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NAG Toolbox

NAG Toolbox: nag_lapack_dstev (f08ja)

Purpose

nag_lapack_dstev (f08ja) computes all the eigenvalues and, optionally, all the eigenvectors of a real nn by nn symmetric tridiagonal matrix AA.

Syntax

[d, e, z, info] = f08ja(jobz, d, e, 'n', n)
[d, e, z, info] = nag_lapack_dstev(jobz, d, e, 'n', n)

Description

nag_lapack_dstev (f08ja) computes all the eigenvalues and, optionally, all the eigenvectors of AA using a combination of the QRQR and QLQL algorithms, with an implicit shift.

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

Parameters

Compulsory Input Parameters

1:     jobz – string (length ≥ 1)
Indicates whether eigenvectors are computed.
jobz = 'N'jobz='N'
Only eigenvalues are computed.
jobz = 'V'jobz='V'
Eigenvalues and eigenvectors are computed.
Constraint: jobz = 'N'jobz='N' or 'V''V'.
2:     d( : :) – double array
Note: the dimension of the array d must be at least max (1,n)max(1,n).
The nn diagonal elements of the tridiagonal matrix AA.
3:     e( : :) – double array
Note: the dimension of the array e must be at least max (1,n1)max(1,n-1).
The (n1)(n-1) subdiagonal elements of the tridiagonal matrix AA.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array d.
nn, the order of the matrix.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

ldz work

Output Parameters

1:     d( : :) – double array
Note: the dimension of the array d must be at least max (1,n)max(1,n).
If INFO = 0INFO=0, the eigenvalues in ascending order.
2:     e( : :) – double array
Note: the dimension of the array e must be at least max (1,n1)max(1,n-1).
The contents of e are destroyed.
3:     z(ldz, : :) – double array
The first dimension, ldz, of the array z will be
  • if jobz = 'V'jobz='V', ldz max (1,n) ldz max(1,n) ;
  • otherwise ldz1ldz1.
The second dimension of the array will be max (1,n)max(1,n) if jobz = 'V'jobz='V', and at least 11 otherwise
If jobz = 'V'jobz='V', then if INFO = 0INFO=0, z contains the orthonormal eigenvectors of the matrix AA, with the iith column of ZZ holding the eigenvector associated with d(i)di.
If jobz = 'N'jobz='N', z is not referenced.
4:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

  info = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: jobz, 2: n, 3: d, 4: e, 5: z, 6: ldz, 7: work, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
  INFO > 0INFO>0
If info = iinfo=i, the algorithm failed to converge; ii off-diagonal elements of e did not converge to zero.

Accuracy

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

Further Comments

The total number of floating point operations is proportional to n2n2 if jobz = 'N'jobz='N' and is proportional to n3n3 if jobz = 'V'jobz='V'.

Example

function nag_lapack_dstev_example
jobz = 'Vectors';
d = [1;
     4;
     9;
     16];
e = [1;
     2;
     3];
[dOut, eOut, z, info] = nag_lapack_dstev(jobz, d, e)
 

dOut =

    0.6476
    3.5470
    8.6578
   17.1477


eOut =

     0
     0
     0


z =

    0.9396    0.3388   -0.0494    0.0034
   -0.3311    0.8628   -0.3781    0.0545
    0.0853   -0.3648   -0.8558    0.3568
   -0.0167    0.0879    0.3497    0.9326


info =

                    0


function f08ja_example
jobz = 'Vectors';
d = [1;
     4;
     9;
     16];
e = [1;
     2;
     3];
[dOut, eOut, z, info] = f08ja(jobz, d, e)
 

dOut =

    0.6476
    3.5470
    8.6578
   17.1477


eOut =

     0
     0
     0


z =

    0.9396    0.3388   -0.0494    0.0034
   -0.3311    0.8628   -0.3781    0.0545
    0.0853   -0.3648   -0.8558    0.3568
   -0.0167    0.0879    0.3497    0.9326


info =

                    0



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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