1 SUBROUTINE DSYTRF( 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 DOUBLE PRECISION A( LDA, * ), WORK( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DSYTRF computes the factorization of a real symmetric matrix A using
21 * the Bunch-Kaufman diagonal pivoting method. The form of the
22 * factorization is
23 *
24 * A = U*D*U**T or A = L*D*L**T
25 *
26 * where U (or L) is a product of permutation and unit upper (lower)
27 * triangular matrices, and D is symmetric 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) DOUBLE PRECISION array, dimension (LDA,N)
43 * On entry, the symmetric 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) DOUBLE PRECISION 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 * If LWORK = -1, then a workspace query is assumed; the routine
75 * only calculates the optimal size of the WORK array, returns
76 * this value as the first entry of the WORK array, and no error
77 * message related to LWORK is issued by XERBLA.
78 *
79 * INFO (output) INTEGER
80 * = 0: successful exit
81 * < 0: if INFO = -i, the i-th argument had an illegal value
82 * > 0: if INFO = i, D(i,i) is exactly zero. The factorization
83 * has been completed, but the block diagonal matrix D is
84 * exactly singular, and division by zero will occur if it
85 * is used to solve a system of equations.
86 *
87 * Further Details
88 * ===============
89 *
90 * If UPLO = 'U', then A = U*D*U**T, where
91 * U = P(n)*U(n)* ... *P(k)U(k)* ...,
92 * i.e., U is a product of terms P(k)*U(k), where k decreases from n to
93 * 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
94 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
95 * defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
96 * that if the diagonal block D(k) is of order s (s = 1 or 2), then
97 *
98 * ( I v 0 ) k-s
99 * U(k) = ( 0 I 0 ) s
100 * ( 0 0 I ) n-k
101 * k-s s n-k
102 *
103 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
104 * If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
105 * and A(k,k), and v overwrites A(1:k-2,k-1:k).
106 *
107 * If UPLO = 'L', then A = L*D*L**T, where
108 * L = P(1)*L(1)* ... *P(k)*L(k)* ...,
109 * i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
110 * n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
111 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
112 * defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
113 * that if the diagonal block D(k) is of order s (s = 1 or 2), then
114 *
115 * ( I 0 0 ) k-1
116 * L(k) = ( 0 I 0 ) s
117 * ( 0 v I ) n-k-s+1
118 * k-1 s n-k-s+1
119 *
120 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
121 * If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
122 * and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
123 *
124 * =====================================================================
125 *
126 * .. Local Scalars ..
127 LOGICAL LQUERY, UPPER
128 INTEGER IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
129 * ..
130 * .. External Functions ..
131 LOGICAL LSAME
132 INTEGER ILAENV
133 EXTERNAL LSAME, ILAENV
134 * ..
135 * .. External Subroutines ..
136 EXTERNAL DLASYF, DSYTF2, XERBLA
137 * ..
138 * .. Intrinsic Functions ..
139 INTRINSIC MAX
140 * ..
141 * .. Executable Statements ..
142 *
143 * Test the input parameters.
144 *
145 INFO = 0
146 UPPER = LSAME( UPLO, 'U' )
147 LQUERY = ( LWORK.EQ.-1 )
148 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
149 INFO = -1
150 ELSE IF( N.LT.0 ) THEN
151 INFO = -2
152 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
153 INFO = -4
154 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
155 INFO = -7
156 END IF
157 *
158 IF( INFO.EQ.0 ) THEN
159 *
160 * Determine the block size
161 *
162 NB = ILAENV( 1, 'DSYTRF', UPLO, N, -1, -1, -1 )
163 LWKOPT = N*NB
164 WORK( 1 ) = LWKOPT
165 END IF
166 *
167 IF( INFO.NE.0 ) THEN
168 CALL XERBLA( 'DSYTRF', -INFO )
169 RETURN
170 ELSE IF( LQUERY ) THEN
171 RETURN
172 END IF
173 *
174 NBMIN = 2
175 LDWORK = N
176 IF( NB.GT.1 .AND. NB.LT.N ) THEN
177 IWS = LDWORK*NB
178 IF( LWORK.LT.IWS ) THEN
179 NB = MAX( LWORK / LDWORK, 1 )
180 NBMIN = MAX( 2, ILAENV( 2, 'DSYTRF', UPLO, N, -1, -1, -1 ) )
181 END IF
182 ELSE
183 IWS = 1
184 END IF
185 IF( NB.LT.NBMIN )
186 $ NB = N
187 *
188 IF( UPPER ) THEN
189 *
190 * Factorize A as U*D*U**T using the upper triangle of A
191 *
192 * K is the main loop index, decreasing from N to 1 in steps of
193 * KB, where KB is the number of columns factorized by DLASYF;
194 * KB is either NB or NB-1, or K for the last block
195 *
196 K = N
197 10 CONTINUE
198 *
199 * If K < 1, exit from loop
200 *
201 IF( K.LT.1 )
202 $ GO TO 40
203 *
204 IF( K.GT.NB ) THEN
205 *
206 * Factorize columns k-kb+1:k of A and use blocked code to
207 * update columns 1:k-kb
208 *
209 CALL DLASYF( UPLO, K, NB, KB, A, LDA, IPIV, WORK, LDWORK,
210 $ IINFO )
211 ELSE
212 *
213 * Use unblocked code to factorize columns 1:k of A
214 *
215 CALL DSYTF2( UPLO, K, A, LDA, IPIV, IINFO )
216 KB = K
217 END IF
218 *
219 * Set INFO on the first occurrence of a zero pivot
220 *
221 IF( INFO.EQ.0 .AND. IINFO.GT.0 )
222 $ INFO = IINFO
223 *
224 * Decrease K and return to the start of the main loop
225 *
226 K = K - KB
227 GO TO 10
228 *
229 ELSE
230 *
231 * Factorize A as L*D*L**T using the lower triangle of A
232 *
233 * K is the main loop index, increasing from 1 to N in steps of
234 * KB, where KB is the number of columns factorized by DLASYF;
235 * KB is either NB or NB-1, or N-K+1 for the last block
236 *
237 K = 1
238 20 CONTINUE
239 *
240 * If K > N, exit from loop
241 *
242 IF( K.GT.N )
243 $ GO TO 40
244 *
245 IF( K.LE.N-NB ) THEN
246 *
247 * Factorize columns k:k+kb-1 of A and use blocked code to
248 * update columns k+kb:n
249 *
250 CALL DLASYF( UPLO, N-K+1, NB, KB, A( K, K ), LDA, IPIV( K ),
251 $ WORK, LDWORK, IINFO )
252 ELSE
253 *
254 * Use unblocked code to factorize columns k:n of A
255 *
256 CALL DSYTF2( UPLO, N-K+1, A( K, K ), LDA, IPIV( K ), IINFO )
257 KB = N - K + 1
258 END IF
259 *
260 * Set INFO on the first occurrence of a zero pivot
261 *
262 IF( INFO.EQ.0 .AND. IINFO.GT.0 )
263 $ INFO = IINFO + K - 1
264 *
265 * Adjust IPIV
266 *
267 DO 30 J = K, K + KB - 1
268 IF( IPIV( J ).GT.0 ) THEN
269 IPIV( J ) = IPIV( J ) + K - 1
270 ELSE
271 IPIV( J ) = IPIV( J ) - K + 1
272 END IF
273 30 CONTINUE
274 *
275 * Increase K and return to the start of the main loop
276 *
277 K = K + KB
278 GO TO 20
279 *
280 END IF
281 *
282 40 CONTINUE
283 WORK( 1 ) = LWKOPT
284 RETURN
285 *
286 * End of DSYTRF
287 *
288 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 DOUBLE PRECISION A( LDA, * ), WORK( * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DSYTRF computes the factorization of a real symmetric matrix A using
21 * the Bunch-Kaufman diagonal pivoting method. The form of the
22 * factorization is
23 *
24 * A = U*D*U**T or A = L*D*L**T
25 *
26 * where U (or L) is a product of permutation and unit upper (lower)
27 * triangular matrices, and D is symmetric 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) DOUBLE PRECISION array, dimension (LDA,N)
43 * On entry, the symmetric 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) DOUBLE PRECISION 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 * If LWORK = -1, then a workspace query is assumed; the routine
75 * only calculates the optimal size of the WORK array, returns
76 * this value as the first entry of the WORK array, and no error
77 * message related to LWORK is issued by XERBLA.
78 *
79 * INFO (output) INTEGER
80 * = 0: successful exit
81 * < 0: if INFO = -i, the i-th argument had an illegal value
82 * > 0: if INFO = i, D(i,i) is exactly zero. The factorization
83 * has been completed, but the block diagonal matrix D is
84 * exactly singular, and division by zero will occur if it
85 * is used to solve a system of equations.
86 *
87 * Further Details
88 * ===============
89 *
90 * If UPLO = 'U', then A = U*D*U**T, where
91 * U = P(n)*U(n)* ... *P(k)U(k)* ...,
92 * i.e., U is a product of terms P(k)*U(k), where k decreases from n to
93 * 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
94 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
95 * defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
96 * that if the diagonal block D(k) is of order s (s = 1 or 2), then
97 *
98 * ( I v 0 ) k-s
99 * U(k) = ( 0 I 0 ) s
100 * ( 0 0 I ) n-k
101 * k-s s n-k
102 *
103 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
104 * If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
105 * and A(k,k), and v overwrites A(1:k-2,k-1:k).
106 *
107 * If UPLO = 'L', then A = L*D*L**T, where
108 * L = P(1)*L(1)* ... *P(k)*L(k)* ...,
109 * i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
110 * n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
111 * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
112 * defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
113 * that if the diagonal block D(k) is of order s (s = 1 or 2), then
114 *
115 * ( I 0 0 ) k-1
116 * L(k) = ( 0 I 0 ) s
117 * ( 0 v I ) n-k-s+1
118 * k-1 s n-k-s+1
119 *
120 * If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
121 * If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
122 * and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
123 *
124 * =====================================================================
125 *
126 * .. Local Scalars ..
127 LOGICAL LQUERY, UPPER
128 INTEGER IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
129 * ..
130 * .. External Functions ..
131 LOGICAL LSAME
132 INTEGER ILAENV
133 EXTERNAL LSAME, ILAENV
134 * ..
135 * .. External Subroutines ..
136 EXTERNAL DLASYF, DSYTF2, XERBLA
137 * ..
138 * .. Intrinsic Functions ..
139 INTRINSIC MAX
140 * ..
141 * .. Executable Statements ..
142 *
143 * Test the input parameters.
144 *
145 INFO = 0
146 UPPER = LSAME( UPLO, 'U' )
147 LQUERY = ( LWORK.EQ.-1 )
148 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
149 INFO = -1
150 ELSE IF( N.LT.0 ) THEN
151 INFO = -2
152 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
153 INFO = -4
154 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
155 INFO = -7
156 END IF
157 *
158 IF( INFO.EQ.0 ) THEN
159 *
160 * Determine the block size
161 *
162 NB = ILAENV( 1, 'DSYTRF', UPLO, N, -1, -1, -1 )
163 LWKOPT = N*NB
164 WORK( 1 ) = LWKOPT
165 END IF
166 *
167 IF( INFO.NE.0 ) THEN
168 CALL XERBLA( 'DSYTRF', -INFO )
169 RETURN
170 ELSE IF( LQUERY ) THEN
171 RETURN
172 END IF
173 *
174 NBMIN = 2
175 LDWORK = N
176 IF( NB.GT.1 .AND. NB.LT.N ) THEN
177 IWS = LDWORK*NB
178 IF( LWORK.LT.IWS ) THEN
179 NB = MAX( LWORK / LDWORK, 1 )
180 NBMIN = MAX( 2, ILAENV( 2, 'DSYTRF', UPLO, N, -1, -1, -1 ) )
181 END IF
182 ELSE
183 IWS = 1
184 END IF
185 IF( NB.LT.NBMIN )
186 $ NB = N
187 *
188 IF( UPPER ) THEN
189 *
190 * Factorize A as U*D*U**T using the upper triangle of A
191 *
192 * K is the main loop index, decreasing from N to 1 in steps of
193 * KB, where KB is the number of columns factorized by DLASYF;
194 * KB is either NB or NB-1, or K for the last block
195 *
196 K = N
197 10 CONTINUE
198 *
199 * If K < 1, exit from loop
200 *
201 IF( K.LT.1 )
202 $ GO TO 40
203 *
204 IF( K.GT.NB ) THEN
205 *
206 * Factorize columns k-kb+1:k of A and use blocked code to
207 * update columns 1:k-kb
208 *
209 CALL DLASYF( UPLO, K, NB, KB, A, LDA, IPIV, WORK, LDWORK,
210 $ IINFO )
211 ELSE
212 *
213 * Use unblocked code to factorize columns 1:k of A
214 *
215 CALL DSYTF2( UPLO, K, A, LDA, IPIV, IINFO )
216 KB = K
217 END IF
218 *
219 * Set INFO on the first occurrence of a zero pivot
220 *
221 IF( INFO.EQ.0 .AND. IINFO.GT.0 )
222 $ INFO = IINFO
223 *
224 * Decrease K and return to the start of the main loop
225 *
226 K = K - KB
227 GO TO 10
228 *
229 ELSE
230 *
231 * Factorize A as L*D*L**T using the lower triangle of A
232 *
233 * K is the main loop index, increasing from 1 to N in steps of
234 * KB, where KB is the number of columns factorized by DLASYF;
235 * KB is either NB or NB-1, or N-K+1 for the last block
236 *
237 K = 1
238 20 CONTINUE
239 *
240 * If K > N, exit from loop
241 *
242 IF( K.GT.N )
243 $ GO TO 40
244 *
245 IF( K.LE.N-NB ) THEN
246 *
247 * Factorize columns k:k+kb-1 of A and use blocked code to
248 * update columns k+kb:n
249 *
250 CALL DLASYF( UPLO, N-K+1, NB, KB, A( K, K ), LDA, IPIV( K ),
251 $ WORK, LDWORK, IINFO )
252 ELSE
253 *
254 * Use unblocked code to factorize columns k:n of A
255 *
256 CALL DSYTF2( UPLO, N-K+1, A( K, K ), LDA, IPIV( K ), IINFO )
257 KB = N - K + 1
258 END IF
259 *
260 * Set INFO on the first occurrence of a zero pivot
261 *
262 IF( INFO.EQ.0 .AND. IINFO.GT.0 )
263 $ INFO = IINFO + K - 1
264 *
265 * Adjust IPIV
266 *
267 DO 30 J = K, K + KB - 1
268 IF( IPIV( J ).GT.0 ) THEN
269 IPIV( J ) = IPIV( J ) + K - 1
270 ELSE
271 IPIV( J ) = IPIV( J ) - K + 1
272 END IF
273 30 CONTINUE
274 *
275 * Increase K and return to the start of the main loop
276 *
277 K = K + KB
278 GO TO 20
279 *
280 END IF
281 *
282 40 CONTINUE
283 WORK( 1 ) = LWKOPT
284 RETURN
285 *
286 * End of DSYTRF
287 *
288 END