1 SUBROUTINE ZHETRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
2 *
3 * -- LAPACK routine (version 3.3.1) --
4 * -- LAPACK is a software package provided by Univ. of Tennessee, --
5 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6 * -- April 2011 --
7 *
8 * .. Scalar Arguments ..
9 CHARACTER UPLO
10 INTEGER INFO, LDA, LWORK, N
11 * ..
12 * .. Array Arguments ..
13 INTEGER IPIV( * )
14 COMPLEX*16 A( LDA, * ), WORK( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * ZHETRF computes the factorization of a complex Hermitian matrix A
21 * using the Bunch-Kaufman diagonal pivoting method. The form of the
22 * factorization is
23 *
24 * A = U*D*U**H or A = L*D*L**H
25 *
26 * where U (or L) is a product of permutation and unit upper (lower)
27 * triangular matrices, and D is Hermitian and block diagonal with
28 * 1-by-1 and 2-by-2 diagonal blocks.
29 *
30 * This is the blocked version of the algorithm, calling Level 3 BLAS.
31 *
32 * Arguments
33 * =========
34 *
35 * UPLO (input) CHARACTER*1
36 * = 'U': Upper triangle of A is stored;
37 * = 'L': Lower triangle of A is stored.
38 *
39 * N (input) INTEGER
40 * The order of the matrix A. N >= 0.
41 *
42 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
43 * On entry, the Hermitian matrix A. If UPLO = 'U', the leading
44 * N-by-N upper triangular part of A contains the upper
45 * triangular part of the matrix A, and the strictly lower
46 * triangular part of A is not referenced. If UPLO = 'L', the
47 * leading N-by-N lower triangular part of A contains the lower
48 * triangular part of the matrix A, and the strictly upper
49 * triangular part of A is not referenced.
50 *
51 * On exit, the block diagonal matrix D and the multipliers used
52 * to obtain the factor U or L (see below for further details).
53 *
54 * LDA (input) INTEGER
55 * The leading dimension of the array A. LDA >= max(1,N).
56 *
57 * IPIV (output) INTEGER array, dimension (N)
58 * Details of the interchanges and the block structure of D.
59 * If IPIV(k) > 0, then rows and columns k and IPIV(k) were
60 * interchanged and D(k,k) is a 1-by-1 diagonal block.
61 * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
62 * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
63 * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
64 * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
65 * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
66 *
67 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
68 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
69 *
70 * LWORK (input) INTEGER
71 * The length of WORK. LWORK >=1. For best performance
72 * LWORK >= N*NB, where NB is the block size returned by ILAENV.
73 *
74 * INFO (output) INTEGER
75 * = 0: successful exit
76 * < 0: if INFO = -i, the i-th argument had an illegal value
77 * > 0: if INFO = i, D(i,i) is exactly zero. The factorization
78 * has been completed, but the block diagonal matrix D is
79 * exactly singular, and division by zero will occur if it
80 * is used to solve a system of equations.
81 *
82 * Further Details
83 * ===============
84 *
85 * If UPLO = 'U', then A = U*D*U**H, where
86 * U = P(n)*U(n)* ... *P(k)U(k)* ...,
87 * i.e., U is a product of terms P(k)*U(k), where k decreases from n to
88 * 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
89 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
90 * defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
91 * that if the diagonal block D(k) is of order s (s = 1 or 2), then
92 *
93 * ( I v 0 ) k-s
94 * U(k) = ( 0 I 0 ) s
95 * ( 0 0 I ) n-k
96 * k-s s n-k
97 *
98 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
99 * If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
100 * and A(k,k), and v overwrites A(1:k-2,k-1:k).
101 *
102 * If UPLO = 'L', then A = L*D*L**H, where
103 * L = P(1)*L(1)* ... *P(k)*L(k)* ...,
104 * i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
105 * n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
106 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
107 * defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
108 * that if the diagonal block D(k) is of order s (s = 1 or 2), then
109 *
110 * ( I 0 0 ) k-1
111 * L(k) = ( 0 I 0 ) s
112 * ( 0 v I ) n-k-s+1
113 * k-1 s n-k-s+1
114 *
115 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
116 * If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
117 * and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
118 *
119 * =====================================================================
120 *
121 * .. Local Scalars ..
122 LOGICAL LQUERY, UPPER
123 INTEGER IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
124 * ..
125 * .. External Functions ..
126 LOGICAL LSAME
127 INTEGER ILAENV
128 EXTERNAL LSAME, ILAENV
129 * ..
130 * .. External Subroutines ..
131 EXTERNAL XERBLA, ZHETF2, ZLAHEF
132 * ..
133 * .. Intrinsic Functions ..
134 INTRINSIC MAX
135 * ..
136 * .. Executable Statements ..
137 *
138 * Test the input parameters.
139 *
140 INFO = 0
141 UPPER = LSAME( UPLO, 'U' )
142 LQUERY = ( LWORK.EQ.-1 )
143 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
144 INFO = -1
145 ELSE IF( N.LT.0 ) THEN
146 INFO = -2
147 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
148 INFO = -4
149 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
150 INFO = -7
151 END IF
152 *
153 IF( INFO.EQ.0 ) THEN
154 *
155 * Determine the block size
156 *
157 NB = ILAENV( 1, 'ZHETRF', UPLO, N, -1, -1, -1 )
158 LWKOPT = N*NB
159 WORK( 1 ) = LWKOPT
160 END IF
161 *
162 IF( INFO.NE.0 ) THEN
163 CALL XERBLA( 'ZHETRF', -INFO )
164 RETURN
165 ELSE IF( LQUERY ) THEN
166 RETURN
167 END IF
168 *
169 NBMIN = 2
170 LDWORK = N
171 IF( NB.GT.1 .AND. NB.LT.N ) THEN
172 IWS = LDWORK*NB
173 IF( LWORK.LT.IWS ) THEN
174 NB = MAX( LWORK / LDWORK, 1 )
175 NBMIN = MAX( 2, ILAENV( 2, 'ZHETRF', UPLO, N, -1, -1, -1 ) )
176 END IF
177 ELSE
178 IWS = 1
179 END IF
180 IF( NB.LT.NBMIN )
181 $ NB = N
182 *
183 IF( UPPER ) THEN
184 *
185 * Factorize A as U*D*U**H using the upper triangle of A
186 *
187 * K is the main loop index, decreasing from N to 1 in steps of
188 * KB, where KB is the number of columns factorized by ZLAHEF;
189 * KB is either NB or NB-1, or K for the last block
190 *
191 K = N
192 10 CONTINUE
193 *
194 * If K < 1, exit from loop
195 *
196 IF( K.LT.1 )
197 $ GO TO 40
198 *
199 IF( K.GT.NB ) THEN
200 *
201 * Factorize columns k-kb+1:k of A and use blocked code to
202 * update columns 1:k-kb
203 *
204 CALL ZLAHEF( UPLO, K, NB, KB, A, LDA, IPIV, WORK, N, IINFO )
205 ELSE
206 *
207 * Use unblocked code to factorize columns 1:k of A
208 *
209 CALL ZHETF2( UPLO, K, A, LDA, IPIV, IINFO )
210 KB = K
211 END IF
212 *
213 * Set INFO on the first occurrence of a zero pivot
214 *
215 IF( INFO.EQ.0 .AND. IINFO.GT.0 )
216 $ INFO = IINFO
217 *
218 * Decrease K and return to the start of the main loop
219 *
220 K = K - KB
221 GO TO 10
222 *
223 ELSE
224 *
225 * Factorize A as L*D*L**H using the lower triangle of A
226 *
227 * K is the main loop index, increasing from 1 to N in steps of
228 * KB, where KB is the number of columns factorized by ZLAHEF;
229 * KB is either NB or NB-1, or N-K+1 for the last block
230 *
231 K = 1
232 20 CONTINUE
233 *
234 * If K > N, exit from loop
235 *
236 IF( K.GT.N )
237 $ GO TO 40
238 *
239 IF( K.LE.N-NB ) THEN
240 *
241 * Factorize columns k:k+kb-1 of A and use blocked code to
242 * update columns k+kb:n
243 *
244 CALL ZLAHEF( UPLO, N-K+1, NB, KB, A( K, K ), LDA, IPIV( K ),
245 $ WORK, N, IINFO )
246 ELSE
247 *
248 * Use unblocked code to factorize columns k:n of A
249 *
250 CALL ZHETF2( UPLO, N-K+1, A( K, K ), LDA, IPIV( K ), IINFO )
251 KB = N - K + 1
252 END IF
253 *
254 * Set INFO on the first occurrence of a zero pivot
255 *
256 IF( INFO.EQ.0 .AND. IINFO.GT.0 )
257 $ INFO = IINFO + K - 1
258 *
259 * Adjust IPIV
260 *
261 DO 30 J = K, K + KB - 1
262 IF( IPIV( J ).GT.0 ) THEN
263 IPIV( J ) = IPIV( J ) + K - 1
264 ELSE
265 IPIV( J ) = IPIV( J ) - K + 1
266 END IF
267 30 CONTINUE
268 *
269 * Increase K and return to the start of the main loop
270 *
271 K = K + KB
272 GO TO 20
273 *
274 END IF
275 *
276 40 CONTINUE
277 WORK( 1 ) = LWKOPT
278 RETURN
279 *
280 * End of ZHETRF
281 *
282 END
2 *
3 * -- LAPACK routine (version 3.3.1) --
4 * -- LAPACK is a software package provided by Univ. of Tennessee, --
5 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6 * -- April 2011 --
7 *
8 * .. Scalar Arguments ..
9 CHARACTER UPLO
10 INTEGER INFO, LDA, LWORK, N
11 * ..
12 * .. Array Arguments ..
13 INTEGER IPIV( * )
14 COMPLEX*16 A( LDA, * ), WORK( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * ZHETRF computes the factorization of a complex Hermitian matrix A
21 * using the Bunch-Kaufman diagonal pivoting method. The form of the
22 * factorization is
23 *
24 * A = U*D*U**H or A = L*D*L**H
25 *
26 * where U (or L) is a product of permutation and unit upper (lower)
27 * triangular matrices, and D is Hermitian and block diagonal with
28 * 1-by-1 and 2-by-2 diagonal blocks.
29 *
30 * This is the blocked version of the algorithm, calling Level 3 BLAS.
31 *
32 * Arguments
33 * =========
34 *
35 * UPLO (input) CHARACTER*1
36 * = 'U': Upper triangle of A is stored;
37 * = 'L': Lower triangle of A is stored.
38 *
39 * N (input) INTEGER
40 * The order of the matrix A. N >= 0.
41 *
42 * A (input/output) COMPLEX*16 array, dimension (LDA,N)
43 * On entry, the Hermitian matrix A. If UPLO = 'U', the leading
44 * N-by-N upper triangular part of A contains the upper
45 * triangular part of the matrix A, and the strictly lower
46 * triangular part of A is not referenced. If UPLO = 'L', the
47 * leading N-by-N lower triangular part of A contains the lower
48 * triangular part of the matrix A, and the strictly upper
49 * triangular part of A is not referenced.
50 *
51 * On exit, the block diagonal matrix D and the multipliers used
52 * to obtain the factor U or L (see below for further details).
53 *
54 * LDA (input) INTEGER
55 * The leading dimension of the array A. LDA >= max(1,N).
56 *
57 * IPIV (output) INTEGER array, dimension (N)
58 * Details of the interchanges and the block structure of D.
59 * If IPIV(k) > 0, then rows and columns k and IPIV(k) were
60 * interchanged and D(k,k) is a 1-by-1 diagonal block.
61 * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
62 * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
63 * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
64 * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
65 * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
66 *
67 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
68 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
69 *
70 * LWORK (input) INTEGER
71 * The length of WORK. LWORK >=1. For best performance
72 * LWORK >= N*NB, where NB is the block size returned by ILAENV.
73 *
74 * INFO (output) INTEGER
75 * = 0: successful exit
76 * < 0: if INFO = -i, the i-th argument had an illegal value
77 * > 0: if INFO = i, D(i,i) is exactly zero. The factorization
78 * has been completed, but the block diagonal matrix D is
79 * exactly singular, and division by zero will occur if it
80 * is used to solve a system of equations.
81 *
82 * Further Details
83 * ===============
84 *
85 * If UPLO = 'U', then A = U*D*U**H, where
86 * U = P(n)*U(n)* ... *P(k)U(k)* ...,
87 * i.e., U is a product of terms P(k)*U(k), where k decreases from n to
88 * 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
89 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
90 * defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
91 * that if the diagonal block D(k) is of order s (s = 1 or 2), then
92 *
93 * ( I v 0 ) k-s
94 * U(k) = ( 0 I 0 ) s
95 * ( 0 0 I ) n-k
96 * k-s s n-k
97 *
98 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
99 * If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
100 * and A(k,k), and v overwrites A(1:k-2,k-1:k).
101 *
102 * If UPLO = 'L', then A = L*D*L**H, where
103 * L = P(1)*L(1)* ... *P(k)*L(k)* ...,
104 * i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
105 * n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
106 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
107 * defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
108 * that if the diagonal block D(k) is of order s (s = 1 or 2), then
109 *
110 * ( I 0 0 ) k-1
111 * L(k) = ( 0 I 0 ) s
112 * ( 0 v I ) n-k-s+1
113 * k-1 s n-k-s+1
114 *
115 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
116 * If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
117 * and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
118 *
119 * =====================================================================
120 *
121 * .. Local Scalars ..
122 LOGICAL LQUERY, UPPER
123 INTEGER IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
124 * ..
125 * .. External Functions ..
126 LOGICAL LSAME
127 INTEGER ILAENV
128 EXTERNAL LSAME, ILAENV
129 * ..
130 * .. External Subroutines ..
131 EXTERNAL XERBLA, ZHETF2, ZLAHEF
132 * ..
133 * .. Intrinsic Functions ..
134 INTRINSIC MAX
135 * ..
136 * .. Executable Statements ..
137 *
138 * Test the input parameters.
139 *
140 INFO = 0
141 UPPER = LSAME( UPLO, 'U' )
142 LQUERY = ( LWORK.EQ.-1 )
143 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
144 INFO = -1
145 ELSE IF( N.LT.0 ) THEN
146 INFO = -2
147 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
148 INFO = -4
149 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
150 INFO = -7
151 END IF
152 *
153 IF( INFO.EQ.0 ) THEN
154 *
155 * Determine the block size
156 *
157 NB = ILAENV( 1, 'ZHETRF', UPLO, N, -1, -1, -1 )
158 LWKOPT = N*NB
159 WORK( 1 ) = LWKOPT
160 END IF
161 *
162 IF( INFO.NE.0 ) THEN
163 CALL XERBLA( 'ZHETRF', -INFO )
164 RETURN
165 ELSE IF( LQUERY ) THEN
166 RETURN
167 END IF
168 *
169 NBMIN = 2
170 LDWORK = N
171 IF( NB.GT.1 .AND. NB.LT.N ) THEN
172 IWS = LDWORK*NB
173 IF( LWORK.LT.IWS ) THEN
174 NB = MAX( LWORK / LDWORK, 1 )
175 NBMIN = MAX( 2, ILAENV( 2, 'ZHETRF', UPLO, N, -1, -1, -1 ) )
176 END IF
177 ELSE
178 IWS = 1
179 END IF
180 IF( NB.LT.NBMIN )
181 $ NB = N
182 *
183 IF( UPPER ) THEN
184 *
185 * Factorize A as U*D*U**H using the upper triangle of A
186 *
187 * K is the main loop index, decreasing from N to 1 in steps of
188 * KB, where KB is the number of columns factorized by ZLAHEF;
189 * KB is either NB or NB-1, or K for the last block
190 *
191 K = N
192 10 CONTINUE
193 *
194 * If K < 1, exit from loop
195 *
196 IF( K.LT.1 )
197 $ GO TO 40
198 *
199 IF( K.GT.NB ) THEN
200 *
201 * Factorize columns k-kb+1:k of A and use blocked code to
202 * update columns 1:k-kb
203 *
204 CALL ZLAHEF( UPLO, K, NB, KB, A, LDA, IPIV, WORK, N, IINFO )
205 ELSE
206 *
207 * Use unblocked code to factorize columns 1:k of A
208 *
209 CALL ZHETF2( UPLO, K, A, LDA, IPIV, IINFO )
210 KB = K
211 END IF
212 *
213 * Set INFO on the first occurrence of a zero pivot
214 *
215 IF( INFO.EQ.0 .AND. IINFO.GT.0 )
216 $ INFO = IINFO
217 *
218 * Decrease K and return to the start of the main loop
219 *
220 K = K - KB
221 GO TO 10
222 *
223 ELSE
224 *
225 * Factorize A as L*D*L**H using the lower triangle of A
226 *
227 * K is the main loop index, increasing from 1 to N in steps of
228 * KB, where KB is the number of columns factorized by ZLAHEF;
229 * KB is either NB or NB-1, or N-K+1 for the last block
230 *
231 K = 1
232 20 CONTINUE
233 *
234 * If K > N, exit from loop
235 *
236 IF( K.GT.N )
237 $ GO TO 40
238 *
239 IF( K.LE.N-NB ) THEN
240 *
241 * Factorize columns k:k+kb-1 of A and use blocked code to
242 * update columns k+kb:n
243 *
244 CALL ZLAHEF( UPLO, N-K+1, NB, KB, A( K, K ), LDA, IPIV( K ),
245 $ WORK, N, IINFO )
246 ELSE
247 *
248 * Use unblocked code to factorize columns k:n of A
249 *
250 CALL ZHETF2( UPLO, N-K+1, A( K, K ), LDA, IPIV( K ), IINFO )
251 KB = N - K + 1
252 END IF
253 *
254 * Set INFO on the first occurrence of a zero pivot
255 *
256 IF( INFO.EQ.0 .AND. IINFO.GT.0 )
257 $ INFO = IINFO + K - 1
258 *
259 * Adjust IPIV
260 *
261 DO 30 J = K, K + KB - 1
262 IF( IPIV( J ).GT.0 ) THEN
263 IPIV( J ) = IPIV( J ) + K - 1
264 ELSE
265 IPIV( J ) = IPIV( J ) - K + 1
266 END IF
267 30 CONTINUE
268 *
269 * Increase K and return to the start of the main loop
270 *
271 K = K + KB
272 GO TO 20
273 *
274 END IF
275 *
276 40 CONTINUE
277 WORK( 1 ) = LWKOPT
278 RETURN
279 *
280 * End of ZHETRF
281 *
282 END