1: *> \brief \b ZHETRF_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ("rook") 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 ZHETRF_ROOK + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhetrf_rook.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhetrf_rook.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhetrf_rook.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZHETRF_ROOK( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, LDA, LWORK, N
26: * ..
27: * .. Array Arguments ..
28: * INTEGER IPIV( * )
29: * COMPLEX*16 A( LDA, * ), WORK( * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> ZHETRF_ROOK computes the factorization of a complex Hermitian matrix A
39: *> using the bounded Bunch-Kaufman ("rook") diagonal pivoting method.
40: *> The form of the factorization is
41: *>
42: *> A = U*D*U**T or A = L*D*L**T
43: *>
44: *> where U (or L) is a product of permutation and unit upper (lower)
45: *> triangular matrices, and D is Hermitian and block diagonal with
46: *> 1-by-1 and 2-by-2 diagonal blocks.
47: *>
48: *> This is the blocked version of the algorithm, calling Level 3 BLAS.
49: *> \endverbatim
50: *
51: * Arguments:
52: * ==========
53: *
54: *> \param[in] UPLO
55: *> \verbatim
56: *> UPLO is CHARACTER*1
57: *> = 'U': Upper triangle of A is stored;
58: *> = 'L': Lower triangle of A is stored.
59: *> \endverbatim
60: *>
61: *> \param[in] N
62: *> \verbatim
63: *> N is INTEGER
64: *> The order of the matrix A. N >= 0.
65: *> \endverbatim
66: *>
67: *> \param[in,out] A
68: *> \verbatim
69: *> A is COMPLEX*16 array, dimension (LDA,N)
70: *> On entry, the Hermitian matrix A. If UPLO = 'U', the leading
71: *> N-by-N upper triangular part of A contains the upper
72: *> triangular part of the matrix A, and the strictly lower
73: *> triangular part of A is not referenced. If UPLO = 'L', the
74: *> leading N-by-N lower triangular part of A contains the lower
75: *> triangular part of the matrix A, and the strictly upper
76: *> triangular part of A is not referenced.
77: *>
78: *> On exit, the block diagonal matrix D and the multipliers used
79: *> to obtain the factor U or L (see below for further details).
80: *> \endverbatim
81: *>
82: *> \param[in] LDA
83: *> \verbatim
84: *> LDA is INTEGER
85: *> The leading dimension of the array A. LDA >= max(1,N).
86: *> \endverbatim
87: *>
88: *> \param[out] IPIV
89: *> \verbatim
90: *> IPIV is INTEGER array, dimension (N)
91: *> Details of the interchanges and the block structure of D.
92: *>
93: *> If UPLO = 'U':
94: *> Only the last KB elements of IPIV are set.
95: *>
96: *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
97: *> interchanged and D(k,k) is a 1-by-1 diagonal block.
98: *>
99: *> If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
100: *> columns k and -IPIV(k) were interchanged and rows and
101: *> columns k-1 and -IPIV(k-1) were inerchaged,
102: *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
103: *>
104: *> If UPLO = 'L':
105: *> Only the first KB elements of IPIV are set.
106: *>
107: *> If IPIV(k) > 0, then rows and columns k and IPIV(k)
108: *> were interchanged and D(k,k) is a 1-by-1 diagonal block.
109: *>
110: *> If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
111: *> columns k and -IPIV(k) were interchanged and rows and
112: *> columns k+1 and -IPIV(k+1) were inerchaged,
113: *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
114: *> \endverbatim
115: *>
116: *> \param[out] WORK
117: *> \verbatim
118: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)).
119: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
120: *> \endverbatim
121: *>
122: *> \param[in] LWORK
123: *> \verbatim
124: *> LWORK is INTEGER
125: *> The length of WORK. LWORK >=1. For best performance
126: *> LWORK >= N*NB, where NB is the block size returned by ILAENV.
127: *>
128: *> If LWORK = -1, then a workspace query is assumed; the routine
129: *> only calculates the optimal size of the WORK array, returns
130: *> this value as the first entry of the WORK array, and no error
131: *> message related to LWORK is issued by XERBLA.
132: *> \endverbatim
133: *>
134: *> \param[out] INFO
135: *> \verbatim
136: *> INFO is INTEGER
137: *> = 0: successful exit
138: *> < 0: if INFO = -i, the i-th argument had an illegal value
139: *> > 0: if INFO = i, D(i,i) is exactly zero. The factorization
140: *> has been completed, but the block diagonal matrix D is
141: *> exactly singular, and division by zero will occur if it
142: *> is used to solve a system of equations.
143: *> \endverbatim
144: *
145: * Authors:
146: * ========
147: *
148: *> \author Univ. of Tennessee
149: *> \author Univ. of California Berkeley
150: *> \author Univ. of Colorado Denver
151: *> \author NAG Ltd.
152: *
153: *> \date June 2016
154: *
155: *> \ingroup complex16HEcomputational
156: *
157: *> \par Further Details:
158: * =====================
159: *>
160: *> \verbatim
161: *>
162: *> If UPLO = 'U', then A = U*D*U**T, where
163: *> U = P(n)*U(n)* ... *P(k)U(k)* ...,
164: *> i.e., U is a product of terms P(k)*U(k), where k decreases from n to
165: *> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
166: *> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
167: *> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
168: *> that if the diagonal block D(k) is of order s (s = 1 or 2), then
169: *>
170: *> ( I v 0 ) k-s
171: *> U(k) = ( 0 I 0 ) s
172: *> ( 0 0 I ) n-k
173: *> k-s s n-k
174: *>
175: *> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
176: *> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
177: *> and A(k,k), and v overwrites A(1:k-2,k-1:k).
178: *>
179: *> If UPLO = 'L', then A = L*D*L**T, where
180: *> L = P(1)*L(1)* ... *P(k)*L(k)* ...,
181: *> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
182: *> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
183: *> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
184: *> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
185: *> that if the diagonal block D(k) is of order s (s = 1 or 2), then
186: *>
187: *> ( I 0 0 ) k-1
188: *> L(k) = ( 0 I 0 ) s
189: *> ( 0 v I ) n-k-s+1
190: *> k-1 s n-k-s+1
191: *>
192: *> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
193: *> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
194: *> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
195: *> \endverbatim
196: *
197: *> \par Contributors:
198: * ==================
199: *>
200: *> \verbatim
201: *>
202: *> June 2016, Igor Kozachenko,
203: *> Computer Science Division,
204: *> University of California, Berkeley
205: *>
206: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
207: *> School of Mathematics,
208: *> University of Manchester
209: *>
210: *> \endverbatim
211: *
212: * =====================================================================
213: SUBROUTINE ZHETRF_ROOK( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
214: *
215: * -- LAPACK computational routine (version 3.6.1) --
216: * -- LAPACK is a software package provided by Univ. of Tennessee, --
217: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
218: * June 2016
219: *
220: * .. Scalar Arguments ..
221: CHARACTER UPLO
222: INTEGER INFO, LDA, LWORK, N
223: * ..
224: * .. Array Arguments ..
225: INTEGER IPIV( * )
226: COMPLEX*16 A( LDA, * ), WORK( * )
227: * ..
228: *
229: * =====================================================================
230: *
231: * .. Local Scalars ..
232: LOGICAL LQUERY, UPPER
233: INTEGER IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
234: * ..
235: * .. External Functions ..
236: LOGICAL LSAME
237: INTEGER ILAENV
238: EXTERNAL LSAME, ILAENV
239: * ..
240: * .. External Subroutines ..
241: EXTERNAL ZLAHEF_ROOK, ZHETF2_ROOK, XERBLA
242: * ..
243: * .. Intrinsic Functions ..
244: INTRINSIC MAX
245: * ..
246: * .. Executable Statements ..
247: *
248: * Test the input parameters.
249: *
250: INFO = 0
251: UPPER = LSAME( UPLO, 'U' )
252: LQUERY = ( LWORK.EQ.-1 )
253: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
254: INFO = -1
255: ELSE IF( N.LT.0 ) THEN
256: INFO = -2
257: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
258: INFO = -4
259: ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
260: INFO = -7
261: END IF
262: *
263: IF( INFO.EQ.0 ) THEN
264: *
265: * Determine the block size
266: *
267: NB = ILAENV( 1, 'ZHETRF_ROOK', UPLO, N, -1, -1, -1 )
268: LWKOPT = MAX( 1, N*NB )
269: WORK( 1 ) = LWKOPT
270: END IF
271: *
272: IF( INFO.NE.0 ) THEN
273: CALL XERBLA( 'ZHETRF_ROOK', -INFO )
274: RETURN
275: ELSE IF( LQUERY ) THEN
276: RETURN
277: END IF
278: *
279: NBMIN = 2
280: LDWORK = N
281: IF( NB.GT.1 .AND. NB.LT.N ) THEN
282: IWS = LDWORK*NB
283: IF( LWORK.LT.IWS ) THEN
284: NB = MAX( LWORK / LDWORK, 1 )
285: NBMIN = MAX( 2, ILAENV( 2, 'ZHETRF_ROOK',
286: $ UPLO, N, -1, -1, -1 ) )
287: END IF
288: ELSE
289: IWS = 1
290: END IF
291: IF( NB.LT.NBMIN )
292: $ NB = N
293: *
294: IF( UPPER ) THEN
295: *
296: * Factorize A as U*D*U**T using the upper triangle of A
297: *
298: * K is the main loop index, decreasing from N to 1 in steps of
299: * KB, where KB is the number of columns factorized by ZLAHEF_ROOK;
300: * KB is either NB or NB-1, or K for the last block
301: *
302: K = N
303: 10 CONTINUE
304: *
305: * If K < 1, exit from loop
306: *
307: IF( K.LT.1 )
308: $ GO TO 40
309: *
310: IF( K.GT.NB ) THEN
311: *
312: * Factorize columns k-kb+1:k of A and use blocked code to
313: * update columns 1:k-kb
314: *
315: CALL ZLAHEF_ROOK( UPLO, K, NB, KB, A, LDA,
316: $ IPIV, WORK, LDWORK, IINFO )
317: ELSE
318: *
319: * Use unblocked code to factorize columns 1:k of A
320: *
321: CALL ZHETF2_ROOK( UPLO, K, A, LDA, IPIV, IINFO )
322: KB = K
323: END IF
324: *
325: * Set INFO on the first occurrence of a zero pivot
326: *
327: IF( INFO.EQ.0 .AND. IINFO.GT.0 )
328: $ INFO = IINFO
329: *
330: * No need to adjust IPIV
331: *
332: * Decrease K and return to the start of the main loop
333: *
334: K = K - KB
335: GO TO 10
336: *
337: ELSE
338: *
339: * Factorize A as L*D*L**T using the lower triangle of A
340: *
341: * K is the main loop index, increasing from 1 to N in steps of
342: * KB, where KB is the number of columns factorized by ZLAHEF_ROOK;
343: * KB is either NB or NB-1, or N-K+1 for the last block
344: *
345: K = 1
346: 20 CONTINUE
347: *
348: * If K > N, exit from loop
349: *
350: IF( K.GT.N )
351: $ GO TO 40
352: *
353: IF( K.LE.N-NB ) THEN
354: *
355: * Factorize columns k:k+kb-1 of A and use blocked code to
356: * update columns k+kb:n
357: *
358: CALL ZLAHEF_ROOK( UPLO, N-K+1, NB, KB, A( K, K ), LDA,
359: $ IPIV( K ), WORK, LDWORK, IINFO )
360: ELSE
361: *
362: * Use unblocked code to factorize columns k:n of A
363: *
364: CALL ZHETF2_ROOK( UPLO, N-K+1, A( K, K ), LDA, IPIV( K ),
365: $ IINFO )
366: KB = N - K + 1
367: END IF
368: *
369: * Set INFO on the first occurrence of a zero pivot
370: *
371: IF( INFO.EQ.0 .AND. IINFO.GT.0 )
372: $ INFO = IINFO + K - 1
373: *
374: * Adjust IPIV
375: *
376: DO 30 J = K, K + KB - 1
377: IF( IPIV( J ).GT.0 ) THEN
378: IPIV( J ) = IPIV( J ) + K - 1
379: ELSE
380: IPIV( J ) = IPIV( J ) - K + 1
381: END IF
382: 30 CONTINUE
383: *
384: * Increase K and return to the start of the main loop
385: *
386: K = K + KB
387: GO TO 20
388: *
389: END IF
390: *
391: 40 CONTINUE
392: WORK( 1 ) = LWKOPT
393: RETURN
394: *
395: * End of ZHETRF_ROOK
396: *
397: END
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