f08jbc computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix . Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
The function may be called by the names: f08jbc, nag_lapackeig_dstevx or nag_dstevx.
f08jbc computes the required eigenvalues and eigenvectors of by reducing the tridiagonal matrix to diagonal form using the algorithm. Bisection is used to determine selected eigenvalues.
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
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
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: – Nag_RangeTypeInput
On entry: if , all eigenvalues will be found.
If , all eigenvalues in the half-open interval will be found.
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. 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).
12: – Integer *Output
On exit: the total number of eigenvalues found. .
If , .
If , .
13: – doubleOutput
On exit: the first m elements contain the selected eigenvalues in ascending order.
14: – 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 , then
if NE_NOERROR, the first m columns of contain the orthonormal eigenvectors of the matrix corresponding to the selected eigenvalues, with the th column of holding the eigenvector associated with ;
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
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; eigenvectors did not converge. Their indices are stored in array jfail.
On entry, , and .
Constraint: if , ;
On entry, , , and .
Constraint: if and , and ;
if and , .
On entry, , and .
Constraint: 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
f08jbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jbc 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.
The total number of floating-point operations is proportional to if and is proportional to if and , otherwise the number of floating-point operations will depend upon the number of computed eigenvectors.
This example finds the eigenvalues in the half-open interval , and the corresponding eigenvectors, of the symmetric tridiagonal matrix