1: *> \brief \b DPBTRF
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DPBTRF( UPLO, N, KD, AB, LDAB, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, KD, LDAB, N
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION AB( LDAB, * )
29: * ..
30: *
31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> DPBTRF computes the Cholesky factorization of a real symmetric
38: *> positive definite band matrix A.
39: *>
40: *> The factorization has the form
41: *> A = U**T * U, if UPLO = 'U', or
42: *> A = L * L**T, 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 DOUBLE PRECISION array, dimension (LDAB,N)
72: *> On entry, the upper or lower triangle of the symmetric 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**T*U or A = L*L**T 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: *
103: *> \author Univ. of Tennessee
104: *> \author Univ. of California Berkeley
105: *> \author Univ. of Colorado Denver
106: *> \author NAG Ltd.
107: *
108: *> \date December 2016
109: *
110: *> \ingroup doubleOTHERcomputational
111: *
112: *> \par Further Details:
113: * =====================
114: *>
115: *> \verbatim
116: *>
117: *> The band storage scheme is illustrated by the following example, when
118: *> N = 6, KD = 2, and UPLO = 'U':
119: *>
120: *> On entry: On exit:
121: *>
122: *> * * a13 a24 a35 a46 * * u13 u24 u35 u46
123: *> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
124: *> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
125: *>
126: *> Similarly, if UPLO = 'L' the format of A is as follows:
127: *>
128: *> On entry: On exit:
129: *>
130: *> a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
131: *> a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
132: *> a31 a42 a53 a64 * * l31 l42 l53 l64 * *
133: *>
134: *> Array elements marked * are not used by the routine.
135: *> \endverbatim
136: *
137: *> \par Contributors:
138: * ==================
139: *>
140: *> Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
141: *
142: * =====================================================================
143: SUBROUTINE DPBTRF( UPLO, N, KD, AB, LDAB, INFO )
144: *
145: * -- LAPACK computational routine (version 3.7.0) --
146: * -- LAPACK is a software package provided by Univ. of Tennessee, --
147: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
148: * December 2016
149: *
150: * .. Scalar Arguments ..
151: CHARACTER UPLO
152: INTEGER INFO, KD, LDAB, N
153: * ..
154: * .. Array Arguments ..
155: DOUBLE PRECISION AB( LDAB, * )
156: * ..
157: *
158: * =====================================================================
159: *
160: * .. Parameters ..
161: DOUBLE PRECISION ONE, ZERO
162: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
163: INTEGER NBMAX, LDWORK
164: PARAMETER ( NBMAX = 32, LDWORK = NBMAX+1 )
165: * ..
166: * .. Local Scalars ..
167: INTEGER I, I2, I3, IB, II, J, JJ, NB
168: * ..
169: * .. Local Arrays ..
170: DOUBLE PRECISION WORK( LDWORK, NBMAX )
171: * ..
172: * .. External Functions ..
173: LOGICAL LSAME
174: INTEGER ILAENV
175: EXTERNAL LSAME, ILAENV
176: * ..
177: * .. External Subroutines ..
178: EXTERNAL DGEMM, DPBTF2, DPOTF2, DSYRK, DTRSM, XERBLA
179: * ..
180: * .. Intrinsic Functions ..
181: INTRINSIC MIN
182: * ..
183: * .. Executable Statements ..
184: *
185: * Test the input parameters.
186: *
187: INFO = 0
188: IF( ( .NOT.LSAME( UPLO, 'U' ) ) .AND.
189: $ ( .NOT.LSAME( UPLO, 'L' ) ) ) THEN
190: INFO = -1
191: ELSE IF( N.LT.0 ) THEN
192: INFO = -2
193: ELSE IF( KD.LT.0 ) THEN
194: INFO = -3
195: ELSE IF( LDAB.LT.KD+1 ) THEN
196: INFO = -5
197: END IF
198: IF( INFO.NE.0 ) THEN
199: CALL XERBLA( 'DPBTRF', -INFO )
200: RETURN
201: END IF
202: *
203: * Quick return if possible
204: *
205: IF( N.EQ.0 )
206: $ RETURN
207: *
208: * Determine the block size for this environment
209: *
210: NB = ILAENV( 1, 'DPBTRF', UPLO, N, KD, -1, -1 )
211: *
212: * The block size must not exceed the semi-bandwidth KD, and must not
213: * exceed the limit set by the size of the local array WORK.
214: *
215: NB = MIN( NB, NBMAX )
216: *
217: IF( NB.LE.1 .OR. NB.GT.KD ) THEN
218: *
219: * Use unblocked code
220: *
221: CALL DPBTF2( UPLO, N, KD, AB, LDAB, INFO )
222: ELSE
223: *
224: * Use blocked code
225: *
226: IF( LSAME( UPLO, 'U' ) ) THEN
227: *
228: * Compute the Cholesky factorization of a symmetric band
229: * matrix, given the upper triangle of the matrix in band
230: * storage.
231: *
232: * Zero the upper triangle of the work array.
233: *
234: DO 20 J = 1, NB
235: DO 10 I = 1, J - 1
236: WORK( I, J ) = ZERO
237: 10 CONTINUE
238: 20 CONTINUE
239: *
240: * Process the band matrix one diagonal block at a time.
241: *
242: DO 70 I = 1, N, NB
243: IB = MIN( NB, N-I+1 )
244: *
245: * Factorize the diagonal block
246: *
247: CALL DPOTF2( UPLO, IB, AB( KD+1, I ), LDAB-1, II )
248: IF( II.NE.0 ) THEN
249: INFO = I + II - 1
250: GO TO 150
251: END IF
252: IF( I+IB.LE.N ) THEN
253: *
254: * Update the relevant part of the trailing submatrix.
255: * If A11 denotes the diagonal block which has just been
256: * factorized, then we need to update the remaining
257: * blocks in the diagram:
258: *
259: * A11 A12 A13
260: * A22 A23
261: * A33
262: *
263: * The numbers of rows and columns in the partitioning
264: * are IB, I2, I3 respectively. The blocks A12, A22 and
265: * A23 are empty if IB = KD. The upper triangle of A13
266: * lies outside the band.
267: *
268: I2 = MIN( KD-IB, N-I-IB+1 )
269: I3 = MIN( IB, N-I-KD+1 )
270: *
271: IF( I2.GT.0 ) THEN
272: *
273: * Update A12
274: *
275: CALL DTRSM( 'Left', 'Upper', 'Transpose',
276: $ 'Non-unit', IB, I2, ONE, AB( KD+1, I ),
277: $ LDAB-1, AB( KD+1-IB, I+IB ), LDAB-1 )
278: *
279: * Update A22
280: *
281: CALL DSYRK( 'Upper', 'Transpose', I2, IB, -ONE,
282: $ 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 DTRSM( 'Left', 'Upper', 'Transpose',
299: $ 'Non-unit', IB, I3, ONE, AB( KD+1, I ),
300: $ LDAB-1, WORK, LDWORK )
301: *
302: * Update A23
303: *
304: IF( I2.GT.0 )
305: $ CALL DGEMM( 'Transpose', 'No Transpose', I2, I3,
306: $ IB, -ONE, AB( KD+1-IB, I+IB ),
307: $ LDAB-1, WORK, LDWORK, ONE,
308: $ AB( 1+IB, I+KD ), LDAB-1 )
309: *
310: * Update A33
311: *
312: CALL DSYRK( 'Upper', 'Transpose', I3, IB, -ONE,
313: $ WORK, LDWORK, ONE, AB( KD+1, I+KD ),
314: $ LDAB-1 )
315: *
316: * Copy the lower triangle of A13 back into place.
317: *
318: DO 60 JJ = 1, I3
319: DO 50 II = JJ, IB
320: AB( II-JJ+1, JJ+I+KD-1 ) = WORK( II, JJ )
321: 50 CONTINUE
322: 60 CONTINUE
323: END IF
324: END IF
325: 70 CONTINUE
326: ELSE
327: *
328: * Compute the Cholesky factorization of a symmetric band
329: * matrix, given the lower triangle of the matrix in band
330: * storage.
331: *
332: * Zero the lower triangle of the work array.
333: *
334: DO 90 J = 1, NB
335: DO 80 I = J + 1, NB
336: WORK( I, J ) = ZERO
337: 80 CONTINUE
338: 90 CONTINUE
339: *
340: * Process the band matrix one diagonal block at a time.
341: *
342: DO 140 I = 1, N, NB
343: IB = MIN( NB, N-I+1 )
344: *
345: * Factorize the diagonal block
346: *
347: CALL DPOTF2( UPLO, IB, AB( 1, I ), LDAB-1, II )
348: IF( II.NE.0 ) THEN
349: INFO = I + II - 1
350: GO TO 150
351: END IF
352: IF( I+IB.LE.N ) THEN
353: *
354: * Update the relevant part of the trailing submatrix.
355: * If A11 denotes the diagonal block which has just been
356: * factorized, then we need to update the remaining
357: * blocks in the diagram:
358: *
359: * A11
360: * A21 A22
361: * A31 A32 A33
362: *
363: * The numbers of rows and columns in the partitioning
364: * are IB, I2, I3 respectively. The blocks A21, A22 and
365: * A32 are empty if IB = KD. The lower triangle of A31
366: * lies outside the band.
367: *
368: I2 = MIN( KD-IB, N-I-IB+1 )
369: I3 = MIN( IB, N-I-KD+1 )
370: *
371: IF( I2.GT.0 ) THEN
372: *
373: * Update A21
374: *
375: CALL DTRSM( 'Right', 'Lower', 'Transpose',
376: $ 'Non-unit', I2, IB, ONE, AB( 1, I ),
377: $ LDAB-1, AB( 1+IB, I ), LDAB-1 )
378: *
379: * Update A22
380: *
381: CALL DSYRK( 'Lower', 'No Transpose', I2, IB, -ONE,
382: $ AB( 1+IB, I ), LDAB-1, ONE,
383: $ AB( 1, I+IB ), LDAB-1 )
384: END IF
385: *
386: IF( I3.GT.0 ) THEN
387: *
388: * Copy the upper triangle of A31 into the work array.
389: *
390: DO 110 JJ = 1, IB
391: DO 100 II = 1, MIN( JJ, I3 )
392: WORK( II, JJ ) = AB( KD+1-JJ+II, JJ+I-1 )
393: 100 CONTINUE
394: 110 CONTINUE
395: *
396: * Update A31 (in the work array).
397: *
398: CALL DTRSM( 'Right', 'Lower', 'Transpose',
399: $ 'Non-unit', I3, IB, ONE, AB( 1, I ),
400: $ LDAB-1, WORK, LDWORK )
401: *
402: * Update A32
403: *
404: IF( I2.GT.0 )
405: $ CALL DGEMM( 'No transpose', 'Transpose', I3, I2,
406: $ IB, -ONE, WORK, LDWORK,
407: $ AB( 1+IB, I ), LDAB-1, ONE,
408: $ AB( 1+KD-IB, I+IB ), LDAB-1 )
409: *
410: * Update A33
411: *
412: CALL DSYRK( 'Lower', 'No Transpose', I3, IB, -ONE,
413: $ WORK, LDWORK, ONE, AB( 1, I+KD ),
414: $ LDAB-1 )
415: *
416: * Copy the upper triangle of A31 back into place.
417: *
418: DO 130 JJ = 1, IB
419: DO 120 II = 1, MIN( JJ, I3 )
420: AB( KD+1-JJ+II, JJ+I-1 ) = WORK( II, JJ )
421: 120 CONTINUE
422: 130 CONTINUE
423: END IF
424: END IF
425: 140 CONTINUE
426: END IF
427: END IF
428: RETURN
429: *
430: 150 CONTINUE
431: RETURN
432: *
433: * End of DPBTRF
434: *
435: END
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