nag_dstevd (f08jcc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dstevd (f08jcc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dstevd (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 QL or QR algorithm.

2  Specification

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

3  Description

nag_dstevd (f08jcc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix T. In other words, it can compute the spectral factorization of T as
T=ZΛZT,
where Λ is a diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the orthogonal matrix whose columns are the eigenvectors zi. Thus
Tzi=λizi,  i=1,2,,n.

4  References

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_DoNothing
Only eigenvalues are computed.
job=Nag_EigVecs
Eigenvalues and eigenvectors are computed.
Constraint: job=Nag_DoNothing or Nag_EigVecs.
3:     nIntegerInput
On entry: n, the order of the matrix T.
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 T.
On exit: the eigenvalues of the matrix T in ascending order.
5:     e[dim]doubleInput/Output
Note: the dimension, dim, of the array e must be at least max1,n.
On entry: the n-1 off-diagonal elements of the tridiagonal matrix T. The nth element of this array is used as workspace.
On exit: e is overwritten with intermediate results.
6:     z[dim]doubleOutput
Note: the dimension, dim, of the array z must be at least
  • max1,pdz×n when job=Nag_EigVecs;
  • 1 when job=Nag_DoNothing.
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_EigVecs, z is overwritten by the orthogonal matrix Z which contains the eigenvectors of T.
If job=Nag_DoNothing, 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_EigVecs, pdz max1,n ;
  • if job=Nag_DoNothing, 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 eigenvectors did not converge.
NE_ENUM_INT_2
On entry, job=value, pdz=value and n=value.
Constraint: if job=Nag_EigVecs, pdz max1,n ;
if job=Nag_DoNothing, 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 T+E, where
E2 = Oε T2 ,
and ε is the machine precision.
If λi is an exact eigenvalue and λ~i is the corresponding computed value, then
λ~i - λi c n ε T2 ,
where cn is a modestly increasing function of n.
If zi is the corresponding exact eigenvector, and z~i is the corresponding computed eigenvector, then the angle θz~i,zi between them is bounded as follows:
θ z~i,zi c n ε T2 min ij λi - λj .
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.

8  Further Comments

There is no complex analogue of this function.

9  Example

This example computes all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix T, where
T = 1.0 1.0 0.0 0.0 1.0 4.0 2.0 0.0 0.0 2.0 9.0 3.0 0.0 0.0 3.0 16.0 .

9.1  Program Text

Program Text (f08jcce.c)

9.2  Program Data

Program Data (f08jcce.d)

9.3  Program Results

Program Results (f08jcce.r)


nag_dstevd (f08jcc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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