f08jcc computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix.
If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the or algorithm.
The function may be called by the names: f08jcc, nag_lapackeig_dstevd or nag_dstevd.
f08jcc computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix .
In other words, it can compute the spectral factorization of as
where is a diagonal matrix whose diagonal elements are the eigenvalues , and is the orthogonal matrix whose columns are the eigenvectors . Thus
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
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.
2: – Nag_JobTypeInput
On entry: indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
3: – IntegerInput
On entry: , the order of the matrix .
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: the eigenvalues of the matrix in ascending order.
5: – doubleInput/Output
Note: the dimension, dim, of the array e
must be at least
On entry: the off-diagonal elements of the tridiagonal matrix . The th element of this array is used as workspace.
On exit: e is overwritten with intermediate results.
6: – doubleOutput
Note: the dimension, dim, of the array z
must be at least
The th element of the matrix is stored in
On exit: if , z is overwritten by the orthogonal matrix which contains the eigenvectors of .
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
if , ;
if , .
8: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
The algorithm failed to converge; off-diagonal elements of e did not converge to zero.
On entry, , and .
Constraint: if , ;
if , .
On entry, .
On entry, . Constraint: .
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.
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.
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.
8Parallelism and Performance
f08jcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jcc 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.
There is no complex analogue of this function.
This example computes all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix , where