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
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