nag_dstev (f08jac) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     jobNag_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:     nIntegerInput
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:     pdzIntegerInput
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:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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.

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  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.

9  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.

9.1  Program Text

Program Text (f08jace.c)

9.2  Program Data

Program Data (f08jace.d)

9.3  Program Results

Program Results (f08jace.r)


nag_dstev (f08jac) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012