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1.1 bertrand 1: SUBROUTINE DPBTRF( UPLO, N, KD, AB, LDAB, INFO )
2: *
3: * -- LAPACK routine (version 3.2) --
4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6: * November 2006
7: *
8: * .. Scalar Arguments ..
9: CHARACTER UPLO
10: INTEGER INFO, KD, LDAB, N
11: * ..
12: * .. Array Arguments ..
13: DOUBLE PRECISION AB( LDAB, * )
14: * ..
15: *
16: * Purpose
17: * =======
18: *
19: * DPBTRF computes the Cholesky factorization of a real symmetric
20: * positive definite band matrix A.
21: *
22: * The factorization has the form
23: * A = U**T * U, if UPLO = 'U', or
24: * A = L * L**T, if UPLO = 'L',
25: * where U is an upper triangular matrix and L is lower triangular.
26: *
27: * Arguments
28: * =========
29: *
30: * UPLO (input) CHARACTER*1
31: * = 'U': Upper triangle of A is stored;
32: * = 'L': Lower triangle of A is stored.
33: *
34: * N (input) INTEGER
35: * The order of the matrix A. N >= 0.
36: *
37: * KD (input) INTEGER
38: * The number of superdiagonals of the matrix A if UPLO = 'U',
39: * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
40: *
41: * AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
42: * On entry, the upper or lower triangle of the symmetric band
43: * matrix A, stored in the first KD+1 rows of the array. The
44: * j-th column of A is stored in the j-th column of the array AB
45: * as follows:
46: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
47: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
48: *
49: * On exit, if INFO = 0, the triangular factor U or L from the
50: * Cholesky factorization A = U**T*U or A = L*L**T of the band
51: * matrix A, in the same storage format as A.
52: *
53: * LDAB (input) INTEGER
54: * The leading dimension of the array AB. LDAB >= KD+1.
55: *
56: * INFO (output) INTEGER
57: * = 0: successful exit
58: * < 0: if INFO = -i, the i-th argument had an illegal value
59: * > 0: if INFO = i, the leading minor of order i is not
60: * positive definite, and the factorization could not be
61: * completed.
62: *
63: * Further Details
64: * ===============
65: *
66: * The band storage scheme is illustrated by the following example, when
67: * N = 6, KD = 2, and UPLO = 'U':
68: *
69: * On entry: On exit:
70: *
71: * * * a13 a24 a35 a46 * * u13 u24 u35 u46
72: * * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
73: * a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
74: *
75: * Similarly, if UPLO = 'L' the format of A is as follows:
76: *
77: * On entry: On exit:
78: *
79: * a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
80: * a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
81: * a31 a42 a53 a64 * * l31 l42 l53 l64 * *
82: *
83: * Array elements marked * are not used by the routine.
84: *
85: * Contributed by
86: * Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
87: *
88: * =====================================================================
89: *
90: * .. Parameters ..
91: DOUBLE PRECISION ONE, ZERO
92: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
93: INTEGER NBMAX, LDWORK
94: PARAMETER ( NBMAX = 32, LDWORK = NBMAX+1 )
95: * ..
96: * .. Local Scalars ..
97: INTEGER I, I2, I3, IB, II, J, JJ, NB
98: * ..
99: * .. Local Arrays ..
100: DOUBLE PRECISION WORK( LDWORK, NBMAX )
101: * ..
102: * .. External Functions ..
103: LOGICAL LSAME
104: INTEGER ILAENV
105: EXTERNAL LSAME, ILAENV
106: * ..
107: * .. External Subroutines ..
108: EXTERNAL DGEMM, DPBTF2, DPOTF2, DSYRK, DTRSM, XERBLA
109: * ..
110: * .. Intrinsic Functions ..
111: INTRINSIC MIN
112: * ..
113: * .. Executable Statements ..
114: *
115: * Test the input parameters.
116: *
117: INFO = 0
118: IF( ( .NOT.LSAME( UPLO, 'U' ) ) .AND.
119: $ ( .NOT.LSAME( UPLO, 'L' ) ) ) THEN
120: INFO = -1
121: ELSE IF( N.LT.0 ) THEN
122: INFO = -2
123: ELSE IF( KD.LT.0 ) THEN
124: INFO = -3
125: ELSE IF( LDAB.LT.KD+1 ) THEN
126: INFO = -5
127: END IF
128: IF( INFO.NE.0 ) THEN
129: CALL XERBLA( 'DPBTRF', -INFO )
130: RETURN
131: END IF
132: *
133: * Quick return if possible
134: *
135: IF( N.EQ.0 )
136: $ RETURN
137: *
138: * Determine the block size for this environment
139: *
140: NB = ILAENV( 1, 'DPBTRF', UPLO, N, KD, -1, -1 )
141: *
142: * The block size must not exceed the semi-bandwidth KD, and must not
143: * exceed the limit set by the size of the local array WORK.
144: *
145: NB = MIN( NB, NBMAX )
146: *
147: IF( NB.LE.1 .OR. NB.GT.KD ) THEN
148: *
149: * Use unblocked code
150: *
151: CALL DPBTF2( UPLO, N, KD, AB, LDAB, INFO )
152: ELSE
153: *
154: * Use blocked code
155: *
156: IF( LSAME( UPLO, 'U' ) ) THEN
157: *
158: * Compute the Cholesky factorization of a symmetric band
159: * matrix, given the upper triangle of the matrix in band
160: * storage.
161: *
162: * Zero the upper triangle of the work array.
163: *
164: DO 20 J = 1, NB
165: DO 10 I = 1, J - 1
166: WORK( I, J ) = ZERO
167: 10 CONTINUE
168: 20 CONTINUE
169: *
170: * Process the band matrix one diagonal block at a time.
171: *
172: DO 70 I = 1, N, NB
173: IB = MIN( NB, N-I+1 )
174: *
175: * Factorize the diagonal block
176: *
177: CALL DPOTF2( UPLO, IB, AB( KD+1, I ), LDAB-1, II )
178: IF( II.NE.0 ) THEN
179: INFO = I + II - 1
180: GO TO 150
181: END IF
182: IF( I+IB.LE.N ) THEN
183: *
184: * Update the relevant part of the trailing submatrix.
185: * If A11 denotes the diagonal block which has just been
186: * factorized, then we need to update the remaining
187: * blocks in the diagram:
188: *
189: * A11 A12 A13
190: * A22 A23
191: * A33
192: *
193: * The numbers of rows and columns in the partitioning
194: * are IB, I2, I3 respectively. The blocks A12, A22 and
195: * A23 are empty if IB = KD. The upper triangle of A13
196: * lies outside the band.
197: *
198: I2 = MIN( KD-IB, N-I-IB+1 )
199: I3 = MIN( IB, N-I-KD+1 )
200: *
201: IF( I2.GT.0 ) THEN
202: *
203: * Update A12
204: *
205: CALL DTRSM( 'Left', 'Upper', 'Transpose',
206: $ 'Non-unit', IB, I2, ONE, AB( KD+1, I ),
207: $ LDAB-1, AB( KD+1-IB, I+IB ), LDAB-1 )
208: *
209: * Update A22
210: *
211: CALL DSYRK( 'Upper', 'Transpose', I2, IB, -ONE,
212: $ AB( KD+1-IB, I+IB ), LDAB-1, ONE,
213: $ AB( KD+1, I+IB ), LDAB-1 )
214: END IF
215: *
216: IF( I3.GT.0 ) THEN
217: *
218: * Copy the lower triangle of A13 into the work array.
219: *
220: DO 40 JJ = 1, I3
221: DO 30 II = JJ, IB
222: WORK( II, JJ ) = AB( II-JJ+1, JJ+I+KD-1 )
223: 30 CONTINUE
224: 40 CONTINUE
225: *
226: * Update A13 (in the work array).
227: *
228: CALL DTRSM( 'Left', 'Upper', 'Transpose',
229: $ 'Non-unit', IB, I3, ONE, AB( KD+1, I ),
230: $ LDAB-1, WORK, LDWORK )
231: *
232: * Update A23
233: *
234: IF( I2.GT.0 )
235: $ CALL DGEMM( 'Transpose', 'No Transpose', I2, I3,
236: $ IB, -ONE, AB( KD+1-IB, I+IB ),
237: $ LDAB-1, WORK, LDWORK, ONE,
238: $ AB( 1+IB, I+KD ), LDAB-1 )
239: *
240: * Update A33
241: *
242: CALL DSYRK( 'Upper', 'Transpose', I3, IB, -ONE,
243: $ WORK, LDWORK, ONE, AB( KD+1, I+KD ),
244: $ LDAB-1 )
245: *
246: * Copy the lower triangle of A13 back into place.
247: *
248: DO 60 JJ = 1, I3
249: DO 50 II = JJ, IB
250: AB( II-JJ+1, JJ+I+KD-1 ) = WORK( II, JJ )
251: 50 CONTINUE
252: 60 CONTINUE
253: END IF
254: END IF
255: 70 CONTINUE
256: ELSE
257: *
258: * Compute the Cholesky factorization of a symmetric band
259: * matrix, given the lower triangle of the matrix in band
260: * storage.
261: *
262: * Zero the lower triangle of the work array.
263: *
264: DO 90 J = 1, NB
265: DO 80 I = J + 1, NB
266: WORK( I, J ) = ZERO
267: 80 CONTINUE
268: 90 CONTINUE
269: *
270: * Process the band matrix one diagonal block at a time.
271: *
272: DO 140 I = 1, N, NB
273: IB = MIN( NB, N-I+1 )
274: *
275: * Factorize the diagonal block
276: *
277: CALL DPOTF2( UPLO, IB, AB( 1, I ), LDAB-1, II )
278: IF( II.NE.0 ) THEN
279: INFO = I + II - 1
280: GO TO 150
281: END IF
282: IF( I+IB.LE.N ) THEN
283: *
284: * Update the relevant part of the trailing submatrix.
285: * If A11 denotes the diagonal block which has just been
286: * factorized, then we need to update the remaining
287: * blocks in the diagram:
288: *
289: * A11
290: * A21 A22
291: * A31 A32 A33
292: *
293: * The numbers of rows and columns in the partitioning
294: * are IB, I2, I3 respectively. The blocks A21, A22 and
295: * A32 are empty if IB = KD. The lower triangle of A31
296: * lies outside the band.
297: *
298: I2 = MIN( KD-IB, N-I-IB+1 )
299: I3 = MIN( IB, N-I-KD+1 )
300: *
301: IF( I2.GT.0 ) THEN
302: *
303: * Update A21
304: *
305: CALL DTRSM( 'Right', 'Lower', 'Transpose',
306: $ 'Non-unit', I2, IB, ONE, AB( 1, I ),
307: $ LDAB-1, AB( 1+IB, I ), LDAB-1 )
308: *
309: * Update A22
310: *
311: CALL DSYRK( 'Lower', 'No Transpose', I2, IB, -ONE,
312: $ AB( 1+IB, I ), LDAB-1, ONE,
313: $ AB( 1, I+IB ), LDAB-1 )
314: END IF
315: *
316: IF( I3.GT.0 ) THEN
317: *
318: * Copy the upper triangle of A31 into the work array.
319: *
320: DO 110 JJ = 1, IB
321: DO 100 II = 1, MIN( JJ, I3 )
322: WORK( II, JJ ) = AB( KD+1-JJ+II, JJ+I-1 )
323: 100 CONTINUE
324: 110 CONTINUE
325: *
326: * Update A31 (in the work array).
327: *
328: CALL DTRSM( 'Right', 'Lower', 'Transpose',
329: $ 'Non-unit', I3, IB, ONE, AB( 1, I ),
330: $ LDAB-1, WORK, LDWORK )
331: *
332: * Update A32
333: *
334: IF( I2.GT.0 )
335: $ CALL DGEMM( 'No transpose', 'Transpose', I3, I2,
336: $ IB, -ONE, WORK, LDWORK,
337: $ AB( 1+IB, I ), LDAB-1, ONE,
338: $ AB( 1+KD-IB, I+IB ), LDAB-1 )
339: *
340: * Update A33
341: *
342: CALL DSYRK( 'Lower', 'No Transpose', I3, IB, -ONE,
343: $ WORK, LDWORK, ONE, AB( 1, I+KD ),
344: $ LDAB-1 )
345: *
346: * Copy the upper triangle of A31 back into place.
347: *
348: DO 130 JJ = 1, IB
349: DO 120 II = 1, MIN( JJ, I3 )
350: AB( KD+1-JJ+II, JJ+I-1 ) = WORK( II, JJ )
351: 120 CONTINUE
352: 130 CONTINUE
353: END IF
354: END IF
355: 140 CONTINUE
356: END IF
357: END IF
358: RETURN
359: *
360: 150 CONTINUE
361: RETURN
362: *
363: * End of DPBTRF
364: *
365: END