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1.8 bertrand 1: *> \brief \b ZPBTRF
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.14 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.8 bertrand 7: *
8: *> \htmlonly
1.14 bertrand 9: *> Download ZPBTRF + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zpbtrf.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zpbtrf.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zpbtrf.f">
1.8 bertrand 15: *> [TXT]</a>
1.14 bertrand 16: *> \endhtmlonly
1.8 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZPBTRF( UPLO, N, KD, AB, LDAB, INFO )
1.14 bertrand 22: *
1.8 bertrand 23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, KD, LDAB, N
26: * ..
27: * .. Array Arguments ..
28: * COMPLEX*16 AB( LDAB, * )
29: * ..
1.14 bertrand 30: *
1.8 bertrand 31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> ZPBTRF computes the Cholesky factorization of a complex Hermitian
38: *> positive definite band matrix A.
39: *>
40: *> The factorization has the form
41: *> A = U**H * U, if UPLO = 'U', or
42: *> A = L * L**H, if UPLO = 'L',
43: *> where U is an upper triangular matrix and L is lower triangular.
44: *> \endverbatim
45: *
46: * Arguments:
47: * ==========
48: *
49: *> \param[in] UPLO
50: *> \verbatim
51: *> UPLO is CHARACTER*1
52: *> = 'U': Upper triangle of A is stored;
53: *> = 'L': Lower triangle of A is stored.
54: *> \endverbatim
55: *>
56: *> \param[in] N
57: *> \verbatim
58: *> N is INTEGER
59: *> The order of the matrix A. N >= 0.
60: *> \endverbatim
61: *>
62: *> \param[in] KD
63: *> \verbatim
64: *> KD is INTEGER
65: *> The number of superdiagonals of the matrix A if UPLO = 'U',
66: *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
67: *> \endverbatim
68: *>
69: *> \param[in,out] AB
70: *> \verbatim
71: *> AB is COMPLEX*16 array, dimension (LDAB,N)
72: *> On entry, the upper or lower triangle of the Hermitian band
73: *> matrix A, stored in the first KD+1 rows of the array. The
74: *> j-th column of A is stored in the j-th column of the array AB
75: *> as follows:
76: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
77: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
78: *>
79: *> On exit, if INFO = 0, the triangular factor U or L from the
80: *> Cholesky factorization A = U**H*U or A = L*L**H of the band
81: *> matrix A, in the same storage format as A.
82: *> \endverbatim
83: *>
84: *> \param[in] LDAB
85: *> \verbatim
86: *> LDAB is INTEGER
87: *> The leading dimension of the array AB. LDAB >= KD+1.
88: *> \endverbatim
89: *>
90: *> \param[out] INFO
91: *> \verbatim
92: *> INFO is INTEGER
93: *> = 0: successful exit
94: *> < 0: if INFO = -i, the i-th argument had an illegal value
95: *> > 0: if INFO = i, the leading minor of order i is not
96: *> positive definite, and the factorization could not be
97: *> completed.
98: *> \endverbatim
99: *
100: * Authors:
101: * ========
102: *
1.14 bertrand 103: *> \author Univ. of Tennessee
104: *> \author Univ. of California Berkeley
105: *> \author Univ. of Colorado Denver
106: *> \author NAG Ltd.
1.8 bertrand 107: *
108: *> \ingroup complex16OTHERcomputational
109: *
110: *> \par Further Details:
111: * =====================
112: *>
113: *> \verbatim
114: *>
115: *> The band storage scheme is illustrated by the following example, when
116: *> N = 6, KD = 2, and UPLO = 'U':
117: *>
118: *> On entry: On exit:
119: *>
120: *> * * a13 a24 a35 a46 * * u13 u24 u35 u46
121: *> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
122: *> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
123: *>
124: *> Similarly, if UPLO = 'L' the format of A is as follows:
125: *>
126: *> On entry: On exit:
127: *>
128: *> a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
129: *> a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
130: *> a31 a42 a53 a64 * * l31 l42 l53 l64 * *
131: *>
132: *> Array elements marked * are not used by the routine.
133: *> \endverbatim
134: *
135: *> \par Contributors:
136: * ==================
137: *>
138: *> Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
139: *
140: * =====================================================================
1.1 bertrand 141: SUBROUTINE ZPBTRF( UPLO, N, KD, AB, LDAB, INFO )
142: *
1.17 ! bertrand 143: * -- LAPACK computational routine --
1.1 bertrand 144: * -- LAPACK is a software package provided by Univ. of Tennessee, --
145: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
146: *
147: * .. Scalar Arguments ..
148: CHARACTER UPLO
149: INTEGER INFO, KD, LDAB, N
150: * ..
151: * .. Array Arguments ..
152: COMPLEX*16 AB( LDAB, * )
153: * ..
154: *
155: * =====================================================================
156: *
157: * .. Parameters ..
158: DOUBLE PRECISION ONE, ZERO
159: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
160: COMPLEX*16 CONE
161: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
162: INTEGER NBMAX, LDWORK
163: PARAMETER ( NBMAX = 32, LDWORK = NBMAX+1 )
164: * ..
165: * .. Local Scalars ..
166: INTEGER I, I2, I3, IB, II, J, JJ, NB
167: * ..
168: * .. Local Arrays ..
169: COMPLEX*16 WORK( LDWORK, NBMAX )
170: * ..
171: * .. External Functions ..
172: LOGICAL LSAME
173: INTEGER ILAENV
174: EXTERNAL LSAME, ILAENV
175: * ..
176: * .. External Subroutines ..
177: EXTERNAL XERBLA, ZGEMM, ZHERK, ZPBTF2, ZPOTF2, ZTRSM
178: * ..
179: * .. Intrinsic Functions ..
180: INTRINSIC MIN
181: * ..
182: * .. Executable Statements ..
183: *
184: * Test the input parameters.
185: *
186: INFO = 0
187: IF( ( .NOT.LSAME( UPLO, 'U' ) ) .AND.
188: $ ( .NOT.LSAME( UPLO, 'L' ) ) ) THEN
189: INFO = -1
190: ELSE IF( N.LT.0 ) THEN
191: INFO = -2
192: ELSE IF( KD.LT.0 ) THEN
193: INFO = -3
194: ELSE IF( LDAB.LT.KD+1 ) THEN
195: INFO = -5
196: END IF
197: IF( INFO.NE.0 ) THEN
198: CALL XERBLA( 'ZPBTRF', -INFO )
199: RETURN
200: END IF
201: *
202: * Quick return if possible
203: *
204: IF( N.EQ.0 )
205: $ RETURN
206: *
207: * Determine the block size for this environment
208: *
209: NB = ILAENV( 1, 'ZPBTRF', UPLO, N, KD, -1, -1 )
210: *
211: * The block size must not exceed the semi-bandwidth KD, and must not
212: * exceed the limit set by the size of the local array WORK.
213: *
214: NB = MIN( NB, NBMAX )
215: *
216: IF( NB.LE.1 .OR. NB.GT.KD ) THEN
217: *
218: * Use unblocked code
219: *
220: CALL ZPBTF2( UPLO, N, KD, AB, LDAB, INFO )
221: ELSE
222: *
223: * Use blocked code
224: *
225: IF( LSAME( UPLO, 'U' ) ) THEN
226: *
227: * Compute the Cholesky factorization of a Hermitian band
228: * matrix, given the upper triangle of the matrix in band
229: * storage.
230: *
231: * Zero the upper triangle of the work array.
232: *
233: DO 20 J = 1, NB
234: DO 10 I = 1, J - 1
235: WORK( I, J ) = ZERO
236: 10 CONTINUE
237: 20 CONTINUE
238: *
239: * Process the band matrix one diagonal block at a time.
240: *
241: DO 70 I = 1, N, NB
242: IB = MIN( NB, N-I+1 )
243: *
244: * Factorize the diagonal block
245: *
246: CALL ZPOTF2( UPLO, IB, AB( KD+1, I ), LDAB-1, II )
247: IF( II.NE.0 ) THEN
248: INFO = I + II - 1
249: GO TO 150
250: END IF
251: IF( I+IB.LE.N ) THEN
252: *
253: * Update the relevant part of the trailing submatrix.
254: * If A11 denotes the diagonal block which has just been
255: * factorized, then we need to update the remaining
256: * blocks in the diagram:
257: *
258: * A11 A12 A13
259: * A22 A23
260: * A33
261: *
262: * The numbers of rows and columns in the partitioning
263: * are IB, I2, I3 respectively. The blocks A12, A22 and
264: * A23 are empty if IB = KD. The upper triangle of A13
265: * lies outside the band.
266: *
267: I2 = MIN( KD-IB, N-I-IB+1 )
268: I3 = MIN( IB, N-I-KD+1 )
269: *
270: IF( I2.GT.0 ) THEN
271: *
272: * Update A12
273: *
274: CALL ZTRSM( 'Left', 'Upper', 'Conjugate transpose',
275: $ 'Non-unit', IB, I2, CONE,
276: $ AB( KD+1, I ), LDAB-1,
277: $ AB( KD+1-IB, I+IB ), LDAB-1 )
278: *
279: * Update A22
280: *
281: CALL ZHERK( 'Upper', 'Conjugate transpose', I2, IB,
282: $ -ONE, AB( KD+1-IB, I+IB ), LDAB-1, ONE,
283: $ AB( KD+1, I+IB ), LDAB-1 )
284: END IF
285: *
286: IF( I3.GT.0 ) THEN
287: *
288: * Copy the lower triangle of A13 into the work array.
289: *
290: DO 40 JJ = 1, I3
291: DO 30 II = JJ, IB
292: WORK( II, JJ ) = AB( II-JJ+1, JJ+I+KD-1 )
293: 30 CONTINUE
294: 40 CONTINUE
295: *
296: * Update A13 (in the work array).
297: *
298: CALL ZTRSM( 'Left', 'Upper', 'Conjugate transpose',
299: $ 'Non-unit', IB, I3, CONE,
300: $ AB( KD+1, I ), LDAB-1, WORK, LDWORK )
301: *
302: * Update A23
303: *
304: IF( I2.GT.0 )
305: $ CALL ZGEMM( 'Conjugate transpose',
306: $ 'No transpose', I2, I3, IB, -CONE,
307: $ AB( KD+1-IB, I+IB ), LDAB-1, WORK,
308: $ LDWORK, CONE, AB( 1+IB, I+KD ),
309: $ LDAB-1 )
310: *
311: * Update A33
312: *
313: CALL ZHERK( 'Upper', 'Conjugate transpose', I3, IB,
314: $ -ONE, WORK, LDWORK, ONE,
315: $ AB( KD+1, I+KD ), LDAB-1 )
316: *
317: * Copy the lower triangle of A13 back into place.
318: *
319: DO 60 JJ = 1, I3
320: DO 50 II = JJ, IB
321: AB( II-JJ+1, JJ+I+KD-1 ) = WORK( II, JJ )
322: 50 CONTINUE
323: 60 CONTINUE
324: END IF
325: END IF
326: 70 CONTINUE
327: ELSE
328: *
329: * Compute the Cholesky factorization of a Hermitian band
330: * matrix, given the lower triangle of the matrix in band
331: * storage.
332: *
333: * Zero the lower triangle of the work array.
334: *
335: DO 90 J = 1, NB
336: DO 80 I = J + 1, NB
337: WORK( I, J ) = ZERO
338: 80 CONTINUE
339: 90 CONTINUE
340: *
341: * Process the band matrix one diagonal block at a time.
342: *
343: DO 140 I = 1, N, NB
344: IB = MIN( NB, N-I+1 )
345: *
346: * Factorize the diagonal block
347: *
348: CALL ZPOTF2( UPLO, IB, AB( 1, I ), LDAB-1, II )
349: IF( II.NE.0 ) THEN
350: INFO = I + II - 1
351: GO TO 150
352: END IF
353: IF( I+IB.LE.N ) THEN
354: *
355: * Update the relevant part of the trailing submatrix.
356: * If A11 denotes the diagonal block which has just been
357: * factorized, then we need to update the remaining
358: * blocks in the diagram:
359: *
360: * A11
361: * A21 A22
362: * A31 A32 A33
363: *
364: * The numbers of rows and columns in the partitioning
365: * are IB, I2, I3 respectively. The blocks A21, A22 and
366: * A32 are empty if IB = KD. The lower triangle of A31
367: * lies outside the band.
368: *
369: I2 = MIN( KD-IB, N-I-IB+1 )
370: I3 = MIN( IB, N-I-KD+1 )
371: *
372: IF( I2.GT.0 ) THEN
373: *
374: * Update A21
375: *
376: CALL ZTRSM( 'Right', 'Lower',
377: $ 'Conjugate transpose', 'Non-unit', I2,
378: $ IB, CONE, AB( 1, I ), LDAB-1,
379: $ AB( 1+IB, I ), LDAB-1 )
380: *
381: * Update A22
382: *
383: CALL ZHERK( 'Lower', 'No transpose', I2, IB, -ONE,
384: $ AB( 1+IB, I ), LDAB-1, ONE,
385: $ AB( 1, I+IB ), LDAB-1 )
386: END IF
387: *
388: IF( I3.GT.0 ) THEN
389: *
390: * Copy the upper triangle of A31 into the work array.
391: *
392: DO 110 JJ = 1, IB
393: DO 100 II = 1, MIN( JJ, I3 )
394: WORK( II, JJ ) = AB( KD+1-JJ+II, JJ+I-1 )
395: 100 CONTINUE
396: 110 CONTINUE
397: *
398: * Update A31 (in the work array).
399: *
400: CALL ZTRSM( 'Right', 'Lower',
401: $ 'Conjugate transpose', 'Non-unit', I3,
402: $ IB, CONE, AB( 1, I ), LDAB-1, WORK,
403: $ LDWORK )
404: *
405: * Update A32
406: *
407: IF( I2.GT.0 )
408: $ CALL ZGEMM( 'No transpose',
409: $ 'Conjugate transpose', I3, I2, IB,
410: $ -CONE, WORK, LDWORK, AB( 1+IB, I ),
411: $ LDAB-1, CONE, AB( 1+KD-IB, I+IB ),
412: $ LDAB-1 )
413: *
414: * Update A33
415: *
416: CALL ZHERK( 'Lower', 'No transpose', I3, IB, -ONE,
417: $ WORK, LDWORK, ONE, AB( 1, I+KD ),
418: $ LDAB-1 )
419: *
420: * Copy the upper triangle of A31 back into place.
421: *
422: DO 130 JJ = 1, IB
423: DO 120 II = 1, MIN( JJ, I3 )
424: AB( KD+1-JJ+II, JJ+I-1 ) = WORK( II, JJ )
425: 120 CONTINUE
426: 130 CONTINUE
427: END IF
428: END IF
429: 140 CONTINUE
430: END IF
431: END IF
432: RETURN
433: *
434: 150 CONTINUE
435: RETURN
436: *
437: * End of ZPBTRF
438: *
439: END