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
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