1 SUBROUTINE DSPGVD( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK,
2 $ LWORK, IWORK, LIWORK, INFO )
3 *
4 * -- LAPACK driver routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 CHARACTER JOBZ, UPLO
11 INTEGER INFO, ITYPE, LDZ, LIWORK, LWORK, N
12 * ..
13 * .. Array Arguments ..
14 INTEGER IWORK( * )
15 DOUBLE PRECISION AP( * ), BP( * ), W( * ), WORK( * ),
16 $ Z( LDZ, * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * DSPGVD computes all the eigenvalues, and optionally, the eigenvectors
23 * of a real generalized symmetric-definite eigenproblem, of the form
24 * A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and
25 * B are assumed to be symmetric, stored in packed format, and B is also
26 * positive definite.
27 * If eigenvectors are desired, it uses a divide and conquer algorithm.
28 *
29 * The divide and conquer algorithm makes very mild assumptions about
30 * floating point arithmetic. It will work on machines with a guard
31 * digit in add/subtract, or on those binary machines without guard
32 * digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
33 * Cray-2. It could conceivably fail on hexadecimal or decimal machines
34 * without guard digits, but we know of none.
35 *
36 * Arguments
37 * =========
38 *
39 * ITYPE (input) INTEGER
40 * Specifies the problem type to be solved:
41 * = 1: A*x = (lambda)*B*x
42 * = 2: A*B*x = (lambda)*x
43 * = 3: B*A*x = (lambda)*x
44 *
45 * JOBZ (input) CHARACTER*1
46 * = 'N': Compute eigenvalues only;
47 * = 'V': Compute eigenvalues and eigenvectors.
48 *
49 * UPLO (input) CHARACTER*1
50 * = 'U': Upper triangles of A and B are stored;
51 * = 'L': Lower triangles of A and B are stored.
52 *
53 * N (input) INTEGER
54 * The order of the matrices A and B. N >= 0.
55 *
56 * AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
57 * On entry, the upper or lower triangle of the symmetric matrix
58 * A, packed columnwise in a linear array. The j-th column of A
59 * is stored in the array AP as follows:
60 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
61 * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
62 *
63 * On exit, the contents of AP are destroyed.
64 *
65 * BP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
66 * On entry, the upper or lower triangle of the symmetric matrix
67 * B, packed columnwise in a linear array. The j-th column of B
68 * is stored in the array BP as follows:
69 * if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
70 * if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
71 *
72 * On exit, the triangular factor U or L from the Cholesky
73 * factorization B = U**T*U or B = L*L**T, in the same storage
74 * format as B.
75 *
76 * W (output) DOUBLE PRECISION array, dimension (N)
77 * If INFO = 0, the eigenvalues in ascending order.
78 *
79 * Z (output) DOUBLE PRECISION array, dimension (LDZ, N)
80 * If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
81 * eigenvectors. The eigenvectors are normalized as follows:
82 * if ITYPE = 1 or 2, Z**T*B*Z = I;
83 * if ITYPE = 3, Z**T*inv(B)*Z = I.
84 * If JOBZ = 'N', then Z is not referenced.
85 *
86 * LDZ (input) INTEGER
87 * The leading dimension of the array Z. LDZ >= 1, and if
88 * JOBZ = 'V', LDZ >= max(1,N).
89 *
90 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
91 * On exit, if INFO = 0, WORK(1) returns the required LWORK.
92 *
93 * LWORK (input) INTEGER
94 * The dimension of the array WORK.
95 * If N <= 1, LWORK >= 1.
96 * If JOBZ = 'N' and N > 1, LWORK >= 2*N.
97 * If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
98 *
99 * If LWORK = -1, then a workspace query is assumed; the routine
100 * only calculates the required sizes of the WORK and IWORK
101 * arrays, returns these values as the first entries of the WORK
102 * and IWORK arrays, and no error message related to LWORK or
103 * LIWORK is issued by XERBLA.
104 *
105 * IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
106 * On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
107 *
108 * LIWORK (input) INTEGER
109 * The dimension of the array IWORK.
110 * If JOBZ = 'N' or N <= 1, LIWORK >= 1.
111 * If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
112 *
113 * If LIWORK = -1, then a workspace query is assumed; the
114 * routine only calculates the required sizes of the WORK and
115 * IWORK arrays, returns these values as the first entries of
116 * the WORK and IWORK arrays, and no error message related to
117 * LWORK or LIWORK is issued by XERBLA.
118 *
119 * INFO (output) INTEGER
120 * = 0: successful exit
121 * < 0: if INFO = -i, the i-th argument had an illegal value
122 * > 0: DPPTRF or DSPEVD returned an error code:
123 * <= N: if INFO = i, DSPEVD failed to converge;
124 * i off-diagonal elements of an intermediate
125 * tridiagonal form did not converge to zero;
126 * > N: if INFO = N + i, for 1 <= i <= N, then the leading
127 * minor of order i of B is not positive definite.
128 * The factorization of B could not be completed and
129 * no eigenvalues or eigenvectors were computed.
130 *
131 * Further Details
132 * ===============
133 *
134 * Based on contributions by
135 * Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
136 *
137 * =====================================================================
138 *
139 * .. Parameters ..
140 DOUBLE PRECISION TWO
141 PARAMETER ( TWO = 2.0D+0 )
142 * ..
143 * .. Local Scalars ..
144 LOGICAL LQUERY, UPPER, WANTZ
145 CHARACTER TRANS
146 INTEGER J, LIWMIN, LWMIN, NEIG
147 * ..
148 * .. External Functions ..
149 LOGICAL LSAME
150 EXTERNAL LSAME
151 * ..
152 * .. External Subroutines ..
153 EXTERNAL DPPTRF, DSPEVD, DSPGST, DTPMV, DTPSV, XERBLA
154 * ..
155 * .. Intrinsic Functions ..
156 INTRINSIC DBLE, MAX
157 * ..
158 * .. Executable Statements ..
159 *
160 * Test the input parameters.
161 *
162 WANTZ = LSAME( JOBZ, 'V' )
163 UPPER = LSAME( UPLO, 'U' )
164 LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
165 *
166 INFO = 0
167 IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
168 INFO = -1
169 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
170 INFO = -2
171 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
172 INFO = -3
173 ELSE IF( N.LT.0 ) THEN
174 INFO = -4
175 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
176 INFO = -9
177 END IF
178 *
179 IF( INFO.EQ.0 ) THEN
180 IF( N.LE.1 ) THEN
181 LIWMIN = 1
182 LWMIN = 1
183 ELSE
184 IF( WANTZ ) THEN
185 LIWMIN = 3 + 5*N
186 LWMIN = 1 + 6*N + 2*N**2
187 ELSE
188 LIWMIN = 1
189 LWMIN = 2*N
190 END IF
191 END IF
192 WORK( 1 ) = LWMIN
193 IWORK( 1 ) = LIWMIN
194 IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
195 INFO = -11
196 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
197 INFO = -13
198 END IF
199 END IF
200 *
201 IF( INFO.NE.0 ) THEN
202 CALL XERBLA( 'DSPGVD', -INFO )
203 RETURN
204 ELSE IF( LQUERY ) THEN
205 RETURN
206 END IF
207 *
208 * Quick return if possible
209 *
210 IF( N.EQ.0 )
211 $ RETURN
212 *
213 * Form a Cholesky factorization of BP.
214 *
215 CALL DPPTRF( UPLO, N, BP, INFO )
216 IF( INFO.NE.0 ) THEN
217 INFO = N + INFO
218 RETURN
219 END IF
220 *
221 * Transform problem to standard eigenvalue problem and solve.
222 *
223 CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
224 CALL DSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, IWORK,
225 $ LIWORK, INFO )
226 LWMIN = MAX( DBLE( LWMIN ), DBLE( WORK( 1 ) ) )
227 LIWMIN = MAX( DBLE( LIWMIN ), DBLE( IWORK( 1 ) ) )
228 *
229 IF( WANTZ ) THEN
230 *
231 * Backtransform eigenvectors to the original problem.
232 *
233 NEIG = N
234 IF( INFO.GT.0 )
235 $ NEIG = INFO - 1
236 IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
237 *
238 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
239 * backtransform eigenvectors: x = inv(L)**T *y or inv(U)*y
240 *
241 IF( UPPER ) THEN
242 TRANS = 'N'
243 ELSE
244 TRANS = 'T'
245 END IF
246 *
247 DO 10 J = 1, NEIG
248 CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
249 $ 1 )
250 10 CONTINUE
251 *
252 ELSE IF( ITYPE.EQ.3 ) THEN
253 *
254 * For B*A*x=(lambda)*x;
255 * backtransform eigenvectors: x = L*y or U**T *y
256 *
257 IF( UPPER ) THEN
258 TRANS = 'T'
259 ELSE
260 TRANS = 'N'
261 END IF
262 *
263 DO 20 J = 1, NEIG
264 CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
265 $ 1 )
266 20 CONTINUE
267 END IF
268 END IF
269 *
270 WORK( 1 ) = LWMIN
271 IWORK( 1 ) = LIWMIN
272 *
273 RETURN
274 *
275 * End of DSPGVD
276 *
277 END
2 $ LWORK, IWORK, LIWORK, INFO )
3 *
4 * -- LAPACK driver routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 CHARACTER JOBZ, UPLO
11 INTEGER INFO, ITYPE, LDZ, LIWORK, LWORK, N
12 * ..
13 * .. Array Arguments ..
14 INTEGER IWORK( * )
15 DOUBLE PRECISION AP( * ), BP( * ), W( * ), WORK( * ),
16 $ Z( LDZ, * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * DSPGVD computes all the eigenvalues, and optionally, the eigenvectors
23 * of a real generalized symmetric-definite eigenproblem, of the form
24 * A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and
25 * B are assumed to be symmetric, stored in packed format, and B is also
26 * positive definite.
27 * If eigenvectors are desired, it uses a divide and conquer algorithm.
28 *
29 * The divide and conquer algorithm makes very mild assumptions about
30 * floating point arithmetic. It will work on machines with a guard
31 * digit in add/subtract, or on those binary machines without guard
32 * digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
33 * Cray-2. It could conceivably fail on hexadecimal or decimal machines
34 * without guard digits, but we know of none.
35 *
36 * Arguments
37 * =========
38 *
39 * ITYPE (input) INTEGER
40 * Specifies the problem type to be solved:
41 * = 1: A*x = (lambda)*B*x
42 * = 2: A*B*x = (lambda)*x
43 * = 3: B*A*x = (lambda)*x
44 *
45 * JOBZ (input) CHARACTER*1
46 * = 'N': Compute eigenvalues only;
47 * = 'V': Compute eigenvalues and eigenvectors.
48 *
49 * UPLO (input) CHARACTER*1
50 * = 'U': Upper triangles of A and B are stored;
51 * = 'L': Lower triangles of A and B are stored.
52 *
53 * N (input) INTEGER
54 * The order of the matrices A and B. N >= 0.
55 *
56 * AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
57 * On entry, the upper or lower triangle of the symmetric matrix
58 * A, packed columnwise in a linear array. The j-th column of A
59 * is stored in the array AP as follows:
60 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
61 * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
62 *
63 * On exit, the contents of AP are destroyed.
64 *
65 * BP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
66 * On entry, the upper or lower triangle of the symmetric matrix
67 * B, packed columnwise in a linear array. The j-th column of B
68 * is stored in the array BP as follows:
69 * if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
70 * if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
71 *
72 * On exit, the triangular factor U or L from the Cholesky
73 * factorization B = U**T*U or B = L*L**T, in the same storage
74 * format as B.
75 *
76 * W (output) DOUBLE PRECISION array, dimension (N)
77 * If INFO = 0, the eigenvalues in ascending order.
78 *
79 * Z (output) DOUBLE PRECISION array, dimension (LDZ, N)
80 * If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
81 * eigenvectors. The eigenvectors are normalized as follows:
82 * if ITYPE = 1 or 2, Z**T*B*Z = I;
83 * if ITYPE = 3, Z**T*inv(B)*Z = I.
84 * If JOBZ = 'N', then Z is not referenced.
85 *
86 * LDZ (input) INTEGER
87 * The leading dimension of the array Z. LDZ >= 1, and if
88 * JOBZ = 'V', LDZ >= max(1,N).
89 *
90 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
91 * On exit, if INFO = 0, WORK(1) returns the required LWORK.
92 *
93 * LWORK (input) INTEGER
94 * The dimension of the array WORK.
95 * If N <= 1, LWORK >= 1.
96 * If JOBZ = 'N' and N > 1, LWORK >= 2*N.
97 * If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
98 *
99 * If LWORK = -1, then a workspace query is assumed; the routine
100 * only calculates the required sizes of the WORK and IWORK
101 * arrays, returns these values as the first entries of the WORK
102 * and IWORK arrays, and no error message related to LWORK or
103 * LIWORK is issued by XERBLA.
104 *
105 * IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
106 * On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
107 *
108 * LIWORK (input) INTEGER
109 * The dimension of the array IWORK.
110 * If JOBZ = 'N' or N <= 1, LIWORK >= 1.
111 * If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
112 *
113 * If LIWORK = -1, then a workspace query is assumed; the
114 * routine only calculates the required sizes of the WORK and
115 * IWORK arrays, returns these values as the first entries of
116 * the WORK and IWORK arrays, and no error message related to
117 * LWORK or LIWORK is issued by XERBLA.
118 *
119 * INFO (output) INTEGER
120 * = 0: successful exit
121 * < 0: if INFO = -i, the i-th argument had an illegal value
122 * > 0: DPPTRF or DSPEVD returned an error code:
123 * <= N: if INFO = i, DSPEVD failed to converge;
124 * i off-diagonal elements of an intermediate
125 * tridiagonal form did not converge to zero;
126 * > N: if INFO = N + i, for 1 <= i <= N, then the leading
127 * minor of order i of B is not positive definite.
128 * The factorization of B could not be completed and
129 * no eigenvalues or eigenvectors were computed.
130 *
131 * Further Details
132 * ===============
133 *
134 * Based on contributions by
135 * Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
136 *
137 * =====================================================================
138 *
139 * .. Parameters ..
140 DOUBLE PRECISION TWO
141 PARAMETER ( TWO = 2.0D+0 )
142 * ..
143 * .. Local Scalars ..
144 LOGICAL LQUERY, UPPER, WANTZ
145 CHARACTER TRANS
146 INTEGER J, LIWMIN, LWMIN, NEIG
147 * ..
148 * .. External Functions ..
149 LOGICAL LSAME
150 EXTERNAL LSAME
151 * ..
152 * .. External Subroutines ..
153 EXTERNAL DPPTRF, DSPEVD, DSPGST, DTPMV, DTPSV, XERBLA
154 * ..
155 * .. Intrinsic Functions ..
156 INTRINSIC DBLE, MAX
157 * ..
158 * .. Executable Statements ..
159 *
160 * Test the input parameters.
161 *
162 WANTZ = LSAME( JOBZ, 'V' )
163 UPPER = LSAME( UPLO, 'U' )
164 LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
165 *
166 INFO = 0
167 IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
168 INFO = -1
169 ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
170 INFO = -2
171 ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
172 INFO = -3
173 ELSE IF( N.LT.0 ) THEN
174 INFO = -4
175 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
176 INFO = -9
177 END IF
178 *
179 IF( INFO.EQ.0 ) THEN
180 IF( N.LE.1 ) THEN
181 LIWMIN = 1
182 LWMIN = 1
183 ELSE
184 IF( WANTZ ) THEN
185 LIWMIN = 3 + 5*N
186 LWMIN = 1 + 6*N + 2*N**2
187 ELSE
188 LIWMIN = 1
189 LWMIN = 2*N
190 END IF
191 END IF
192 WORK( 1 ) = LWMIN
193 IWORK( 1 ) = LIWMIN
194 IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
195 INFO = -11
196 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
197 INFO = -13
198 END IF
199 END IF
200 *
201 IF( INFO.NE.0 ) THEN
202 CALL XERBLA( 'DSPGVD', -INFO )
203 RETURN
204 ELSE IF( LQUERY ) THEN
205 RETURN
206 END IF
207 *
208 * Quick return if possible
209 *
210 IF( N.EQ.0 )
211 $ RETURN
212 *
213 * Form a Cholesky factorization of BP.
214 *
215 CALL DPPTRF( UPLO, N, BP, INFO )
216 IF( INFO.NE.0 ) THEN
217 INFO = N + INFO
218 RETURN
219 END IF
220 *
221 * Transform problem to standard eigenvalue problem and solve.
222 *
223 CALL DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
224 CALL DSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, IWORK,
225 $ LIWORK, INFO )
226 LWMIN = MAX( DBLE( LWMIN ), DBLE( WORK( 1 ) ) )
227 LIWMIN = MAX( DBLE( LIWMIN ), DBLE( IWORK( 1 ) ) )
228 *
229 IF( WANTZ ) THEN
230 *
231 * Backtransform eigenvectors to the original problem.
232 *
233 NEIG = N
234 IF( INFO.GT.0 )
235 $ NEIG = INFO - 1
236 IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
237 *
238 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
239 * backtransform eigenvectors: x = inv(L)**T *y or inv(U)*y
240 *
241 IF( UPPER ) THEN
242 TRANS = 'N'
243 ELSE
244 TRANS = 'T'
245 END IF
246 *
247 DO 10 J = 1, NEIG
248 CALL DTPSV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
249 $ 1 )
250 10 CONTINUE
251 *
252 ELSE IF( ITYPE.EQ.3 ) THEN
253 *
254 * For B*A*x=(lambda)*x;
255 * backtransform eigenvectors: x = L*y or U**T *y
256 *
257 IF( UPPER ) THEN
258 TRANS = 'T'
259 ELSE
260 TRANS = 'N'
261 END IF
262 *
263 DO 20 J = 1, NEIG
264 CALL DTPMV( UPLO, TRANS, 'Non-unit', N, BP, Z( 1, J ),
265 $ 1 )
266 20 CONTINUE
267 END IF
268 END IF
269 *
270 WORK( 1 ) = LWMIN
271 IWORK( 1 ) = LIWMIN
272 *
273 RETURN
274 *
275 * End of DSPGVD
276 *
277 END