The function may be called by the names: f08ucc, nag_lapackeig_dsbgvd or nag_dsbgvd.
3Description
The generalized symmetric-definite band problem
is first reduced to a standard band symmetric problem
where is a symmetric band matrix, using Wilkinson's modification to Crawford's algorithm (see Crawford (1973) and Wilkinson (1977)).
The symmetric eigenvalue problem is then solved for the eigenvalues and the eigenvectors, if required, which are then backtransformed to the eigenvectors of the original problem.
The eigenvectors are normalized so that the matrix of eigenvectors, , satisfies
where is the diagonal matrix whose diagonal elements are the eigenvalues.
4References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM16 41–44
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1977) Some recent advances in numerical linear algebra The State of the Art in Numerical Analysis (ed D A H Jacobs) Academic Press
5Arguments
1: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
2: – Nag_JobTypeInput
On entry: indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
Constraint:
or .
3: – Nag_UploTypeInput
On entry: if , the upper triangles of and are stored.
If , the lower triangles of and are stored.
Constraint:
or .
4: – IntegerInput
On entry: , the order of the matrices and .
Constraint:
.
5: – IntegerInput
On entry: if , the number of superdiagonals, , of the matrix .
If , the number of subdiagonals, , of the matrix .
Constraint:
.
6: – IntegerInput
On entry: if , the number of superdiagonals, , of the matrix .
If , the number of subdiagonals, , of the matrix .
Constraint:
.
7: – doubleInput/Output
Note: the dimension, dim, of the array ab
must be at least
.
On entry: the upper or lower triangle of the symmetric band matrix .
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of , depends on the order and uplo arguments as follows:
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array
ab.
Constraint:
.
9: – doubleInput/Output
Note: the dimension, dim, of the array bb
must be at least
.
On entry: the upper or lower triangle of the symmetric band matrix .
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of , depends on the order and uplo arguments as follows:
if and ,
is stored in , for and ;
if and ,
is stored in , for and ;
if and ,
is stored in , for and ;
if and ,
is stored in , for and .
On exit: the factor from the split Cholesky factorization , as returned by f08ufc.
10: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array
bb.
Constraint:
.
11: – doubleOutput
On exit: the eigenvalues in ascending order.
12: – doubleOutput
Note: the dimension, dim, of the array z
must be at least
when
;
otherwise.
The th element of the matrix is stored in
when ;
when .
On exit: if , z contains the matrix of eigenvectors, with the th column of holding the eigenvector associated with . The eigenvectors are normalized so that .
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
if , ;
otherwise .
14: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_CONVERGENCE
The algorithm failed to converge; off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
NE_ENUM_INT_2
On entry, , and .
Constraint: if , ;
otherwise .
NE_INT
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
NE_INT_2
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MAT_NOT_POS_DEF
If , for , f08ufc returned : is not positive definite. The factorization of could not be completed and no eigenvalues or eigenvectors were computed.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
7Accuracy
If is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08ucc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08ucc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The total number of floating-point operations is proportional to if and, assuming that , is approximately proportional to otherwise.