f08jvc computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix, or of a complex full or banded Hermitian matrix which has been reduced to tridiagonal form.
The function may be called by the names: f08jvc, nag_lapackeig_zstedc or nag_zstedc.
3Description
f08jvc computes all the eigenvalues and, optionally, the eigenvectors of a real symmetric tridiagonal matrix . That is, the function computes the spectral factorization of given by
where is a diagonal matrix whose diagonal elements are the eigenvalues, , of and is an orthogonal matrix whose columns are the eigenvectors, , of . Thus
The function may also be used to compute all the eigenvalues and eigenvectors of a complex full, or banded, Hermitian matrix which has been reduced to real tridiagonal form as
where is unitary. The spectral factorization of is then given by
In this case must be formed explicitly and passed to f08jvc in the array z, and the function called with . Functions which may be called to form and are
When only eigenvalues are required then this function calls f08jfc to compute the eigenvalues of the tridiagonal matrix , but when eigenvectors of are also required and the matrix is not too small, then a divide and conquer method is used, which can be much faster than f08jsc, although more storage is required.
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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
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_ComputeEigVecsTypeInput
On entry: indicates whether the eigenvectors are to be computed.
Only the eigenvalues are computed (and the array z is not referenced).
The eigenvalues and eigenvectors of are computed (and the array z must contain the matrix on entry).
The eigenvalues and eigenvectors of are computed (and the array z is initialized by the function).
Constraint:
, or .
3: – IntegerInput
On entry: , the order of the symmetric tridiagonal matrix .
Constraint:
.
4: – doubleInput/Output
Note: the dimension, dim, of the array d
must be at least
.
On entry: the diagonal elements of the tridiagonal matrix.
On exit: if NE_NOERROR, the eigenvalues in ascending order.
5: – doubleInput/Output
Note: the dimension, dim, of the array e
must be at least
.
On entry: the subdiagonal elements of the tridiagonal matrix.
Note: the dimension, dim, of the array z
must be at least
when
or ;
otherwise.
if then theth element of the matrix is stored in
when ;
when .
On entry: if , z must contain the unitary matrix used in the reduction to tridiagonal form.
On exit: if , z contains the orthonormal eigenvectors of the original Hermitian matrix , and if , z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix .
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
if or , ;
otherwise .
8: – 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 compute an eigenvalue while working on the submatrix lying in rows and columns through .
NE_ENUM_INT_2
On entry, , and .
Constraint: if or , ;
otherwise .
NE_INT
On entry, .
Constraint: .
On entry, . 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_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
The computed eigenvalues and eigenvectors are exact for a nearby matrix , where
and is the machine precision.
If is an exact eigenvalue and is the corresponding computed value, then
where is a modestly increasing function of .
If is the corresponding exact eigenvector, and is the corresponding computed eigenvector, then the angle between them is bounded as follows:
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.
Background information to multithreading can be found in the Multithreading documentation.
f08jvc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jvc 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
If only eigenvalues are required, the total number of floating-point operations is approximately proportional to . When eigenvectors are required the number of operations is bounded above by approximately the same number of operations as f08jsc, but for large matrices f08jvc is usually much faster.