1SSPGVD(1) LAPACK driver routine (version 3.2) SSPGVD(1)
2
6 SSPGVD - computes all the eigenvalues, and optionally, the eigenvectors
7 of a 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 SSPGVD( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK,
12 LWORK, IWORK, LIWORK, INFO )
13 CHARACTER JOBZ, UPLO
15 INTEGER INFO, ITYPE, LDZ, LIWORK, LWORK, N
17 INTEGER IWORK( * )
19 REAL AP( * ), BP( * ), W( * ), WORK( * ), Z( LDZ, * )
21
23 SSPGVD computes all the eigenvalues, and optionally, the eigenvectors
24 of a real generalized symmetric-definite eigenproblem, of the form
25 A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B
26 are assumed to be symmetric, stored in packed format, and B is also
27 positive definite.
28 If eigenvectors are desired, it uses a divide and conquer algorithm.
29 The divide and conquer algorithm makes very mild assumptions about
30 floating point arithmetic. It will work on machines with a guard digit
31 in add/subtract, or on those binary machines without guard digits which
32 subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could
33 conceivably fail on hexadecimal or decimal machines without guard dig‐
34 its, but we know of none.
35
37 ITYPE (input) INTEGER
38 Specifies the problem type to be solved:
39 = 1: A*x = (lambda)*B*x
40 = 2: A*B*x = (lambda)*x
41 = 3: B*A*x = (lambda)*x
42 JOBZ (input) CHARACTER*1
44 = 'N': Compute eigenvalues only;
45 = 'V': Compute eigenvalues and eigenvectors.
46 UPLO (input) CHARACTER*1
48 = 'U': Upper triangles of A and B are stored;
49 = 'L': Lower triangles of A and B are stored.
50 N (input) INTEGER
52 The order of the matrices A and B. N >= 0.
53 AP (input/output) REAL array, dimension (N*(N+1)/2)
55 On entry, the upper or lower triangle of the symmetric matrix
56 A, packed columnwise in a linear array. The j-th column of A
57 is stored in the array AP as follows: if UPLO = 'U', AP(i +
58 (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
59 (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, the contents
60 of AP are destroyed.
61 BP (input/output) REAL array, dimension (N*(N+1)/2)
63 On entry, the upper or lower triangle of the symmetric matrix
64 B, packed columnwise in a linear array. The j-th column of B
65 is stored in the array BP as follows: if UPLO = 'U', BP(i +
66 (j-1)*j/2) = B(i,j) for 1<=i<=j; if UPLO = 'L', BP(i +
67 (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. On exit, the triangular
68 factor U or L from the Cholesky factorization B = U**T*U or B =
69 L*L**T, in the same storage format as B.
70 W (output) REAL array, dimension (N)
72 If INFO = 0, the eigenvalues in ascending order.
73 Z (output) REAL array, dimension (LDZ, N)
75 If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
76 eigenvectors. The eigenvectors are normalized as follows: if
77 ITYPE = 1 or 2, Z**T*B*Z = I; if ITYPE = 3, Z**T*inv(B)*Z = I.
78 If JOBZ = 'N', then Z is not referenced.
79 LDZ (input) INTEGER
81 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
82 'V', LDZ >= max(1,N).
83 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
85 On exit, if INFO = 0, WORK(1) returns the required LWORK.
86 LWORK (input) INTEGER
88 The dimension of the array WORK. If N <= 1,
89 LWORK >= 1. If JOBZ = 'N' and N > 1, LWORK >= 2*N. If JOBZ =
90 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2. If LWORK = -1, then
91 a workspace query is assumed; the routine only calculates the
92 required sizes of the WORK and IWORK arrays, returns these val‐
93 ues as the first entries of the WORK and IWORK arrays, and no
94 error message related to LWORK or LIWORK is issued by XERBLA.
95 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
97 On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
98 LIWORK (input) INTEGER
100 The dimension of the array IWORK. If JOBZ = 'N' or N <= 1,
101 LIWORK >= 1. If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N. If
102 LIWORK = -1, then a workspace query is assumed; the routine
103 only calculates the required sizes of the WORK and IWORK
104 arrays, returns these values as the first entries of the WORK
105 and IWORK arrays, and no error message related to LWORK or
106 LIWORK is issued by XERBLA.
107 INFO (output) INTEGER
109 = 0: successful exit
110 < 0: if INFO = -i, the i-th argument had an illegal value
111 > 0: SPPTRF or SSPEVD returned an error code:
112 <= N: if INFO = i, SSPEVD failed to converge; i off-diagonal
113 elements of an intermediate tridiagonal form did not converge
114 to zero; > N: if INFO = N + i, for 1 <= i <= N, then the
115 leading minor of order i of B is not positive definite. The
116 factorization of B could not be completed and no eigenvalues or
117 eigenvectors were computed.
118
120 Based on contributions by
121 Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
122 LAPACK driver routine (version 3.N2o)vember 2008 SSPGVD(1)