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NAG Toolbox: nag_lapack_dspev (f08ga)

Purpose

nag_lapack_dspev (f08ga) computes all the eigenvalues and, optionally, all the eigenvectors of a real nn by nn symmetric matrix AA in packed storage.

Syntax

[ap, w, z, info] = f08ga(jobz, uplo, n, ap)
[ap, w, z, info] = nag_lapack_dspev(jobz, uplo, n, ap)

Description

The symmetric matrix AA is first reduced to tridiagonal form, using orthogonal similarity transformations, and then the QRQR algorithm is applied to the tridiagonal matrix to compute the eigenvalues and (optionally) the eigenvectors.

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:     uplo – string (length ≥ 1)
If uplo = 'U'uplo='U', the upper triangular part of AA is stored.
If uplo = 'L'uplo='L', the lower triangular part of AA is stored.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
3:     n – int64int32nag_int scalar
nn, the order of the matrix AA.
Constraint: n0n0.
4:     ap( : :) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)max(1,n×(n+1)/2).
The upper or lower triangle of the nn by nn symmetric matrix AA, packed by columns.
More precisely,
  • if uplo = 'U'uplo='U', the upper triangle of AA must be stored with element AijAij in ap(i + j(j1) / 2)api+j(j-1)/2 for ijij;
  • if uplo = 'L'uplo='L', the lower triangle of AA must be stored with element AijAij in ap(i + (2nj)(j1) / 2)api+(2n-j)(j-1)/2 for ijij.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

ldz work

Output Parameters

1:     ap( : :) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)max(1,n×(n+1)/2).
ap stores the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of AA.
2:     w(n) – double array
The eigenvalues in ascending order.
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', z contains the orthonormal eigenvectors of the matrix AA, with the iith column of ZZ holding the eigenvector associated with w(i)wi.
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: uplo, 3: n, 4: ap, 5: w, 6: z, 7: ldz, 8: work, 9: 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 an intermediate tridiagonal form 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 n3n3.
The complex analogue of this function is nag_lapack_zhpev (f08gn).

Example

function nag_lapack_dspev_example
jobz = 'No vectors';
uplo = 'U';
n = int64(4);
ap = [1;
     2;
     2;
     3;
     3;
     3;
     4;
     4;
     4;
     4];
[apOut, w, z, info] = nag_lapack_dspev(jobz, uplo, n, ap)
 

apOut =

   -0.3571
    0.1237
   -0.9762
    0.6262
   -1.2472
    7.3333
    0.3660
    0.3660
   -6.9282
    4.0000


w =

   -2.0531
   -0.5146
   -0.2943
   12.8621


z =

     0


info =

                    0


function f08ga_example
jobz = 'No vectors';
uplo = 'U';
n = int64(4);
ap = [1;
     2;
     2;
     3;
     3;
     3;
     4;
     4;
     4;
     4];
[apOut, w, z, info] = f08ga(jobz, uplo, n, ap)
 

apOut =

   -0.3571
    0.1237
   -0.9762
    0.6262
   -1.2472
    7.3333
    0.3660
    0.3660
   -6.9282
    4.0000


w =

   -2.0531
   -0.5146
   -0.2943
   12.8621


z =

     0


info =

                    0



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

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