1DSYGVD(1) LAPACK driver routine (version 3.1) DSYGVD(1)
2
6 DSYGVD - all the eigenvalues, and optionally, the eigenvectors of a
7 real generalized symmetric-definite eigenproblem, of the form
8 A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x
9
11 SUBROUTINE DSYGVD( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
12 LWORK, IWORK, LIWORK, INFO )
13 CHARACTER JOBZ, UPLO
15 INTEGER INFO, ITYPE, LDA, LDB, LIWORK, LWORK, N
17 INTEGER IWORK( * )
19 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( *
21 )
22
24 DSYGVD computes all the eigenvalues, and optionally, the eigenvectors
25 of a real generalized symmetric-definite eigenproblem, of the form
26 A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B
27 are assumed to be symmetric and B is also positive definite. If eigen‐
28 vectors are desired, it uses a divide and conquer algorithm.
29 The divide and conquer algorithm makes very mild assumptions about
31 floating point arithmetic. It will work on machines with a guard digit
32 in add/subtract, or on those binary machines without guard digits which
33 subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could
34 conceivably fail on hexadecimal or decimal machines without guard dig‐
35 its, but we know of none.
36
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 JOBZ (input) CHARACTER*1
46 = 'N': Compute eigenvalues only;
47 = 'V': Compute eigenvalues and eigenvectors.
48 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 N (input) INTEGER
54 The order of the matrices A and B. N >= 0.
55 A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
57 On entry, the symmetric matrix A. If UPLO = 'U', the leading
58 N-by-N upper triangular part of A contains the upper triangular
59 part of the matrix A. If UPLO = 'L', the leading N-by-N lower
60 triangular part of A contains the lower triangular part of the
61 matrix A.
62 On exit, if JOBZ = 'V', then if INFO = 0, A contains the matrix
64 Z of eigenvectors. The eigenvectors are normalized as follows:
65 if ITYPE = 1 or 2, Z**T*B*Z = I; if ITYPE = 3, Z**T*inv(B)*Z =
66 I. If JOBZ = 'N', then on exit the upper triangle (if
67 UPLO='U') or the lower triangle (if UPLO='L') of A, including
68 the diagonal, is destroyed.
69 LDA (input) INTEGER
71 The leading dimension of the array A. LDA >= max(1,N).
72 B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
74 On entry, the symmetric matrix B. If UPLO = 'U', the leading
75 N-by-N upper triangular part of B contains the upper triangular
76 part of the matrix B. If UPLO = 'L', the leading N-by-N lower
77 triangular part of B contains the lower triangular part of the
78 matrix B.
79 On exit, if INFO <= N, the part of B containing the matrix is
81 overwritten by the triangular factor U or L from the Cholesky
82 factorization B = U**T*U or B = L*L**T.
83 LDB (input) INTEGER
85 The leading dimension of the array B. LDB >= max(1,N).
86 W (output) DOUBLE PRECISION array, dimension (N)
88 If INFO = 0, the eigenvalues in ascending order.
89 WORK (workspace/output) DOUBLE PRECISION array, dimension
91 (MAX(1,LWORK))
92 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
93 LWORK (input) INTEGER
95 The dimension of the array WORK. If N <= 1,
96 LWORK >= 1. If JOBZ = 'N' and N > 1, LWORK >= 2*N+1. If JOBZ
97 = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
98 If LWORK = -1, then a workspace query is assumed; the routine
100 only calculates the optimal sizes of the WORK and IWORK arrays,
101 returns these values as the first entries of the WORK and IWORK
102 arrays, and no error message related to LWORK or LIWORK is
103 issued by XERBLA.
104 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
106 On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
107 LIWORK (input) INTEGER
109 The dimension of the array IWORK. If N <= 1,
110 LIWORK >= 1. If JOBZ = 'N' and N > 1, LIWORK >= 1. If JOBZ
111 = 'V' and N > 1, LIWORK >= 3 + 5*N.
112 If LIWORK = -1, then a workspace query is assumed; the routine
114 only calculates the optimal sizes of the WORK and IWORK arrays,
115 returns these values as the first entries of the WORK and IWORK
116 arrays, and no error message related to LWORK or LIWORK is
117 issued by XERBLA.
118 INFO (output) INTEGER
120 = 0: successful exit
121 < 0: if INFO = -i, the i-th argument had an illegal value
122 > 0: DPOTRF or DSYEVD returned an error code:
123 <= N: if INFO = i and JOBZ = 'N', then the algorithm failed to
124 converge; i off-diagonal elements of an intermediate tridiago‐
125 nal form did not converge to zero; if INFO = i and JOBZ = 'V',
126 then the algorithm failed to compute an eigenvalue while work‐
127 ing on the submatrix lying in rows and columns INFO/(N+1)
128 through mod(INFO,N+1); > N: if INFO = N + i, for 1 <= i <= N,
129 then the leading minor of order i of B is not positive defi‐
130 nite. The factorization of B could not be completed and no ei‐
131 genvalues or eigenvectors were computed.
132
134 Based on contributions by
135 Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
136 Modified so that no backsubstitution is performed if DSYEVD fails to
138 converge (NEIG in old code could be greater than N causing out of
139 bounds reference to A - reported by Ralf Meyer). Also corrected the
140 description of INFO and the test on ITYPE. Sven, 16 Feb 05.
141 LAPACK driver routine (version 3.N1o)vember 2006 DSYGVD(1)