NAG Library Function Document

nag_dstev (f08jac)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_dstev (f08jac) computes all the eigenvalues and, optionally, all the eigenvectors of a real n by n symmetric tridiagonal matrix A.

2
Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dstev (Nag_OrderType order, Nag_JobType job, Integer n, double d[], double e[], double z[], Integer pdz, NagError *fail)

3
Description

nag_dstev (f08jac) 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 http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5
Arguments

1:     order 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 order=Nag_RowMajor. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     job Nag_JobTypeInput
On entry: indicates whether eigenvectors are computed.
job=Nag_EigVals
Only eigenvalues are computed.
job=Nag_DoBoth
Eigenvalues and eigenvectors are computed.
Constraint: job=Nag_EigVals or Nag_DoBoth.
3:     n IntegerInput
On entry: n, the order of the matrix.
Constraint: n0.
4:     d[dim] doubleInput/Output
Note: the dimension, dim, of the array d must be at least max1,n.
On entry: the n diagonal elements of the tridiagonal matrix A.
On exit: if fail.code= NE_NOERROR, the eigenvalues in ascending order.
5:     e[dim] doubleInput/Output
Note: the dimension, dim, of the array e must be at least max1,n-1.
On entry: the n-1 subdiagonal elements of the tridiagonal matrix A.
On exit: the contents of e are destroyed.
6:     z[dim] doubleOutput
Note: the dimension, dim, of the array z must be at least
  • max1,pdz×n when job=Nag_DoBoth;
  • 1 otherwise.
The i,jth element of the matrix Z is stored in
  • z[j-1×pdz+i-1] when order=Nag_ColMajor;
  • z[i-1×pdz+j-1] when order=Nag_RowMajor.
On exit: if job=Nag_DoBoth, then if fail.code= NE_NOERROR, z contains the orthonormal eigenvectors of the matrix A, with the ith column of Z holding the eigenvector associated with d[i-1].
If job=Nag_EigVals, z is not referenced.
7:     pdz IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
  • if job=Nag_DoBoth, pdz max1,n ;
  • otherwise pdz1.
8:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_CONVERGENCE
The algorithm failed to converge; value off-diagonal elements of e did not converge to zero.
NE_ENUM_INT_2
On entry, job=value, pdz=value and n=value.
Constraint: if job=Nag_DoBoth, pdz max1,n ;
otherwise pdz1.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pdz=value.
Constraint: pdz>0.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7
Accuracy

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

8
Parallelism and Performance

nag_dstev (f08jac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dstev (f08jac) 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.

9
Further Comments

The total number of floating-point operations is proportional to n2 if job=Nag_EigVals and is proportional to n3 if job=Nag_DoBoth.

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 (f08jace.c)

10.2
Program Data

Program Data (f08jace.d)

10.3
Program Results

Program Results (f08jace.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017