NAG FL Interface
f08jaf (dstev)

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1 Purpose

f08jaf computes all the eigenvalues and, optionally, all the eigenvectors of a real n×n symmetric tridiagonal matrix A.

2 Specification

Fortran Interface
Subroutine f08jaf ( jobz, n, d, e, z, ldz, work, info)
Integer, Intent (In) :: n, ldz
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (Inout) :: d(*), e(*), z(ldz,*), work(*)
Character (1), Intent (In) :: jobz
C Header Interface
#include <nag.h>
void  f08jaf_ (const char *jobz, const Integer *n, double d[], double e[], double z[], const Integer *ldz, double work[], Integer *info, const Charlen length_jobz)
The routine may be called by the names f08jaf, nagf_lapackeig_dstev or its LAPACK name dstev.

3 Description

f08jaf computes all the eigenvalues and, optionally, all the eigenvectors of A using a combination of the QR and QL algorithms, with an implicit shift.

4 References

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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: jobz Character(1) Input
On entry: indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
Constraint: jobz='N' or 'V'.
2: n Integer Input
On entry: n, the order of the matrix.
Constraint: n0.
3: d(*) Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array d must be at least max(1,n).
On entry: the n diagonal elements of the tridiagonal matrix A.
On exit: if info=0, the eigenvalues in ascending order.
4: e(*) Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array e must be at least max(1,n-1).
On entry: the (n-1) subdiagonal elements of the tridiagonal matrix A.
On exit: the contents of e are destroyed.
5: z(ldz,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array z must be at least max(1,n) if jobz='V', and at least 1 otherwise.
On exit: if jobz='V', then if info=0, z contains the orthonormal eigenvectors of the matrix A, with the ith column of Z holding the eigenvector associated with d(i).
If jobz='N', z is not referenced.
6: ldz Integer Input
On entry: the first dimension of the array z as declared in the (sub)program from which f08jaf is called.
  • if jobz='V', ldz max(1,n) ;
  • otherwise ldz1.
7: work(*) Real (Kind=nag_wp) array Workspace
Note: the dimension of the array work must be at least max(1,2×n-2).
On exit: if jobz='N', work is not referenced.
8: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
The algorithm failed to converge; value off-diagonal elements of e did not converge to zero.

7 Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix (A+E), where
E2 = O(ε) A2 ,  
and ε is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08jaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jaf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of floating-point operations is proportional to n2 if jobz='N' and is proportional to n3 if jobz='V'.

10 Example

This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
A = ( 1 1 0 0 1 4 2 0 0 2 9 3 0 0 3 16 ) ,  
together with approximate error bounds for the computed eigenvalues and eigenvectors.

10.1 Program Text

Program Text (f08jafe.f90)

10.2 Program Data

Program Data (f08jafe.d)

10.3 Program Results

Program Results (f08jafe.r)