T*D*U using the upper triangle of A
224: * .....................................................
225: *
226: * Copy first row A(1, 1:N) into H(1:n) (stored in WORK(1:N))
227: *
228: CALL ZCOPY( N, A( 1, 1 ), LDA, WORK( 1 ), 1 )
229: *
230: * J is the main loop index, increasing from 1 to N in steps of
231: * JB, where JB is the number of columns factorized by ZLASYF;
232: * JB is either NB, or N-J+1 for the last block
233: *
234: J = 0
235: 10 CONTINUE
236: IF( J.GE.N )
237: $ GO TO 20
238: *
239: * each step of the main loop
240: * J is the last column of the previous panel
241: * J1 is the first column of the current panel
242: * K1 identifies if the previous column of the panel has been
243: * explicitly stored, e.g., K1=1 for the first panel, and
244: * K1=0 for the rest
245: *
246: J1 = J + 1
247: JB = MIN( N-J1+1, NB )
248: K1 = MAX(1, J)-J
249: *
250: * Panel factorization
251: *
252: CALL ZLASYF_AA( UPLO, 2-K1, N-J, JB,
253: $ A( MAX(1, J), J+1 ), LDA,
254: $ IPIV( J+1 ), WORK, N, WORK( N*NB+1 ) )
255: *
256: * Adjust IPIV and apply it back (J-th step picks (J+1)-th pivot)
257: *
258: DO J2 = J+2, MIN(N, J+JB+1)
259: IPIV( J2 ) = IPIV( J2 ) + J
260: IF( (J2.NE.IPIV(J2)) .AND. ((J1-K1).GT.2) ) THEN
261: CALL ZSWAP( J1-K1-2, A( 1, J2 ), 1,
262: $ A( 1, IPIV(J2) ), 1 )
263: END IF
264: END DO
265: J = J + JB
266: *
267: * Trailing submatrix update, where
268: * the row A(J1-1, J2-1:N) stores U(J1, J2+1:N) and
269: * WORK stores the current block of the auxiriarly matrix H
270: *
271: IF( J.LT.N ) THEN
272: *
273: * If first panel and JB=1 (NB=1), then nothing to do
274: *
275: IF( J1.GT.1 .OR. JB.GT.1 ) THEN
276: *
277: * Merge rank-1 update with BLAS-3 update
278: *
279: ALPHA = A( J, J+1 )
280: A( J, J+1 ) = ONE
281: CALL ZCOPY( N-J, A( J-1, J+1 ), LDA,
282: $ WORK( (J+1-J1+1)+JB*N ), 1 )
283: CALL ZSCAL( N-J, ALPHA, WORK( (J+1-J1+1)+JB*N ), 1 )
284: *
285: * K1 identifies if the previous column of the panel has been
286: * explicitly stored, e.g., K1=1 and K2= 0 for the first panel,
287: * while K1=0 and K2=1 for the rest
288: *
289: IF( J1.GT.1 ) THEN
290: *
291: * Not first panel
292: *
293: K2 = 1
294: ELSE
295: *
296: * First panel
297: *
298: K2 = 0
299: *
300: * First update skips the first column
301: *
302: JB = JB - 1
303: END IF
304: *
305: DO J2 = J+1, N, NB
306: NJ = MIN( NB, N-J2+1 )
307: *
308: * Update (J2, J2) diagonal block with ZGEMV
309: *
310: J3 = J2
311: DO MJ = NJ-1, 1, -1
312: CALL ZGEMV( 'No transpose', MJ, JB+1,
313: $ -ONE, WORK( J3-J1+1+K1*N ), N,
314: $ A( J1-K2, J3 ), 1,
315: $ ONE, A( J3, J3 ), LDA )
316: J3 = J3 + 1
317: END DO
318: *
319: * Update off-diagonal block of J2-th block row with ZGEMM
320: *
321: CALL ZGEMM( 'Transpose', 'Transpose',
322: $ NJ, N-J3+1, JB+1,
323: $ -ONE, A( J1-K2, J2 ), LDA,
324: $ WORK( J3-J1+1+K1*N ), N,
325: $ ONE, A( J2, J3 ), LDA )
326: END DO
327: *
328: * Recover T( J, J+1 )
329: *
330: A( J, J+1 ) = ALPHA
331: END IF
332: *
333: * WORK(J+1, 1) stores H(J+1, 1)
334: *
335: CALL ZCOPY( N-J, A( J+1, J+1 ), LDA, WORK( 1 ), 1 )
336: END IF
337: GO TO 10
338: ELSE
339: *
340: * .....................................................
341: * Factorize A as L*D*L**T using the lower triangle of A
342: * .....................................................
343: *
344: * copy first column A(1:N, 1) into H(1:N, 1)
345: * (stored in WORK(1:N))
346: *
347: CALL ZCOPY( N, A( 1, 1 ), 1, WORK( 1 ), 1 )
348: *
349: * J is the main loop index, increasing from 1 to N in steps of
350: * JB, where JB is the number of columns factorized by ZLASYF;
351: * JB is either NB, or N-J+1 for the last block
352: *
353: J = 0
354: 11 CONTINUE
355: IF( J.GE.N )
356: $ GO TO 20
357: *
358: * each step of the main loop
359: * J is the last column of the previous panel
360: * J1 is the first column of the current panel
361: * K1 identifies if the previous column of the panel has been
362: * explicitly stored, e.g., K1=1 for the first panel, and
363: * K1=0 for the rest
364: *
365: J1 = J+1
366: JB = MIN( N-J1+1, NB )
367: K1 = MAX(1, J)-J
368: *
369: * Panel factorization
370: *
371: CALL ZLASYF_AA( UPLO, 2-K1, N-J, JB,
372: $ A( J+1, MAX(1, J) ), LDA,
373: $ IPIV( J+1 ), WORK, N, WORK( N*NB+1 ) )
374: *
375: * Adjust IPIV and apply it back (J-th step picks (J+1)-th pivot)
376: *
377: DO J2 = J+2, MIN(N, J+JB+1)
378: IPIV( J2 ) = IPIV( J2 ) + J
379: IF( (J2.NE.IPIV(J2)) .AND. ((J1-K1).GT.2) ) THEN
380: CALL ZSWAP( J1-K1-2, A( J2, 1 ), LDA,
381: $ A( IPIV(J2), 1 ), LDA )
382: END IF
383: END DO
384: J = J + JB
385: *
386: * Trailing submatrix update, where
387: * A(J2+1, J1-1) stores L(J2+1, J1) and
388: * WORK(J2+1, 1) stores H(J2+1, 1)
389: *
390: IF( J.LT.N ) THEN
391: *
392: * if first panel and JB=1 (NB=1), then nothing to do
393: *
394: IF( J1.GT.1 .OR. JB.GT.1 ) THEN
395: *
396: * Merge rank-1 update with BLAS-3 update
397: *
398: ALPHA = A( J+1, J )
399: A( J+1, J ) = ONE
400: CALL ZCOPY( N-J, A( J+1, J-1 ), 1,
401: $ WORK( (J+1-J1+1)+JB*N ), 1 )
402: CALL ZSCAL( N-J, ALPHA, WORK( (J+1-J1+1)+JB*N ), 1 )
403: *
404: * K1 identifies if the previous column of the panel has been
405: * explicitly stored, e.g., K1=1 and K2= 0 for the first panel,
406: * while K1=0 and K2=1 for the rest
407: *
408: IF( J1.GT.1 ) THEN
409: *
410: * Not first panel
411: *
412: K2 = 1
413: ELSE
414: *
415: * First panel
416: *
417: K2 = 0
418: *
419: * First update skips the first column
420: *
421: JB = JB - 1
422: END IF
423: *
424: DO J2 = J+1, N, NB
425: NJ = MIN( NB, N-J2+1 )
426: *
427: * Update (J2, J2) diagonal block with ZGEMV
428: *
429: J3 = J2
430: DO MJ = NJ-1, 1, -1
431: CALL ZGEMV( 'No transpose', MJ, JB+1,
432: $ -ONE, WORK( J3-J1+1+K1*N ), N,
433: $ A( J3, J1-K2 ), LDA,
434: $ ONE, A( J3, J3 ), 1 )
435: J3 = J3 + 1
436: END DO
437: *
438: * Update off-diagonal block in J2-th block column with ZGEMM
439: *
440: CALL ZGEMM( 'No transpose', 'Transpose',
441: $ N-J3+1, NJ, JB+1,
442: $ -ONE, WORK( J3-J1+1+K1*N ), N,
443: $ A( J2, J1-K2 ), LDA,
444: $ ONE, A( J3, J2 ), LDA )
445: END DO
446: *
447: * Recover T( J+1, J )
448: *
449: A( J+1, J ) = ALPHA
450: END IF
451: *
452: * WORK(J+1, 1) stores H(J+1, 1)
453: *
454: CALL ZCOPY( N-J, A( J+1, J+1 ), 1, WORK( 1 ), 1 )
455: END IF
456: GO TO 11
457: END IF
458: *
459: 20 CONTINUE
460: WORK( 1 ) = LWKOPT
461: RETURN
462: *
463: * End of ZSYTRF_AA
464: *
465: END
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