1SSYGVX(1) LAPACK driver routine (version 3.1) SSYGVX(1)
2
6 SSYGVX - selected eigenvalues, and optionally, eigenvectors of a real
7 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 SSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, VL, VU,
12 IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK,
13 IFAIL, INFO )
14 CHARACTER JOBZ, RANGE, UPLO
16 INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
18 REAL ABSTOL, VL, VU
20 INTEGER IFAIL( * ), IWORK( * )
22 REAL A( LDA, * ), B( LDB, * ), W( * ), WORK( * ), Z( LDZ,
24 * )
25
27 SSYGVX computes selected eigenvalues, and optionally, eigenvectors of a
28 real generalized symmetric-definite eigenproblem, of the form
29 A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B
30 are assumed to be symmetric and B is also positive definite. Eigenval‐
31 ues and eigenvectors can be selected by specifying either a range of
32 values or a range of indices for the desired eigenvalues.
33
36 ITYPE (input) INTEGER
37 Specifies the problem type to be solved:
38 = 1: A*x = (lambda)*B*x
39 = 2: A*B*x = (lambda)*x
40 = 3: B*A*x = (lambda)*x
41 JOBZ (input) CHARACTER*1
43 = 'N': Compute eigenvalues only;
44 = 'V': Compute eigenvalues and eigenvectors.
45 RANGE (input) CHARACTER*1
47 = 'A': all eigenvalues will be found.
48 = 'V': all eigenvalues in the half-open interval (VL,VU] will
49 be found. = 'I': the IL-th through IU-th eigenvalues will be
50 found.
51 UPLO (input) CHARACTER*1
53 = 'U': Upper triangle of A and B are stored;
54 = 'L': Lower triangle of A and B are stored.
55 N (input) INTEGER
57 The order of the matrix pencil (A,B). N >= 0.
58 A (input/output) REAL array, dimension (LDA, N)
60 On entry, the symmetric matrix A. If UPLO = 'U', the leading
61 N-by-N upper triangular part of A contains the upper triangular
62 part of the matrix A. If UPLO = 'L', the leading N-by-N lower
63 triangular part of A contains the lower triangular part of the
64 matrix A.
65 On exit, the lower triangle (if UPLO='L') or the upper triangle
67 (if UPLO='U') of A, including the diagonal, is destroyed.
68 LDA (input) INTEGER
70 The leading dimension of the array A. LDA >= max(1,N).
71 B (input/output) REAL array, dimension (LDA, N)
73 On entry, the symmetric matrix B. If UPLO = 'U', the leading
74 N-by-N upper triangular part of B contains the upper triangular
75 part of the matrix B. If UPLO = 'L', the leading N-by-N lower
76 triangular part of B contains the lower triangular part of the
77 matrix B.
78 On exit, if INFO <= N, the part of B containing the matrix is
80 overwritten by the triangular factor U or L from the Cholesky
81 factorization B = U**T*U or B = L*L**T.
82 LDB (input) INTEGER
84 The leading dimension of the array B. LDB >= max(1,N).
85 VL (input) REAL
87 VU (input) REAL If RANGE='V', the lower and upper bounds
88 of the interval to be searched for eigenvalues. VL < VU. Not
89 referenced if RANGE = 'A' or 'I'.
90 IL (input) INTEGER
92 IU (input) INTEGER If RANGE='I', the indices (in ascending
93 order) of the smallest and largest eigenvalues to be returned.
94 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
95 referenced if RANGE = 'A' or 'V'.
96 ABSTOL (input) REAL
98 The absolute error tolerance for the eigenvalues. An approxi‐
99 mate eigenvalue is accepted as converged when it is determined
100 to lie in an interval [a,b] of width less than or equal to
101 ABSTOL + EPS * max( |a|,|b| ) ,
103 where EPS is the machine precision. If ABSTOL is less than or
105 equal to zero, then EPS*|T| will be used in its place, where
106 |T| is the 1-norm of the tridiagonal matrix obtained by reduc‐
107 ing A to tridiagonal form.
108 Eigenvalues will be computed most accurately when ABSTOL is set
110 to twice the underflow threshold 2*DLAMCH('S'), not zero. If
111 this routine returns with INFO>0, indicating that some eigen‐
112 vectors did not converge, try setting ABSTOL to 2*SLAMCH('S').
113 M (output) INTEGER
115 The total number of eigenvalues found. 0 <= M <= N. If RANGE
116 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
117 W (output) REAL array, dimension (N)
119 On normal exit, the first M elements contain the selected ei‐
120 genvalues in ascending order.
121 Z (output) REAL array, dimension (LDZ, max(1,M))
123 If JOBZ = 'N', then Z is not referenced. If JOBZ = 'V', then
124 if INFO = 0, the first M columns of Z contain the orthonormal
125 eigenvectors of the matrix A corresponding to the selected ei‐
126 genvalues, with the i-th column of Z holding the eigenvector
127 associated with W(i). The eigenvectors are normalized as fol‐
128 lows: if ITYPE = 1 or 2, Z**T*B*Z = I; if ITYPE = 3,
129 Z**T*inv(B)*Z = I.
130 If an eigenvector fails to converge, then that column of Z con‐
132 tains the latest approximation to the eigenvector, and the
133 index of the eigenvector is returned in IFAIL. Note: the user
134 must ensure that at least max(1,M) columns are supplied in the
135 array Z; if RANGE = 'V', the exact value of M is not known in
136 advance and an upper bound must be used.
137 LDZ (input) INTEGER
139 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
140 'V', LDZ >= max(1,N).
141 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
143 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
144 LWORK (input) INTEGER
146 The length of the array WORK. LWORK >= max(1,8*N). For opti‐
147 mal efficiency, LWORK >= (NB+3)*N, where NB is the blocksize
148 for SSYTRD returned by ILAENV.
149 If LWORK = -1, then a workspace query is assumed; the routine
151 only calculates the optimal size of the WORK array, returns
152 this value as the first entry of the WORK array, and no error
153 message related to LWORK is issued by XERBLA.
154 IWORK (workspace) INTEGER array, dimension (5*N)
156 IFAIL (output) INTEGER array, dimension (N)
158 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
159 are zero. If INFO > 0, then IFAIL contains the indices of the
160 eigenvectors that failed to converge. If JOBZ = 'N', then
161 IFAIL is not referenced.
162 INFO (output) INTEGER
164 = 0: successful exit
165 < 0: if INFO = -i, the i-th argument had an illegal value
166 > 0: SPOTRF or SSYEVX returned an error code:
167 <= N: if INFO = i, SSYEVX failed to converge; i eigenvectors
168 failed to converge. Their indices are stored in array IFAIL.
169 > N: if INFO = N + i, for 1 <= i <= N, then the leading minor
170 of order i of B is not positive definite. The factorization of
171 B could not be completed and no eigenvalues or eigenvectors
172 were computed.
173
175 Based on contributions by
176 Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
177 LAPACK driver routine (version 3.N1o)vember 2006 SSYGVX(1)