1SSPGVD(1) LAPACK driver routine (version 3.1) SSPGVD(1)
2
6 SSPGVD - 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 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
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 AP (input/output) REAL 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: if UPLO = 'U', AP(i +
60 (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
61 (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
62 On exit, the contents of AP are destroyed.
64 BP (input/output) REAL 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: if UPLO = 'U', BP(i +
69 (j-1)*j/2) = B(i,j) for 1<=i<=j; if UPLO = 'L', BP(i +
70 (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
71 On exit, the triangular factor U or L from the Cholesky factor‐
73 ization B = U**T*U or B = L*L**T, in the same storage format as
74 B.
75 W (output) REAL array, dimension (N)
77 If INFO = 0, the eigenvalues in ascending order.
78 Z (output) REAL 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: if
82 ITYPE = 1 or 2, Z**T*B*Z = I; if ITYPE = 3, Z**T*inv(B)*Z = I.
83 If JOBZ = 'N', then Z is not referenced.
84 LDZ (input) INTEGER
86 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
87 'V', LDZ >= max(1,N).
88 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
90 On exit, if INFO = 0, WORK(1) returns the required LWORK.
91 LWORK (input) INTEGER
93 The dimension of the array WORK. If N <= 1,
94 LWORK >= 1. If JOBZ = 'N' and N > 1, LWORK >= 2*N. If JOBZ =
95 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
96 If LWORK = -1, then a workspace query is assumed; the routine
98 only calculates the required sizes of the WORK and IWORK
99 arrays, returns these values as the first entries of the WORK
100 and IWORK arrays, and no error message related to LWORK or
101 LIWORK is issued by XERBLA.
102 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
104 On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
105 LIWORK (input) INTEGER
107 The dimension of the array IWORK. If JOBZ = 'N' or N <= 1,
108 LIWORK >= 1. If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
109 If LIWORK = -1, then a workspace query is assumed; the routine
111 only calculates the required sizes of the WORK and IWORK
112 arrays, returns these values as the first entries of the WORK
113 and IWORK arrays, and no error message related to LWORK or
114 LIWORK is issued by XERBLA.
115 INFO (output) INTEGER
117 = 0: successful exit
118 < 0: if INFO = -i, the i-th argument had an illegal value
119 > 0: SPPTRF or SSPEVD returned an error code:
120 <= N: if INFO = i, SSPEVD failed to converge; i off-diagonal
121 elements of an intermediate tridiagonal form did not converge
122 to zero; > N: if INFO = N + i, for 1 <= i <= N, then the
123 leading minor of order i of B is not positive definite. The
124 factorization of B could not be completed and no eigenvalues or
125 eigenvectors were computed.
126
128 Based on contributions by
129 Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
130 LAPACK driver routine (version 3.N1o)vember 2006 SSPGVD(1)