f08upc computes selected the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form
where and are Hermitian and banded, and is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.
The function may be called by the names: f08upc, nag_lapackeig_zhbgvx or nag_zhbgvx.
3Description
The generalized Hermitian-definite band problem
is first reduced to a standard band Hermitian problem
where is a Hermitian band matrix, using Wilkinson's modification to Crawford's algorithm (see Crawford (1973) and Wilkinson (1977)). The Hermitian eigenvalue problem is then solved for the required eigenvalues and eigenvectors, and the eigenvectors are then backtransformed to the eigenvectors of the original problem.
The eigenvectors are normalized so that
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
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput.11 873–912
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_RangeTypeInput
On entry: if , all eigenvalues will be found.
If , all eigenvalues in the half-open interval will be found.
On entry: if , the upper triangles of and are stored.
If , the lower triangles of and are stored.
Constraint:
or .
5: – IntegerInput
On entry: , the order of the matrices and .
Constraint:
.
6: – IntegerInput
On entry: if , the number of superdiagonals, , of the matrix .
If , the number of subdiagonals, , of the matrix .
Constraint:
.
7: – IntegerInput
On entry: if , the number of superdiagonals, , of the matrix .
If , the number of subdiagonals, , of the matrix .
Constraint:
.
8: – ComplexInput/Output
Note: the dimension, dim, of the array ab
must be at least
.
On entry: the upper or lower triangle of the Hermitian 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:
.
10: – ComplexInput/Output
Note: the dimension, dim, of the array bb
must be at least
.
On entry: the upper or lower triangle of the Hermitian positive definite 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 f08utc.
11: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array
bb.
Constraint:
.
12: – ComplexOutput
Note: the dimension, dim, of the array q
must be at least
when
;
otherwise.
The th element of the matrix is stored in
when ;
when .
On exit: if , the matrix, used in the reduction of the standard form, i.e., , from symmetric banded to tridiagonal form.
On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval of width less than or equal to
where is the machine precision. If abstol is less than or equal to zero, then will be used in its place, where is the tridiagonal matrix obtained by reducing to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold , not zero. If this function returns with NE_CONVERGENCE, indicating that some eigenvectors did not converge, try setting abstol to . See Demmel and Kahan (1990).
19: – Integer *Output
On exit: the total number of eigenvalues found. .
If , .
If , .
20: – doubleOutput
On exit: the eigenvalues in ascending order.
21: – ComplexOutput
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 , then
if NE_NOERROR, the first m columns of contain the eigenvectors corresponding to the selected eigenvalues, with the th column of holding the eigenvector associated with . The eigenvectors are normalized so that ;
if an eigenvector fails to converge ( NE_CONVERGENCE), then that column of contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
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; eigenvectors did not converge. Their indices are stored in array jfail.
NE_ENUM_INT_2
On entry, , and .
Constraint: if , ;
otherwise .
On entry, , and .
Constraint: if , ;
otherwise .
NE_ENUM_INT_3
On entry, , , and .
Constraint: if and , and ;
if and , .
NE_ENUM_REAL_2
On entry, , and .
Constraint: if , .
NE_INT
On entry, .
Constraint: .
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 , f08utc 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.
f08upc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08upc 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 , and assuming that , is approximately proportional to if . Otherwise the number of floating-point operations depends upon the number of eigenvectors computed.