1: SUBROUTINE ZPBTRF( 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: COMPLEX*16 AB( LDAB, * )
14: * ..
15: *
16: * Purpose
17: * =======
18: *
19: * ZPBTRF computes the Cholesky factorization of a complex Hermitian
20: * positive definite band matrix A.
21: *
22: * The factorization has the form
23: * A = U**H * U, if UPLO = 'U', or
24: * A = L * L**H, 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) COMPLEX*16 array, dimension (LDAB,N)
42: * On entry, the upper or lower triangle of the Hermitian 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**H*U or A = L*L**H 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: COMPLEX*16 CONE
94: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
95: INTEGER NBMAX, LDWORK
96: PARAMETER ( NBMAX = 32, LDWORK = NBMAX+1 )
97: * ..
98: * .. Local Scalars ..
99: INTEGER I, I2, I3, IB, II, J, JJ, NB
100: * ..
101: * .. Local Arrays ..
102: COMPLEX*16 WORK( LDWORK, NBMAX )
103: * ..
104: * .. External Functions ..
105: LOGICAL LSAME
106: INTEGER ILAENV
107: EXTERNAL LSAME, ILAENV
108: * ..
109: * .. External Subroutines ..
110: EXTERNAL XERBLA, ZGEMM, ZHERK, ZPBTF2, ZPOTF2, ZTRSM
111: * ..
112: * .. Intrinsic Functions ..
113: INTRINSIC MIN
114: * ..
115: * .. Executable Statements ..
116: *
117: * Test the input parameters.
118: *
119: INFO = 0
120: IF( ( .NOT.LSAME( UPLO, 'U' ) ) .AND.
121: $ ( .NOT.LSAME( UPLO, 'L' ) ) ) THEN
122: INFO = -1
123: ELSE IF( N.LT.0 ) THEN
124: INFO = -2
125: ELSE IF( KD.LT.0 ) THEN
126: INFO = -3
127: ELSE IF( LDAB.LT.KD+1 ) THEN
128: INFO = -5
129: END IF
130: IF( INFO.NE.0 ) THEN
131: CALL XERBLA( 'ZPBTRF', -INFO )
132: RETURN
133: END IF
134: *
135: * Quick return if possible
136: *
137: IF( N.EQ.0 )
138: $ RETURN
139: *
140: * Determine the block size for this environment
141: *
142: NB = ILAENV( 1, 'ZPBTRF', UPLO, N, KD, -1, -1 )
143: *
144: * The block size must not exceed the semi-bandwidth KD, and must not
145: * exceed the limit set by the size of the local array WORK.
146: *
147: NB = MIN( NB, NBMAX )
148: *
149: IF( NB.LE.1 .OR. NB.GT.KD ) THEN
150: *
151: * Use unblocked code
152: *
153: CALL ZPBTF2( UPLO, N, KD, AB, LDAB, INFO )
154: ELSE
155: *
156: * Use blocked code
157: *
158: IF( LSAME( UPLO, 'U' ) ) THEN
159: *
160: * Compute the Cholesky factorization of a Hermitian band
161: * matrix, given the upper triangle of the matrix in band
162: * storage.
163: *
164: * Zero the upper triangle of the work array.
165: *
166: DO 20 J = 1, NB
167: DO 10 I = 1, J - 1
168: WORK( I, J ) = ZERO
169: 10 CONTINUE
170: 20 CONTINUE
171: *
172: * Process the band matrix one diagonal block at a time.
173: *
174: DO 70 I = 1, N, NB
175: IB = MIN( NB, N-I+1 )
176: *
177: * Factorize the diagonal block
178: *
179: CALL ZPOTF2( UPLO, IB, AB( KD+1, I ), LDAB-1, II )
180: IF( II.NE.0 ) THEN
181: INFO = I + II - 1
182: GO TO 150
183: END IF
184: IF( I+IB.LE.N ) THEN
185: *
186: * Update the relevant part of the trailing submatrix.
187: * If A11 denotes the diagonal block which has just been
188: * factorized, then we need to update the remaining
189: * blocks in the diagram:
190: *
191: * A11 A12 A13
192: * A22 A23
193: * A33
194: *
195: * The numbers of rows and columns in the partitioning
196: * are IB, I2, I3 respectively. The blocks A12, A22 and
197: * A23 are empty if IB = KD. The upper triangle of A13
198: * lies outside the band.
199: *
200: I2 = MIN( KD-IB, N-I-IB+1 )
201: I3 = MIN( IB, N-I-KD+1 )
202: *
203: IF( I2.GT.0 ) THEN
204: *
205: * Update A12
206: *
207: CALL ZTRSM( 'Left', 'Upper', 'Conjugate transpose',
208: $ 'Non-unit', IB, I2, CONE,
209: $ AB( KD+1, I ), LDAB-1,
210: $ AB( KD+1-IB, I+IB ), LDAB-1 )
211: *
212: * Update A22
213: *
214: CALL ZHERK( 'Upper', 'Conjugate transpose', I2, IB,
215: $ -ONE, AB( KD+1-IB, I+IB ), LDAB-1, ONE,
216: $ AB( KD+1, I+IB ), LDAB-1 )
217: END IF
218: *
219: IF( I3.GT.0 ) THEN
220: *
221: * Copy the lower triangle of A13 into the work array.
222: *
223: DO 40 JJ = 1, I3
224: DO 30 II = JJ, IB
225: WORK( II, JJ ) = AB( II-JJ+1, JJ+I+KD-1 )
226: 30 CONTINUE
227: 40 CONTINUE
228: *
229: * Update A13 (in the work array).
230: *
231: CALL ZTRSM( 'Left', 'Upper', 'Conjugate transpose',
232: $ 'Non-unit', IB, I3, CONE,
233: $ AB( KD+1, I ), LDAB-1, WORK, LDWORK )
234: *
235: * Update A23
236: *
237: IF( I2.GT.0 )
238: $ CALL ZGEMM( 'Conjugate transpose',
239: $ 'No transpose', I2, I3, IB, -CONE,
240: $ AB( KD+1-IB, I+IB ), LDAB-1, WORK,
241: $ LDWORK, CONE, AB( 1+IB, I+KD ),
242: $ LDAB-1 )
243: *
244: * Update A33
245: *
246: CALL ZHERK( 'Upper', 'Conjugate transpose', I3, IB,
247: $ -ONE, WORK, LDWORK, ONE,
248: $ AB( KD+1, I+KD ), LDAB-1 )
249: *
250: * Copy the lower triangle of A13 back into place.
251: *
252: DO 60 JJ = 1, I3
253: DO 50 II = JJ, IB
254: AB( II-JJ+1, JJ+I+KD-1 ) = WORK( II, JJ )
255: 50 CONTINUE
256: 60 CONTINUE
257: END IF
258: END IF
259: 70 CONTINUE
260: ELSE
261: *
262: * Compute the Cholesky factorization of a Hermitian band
263: * matrix, given the lower triangle of the matrix in band
264: * storage.
265: *
266: * Zero the lower triangle of the work array.
267: *
268: DO 90 J = 1, NB
269: DO 80 I = J + 1, NB
270: WORK( I, J ) = ZERO
271: 80 CONTINUE
272: 90 CONTINUE
273: *
274: * Process the band matrix one diagonal block at a time.
275: *
276: DO 140 I = 1, N, NB
277: IB = MIN( NB, N-I+1 )
278: *
279: * Factorize the diagonal block
280: *
281: CALL ZPOTF2( UPLO, IB, AB( 1, I ), LDAB-1, II )
282: IF( II.NE.0 ) THEN
283: INFO = I + II - 1
284: GO TO 150
285: END IF
286: IF( I+IB.LE.N ) THEN
287: *
288: * Update the relevant part of the trailing submatrix.
289: * If A11 denotes the diagonal block which has just been
290: * factorized, then we need to update the remaining
291: * blocks in the diagram:
292: *
293: * A11
294: * A21 A22
295: * A31 A32 A33
296: *
297: * The numbers of rows and columns in the partitioning
298: * are IB, I2, I3 respectively. The blocks A21, A22 and
299: * A32 are empty if IB = KD. The lower triangle of A31
300: * lies outside the band.
301: *
302: I2 = MIN( KD-IB, N-I-IB+1 )
303: I3 = MIN( IB, N-I-KD+1 )
304: *
305: IF( I2.GT.0 ) THEN
306: *
307: * Update A21
308: *
309: CALL ZTRSM( 'Right', 'Lower',
310: $ 'Conjugate transpose', 'Non-unit', I2,
311: $ IB, CONE, AB( 1, I ), LDAB-1,
312: $ AB( 1+IB, I ), LDAB-1 )
313: *
314: * Update A22
315: *
316: CALL ZHERK( 'Lower', 'No transpose', I2, IB, -ONE,
317: $ AB( 1+IB, I ), LDAB-1, ONE,
318: $ AB( 1, I+IB ), LDAB-1 )
319: END IF
320: *
321: IF( I3.GT.0 ) THEN
322: *
323: * Copy the upper triangle of A31 into the work array.
324: *
325: DO 110 JJ = 1, IB
326: DO 100 II = 1, MIN( JJ, I3 )
327: WORK( II, JJ ) = AB( KD+1-JJ+II, JJ+I-1 )
328: 100 CONTINUE
329: 110 CONTINUE
330: *
331: * Update A31 (in the work array).
332: *
333: CALL ZTRSM( 'Right', 'Lower',
334: $ 'Conjugate transpose', 'Non-unit', I3,
335: $ IB, CONE, AB( 1, I ), LDAB-1, WORK,
336: $ LDWORK )
337: *
338: * Update A32
339: *
340: IF( I2.GT.0 )
341: $ CALL ZGEMM( 'No transpose',
342: $ 'Conjugate transpose', I3, I2, IB,
343: $ -CONE, WORK, LDWORK, AB( 1+IB, I ),
344: $ LDAB-1, CONE, AB( 1+KD-IB, I+IB ),
345: $ LDAB-1 )
346: *
347: * Update A33
348: *
349: CALL ZHERK( 'Lower', 'No transpose', I3, IB, -ONE,
350: $ WORK, LDWORK, ONE, AB( 1, I+KD ),
351: $ LDAB-1 )
352: *
353: * Copy the upper triangle of A31 back into place.
354: *
355: DO 130 JJ = 1, IB
356: DO 120 II = 1, MIN( JJ, I3 )
357: AB( KD+1-JJ+II, JJ+I-1 ) = WORK( II, JJ )
358: 120 CONTINUE
359: 130 CONTINUE
360: END IF
361: END IF
362: 140 CONTINUE
363: END IF
364: END IF
365: RETURN
366: *
367: 150 CONTINUE
368: RETURN
369: *
370: * End of ZPBTRF
371: *
372: END
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