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NAG Toolbox

NAG Toolbox: nag_lapack_dspevd (f08gc)

Purpose

nag_lapack_dspevd (f08gc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric matrix held in packed storage. 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 QLQL or QRQR algorithm.

Syntax

[ap, w, z, info] = f08gc(job, uplo, n, ap)
[ap, w, z, info] = nag_lapack_dspevd(job, uplo, n, ap)

Description

nag_lapack_dspevd (f08gc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric matrix AA (held in packed storage). In other words, it can compute the spectral factorization of AA as
A = ZΛZT,
A=ZΛZT,
where ΛΛ is a diagonal matrix whose diagonal elements are the eigenvalues λiλi, and ZZ is the orthogonal matrix whose columns are the eigenvectors zizi. Thus
Azi = λizi,  i = 1,2,,n.
Azi=λizi,  i=1,2,,n.

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:     job – string (length ≥ 1)
Indicates whether eigenvectors are computed.
job = 'N'job='N'
Only eigenvalues are computed.
job = 'V'job='V'
Eigenvalues and eigenvectors are computed.
Constraint: job = 'N'job='N' or 'V''V'.
2:     uplo – string (length ≥ 1)
Indicates whether the upper or lower triangular part of AA is stored.
uplo = 'U'uplo='U'
The upper triangular part of AA is stored.
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 lwork iwork liwork

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( : :) – double array
Note: the dimension of the array w must be at least max (1,n)max(1,n).
The eigenvalues of the matrix AA in ascending order.
3:     z(ldz, : :) – double array
The first dimension, ldz, of the array z will be
  • if job = 'V'job='V', ldz max (1,n) ldz max(1,n) ;
  • if job = 'N'job='N', ldz1ldz1.
The second dimension of the array will be max (1,n)max(1,n) if job = 'V'job='V' and at least 11 if job = 'N'job='N'
If job = 'V'job='V', z stores the orthogonal matrix ZZ which contains the eigenvectors of AA.
If job = 'N'job='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: job, 2: uplo, 3: n, 4: ap, 5: w, 6: z, 7: ldz, 8: work, 9: lwork, 10: iwork, 11: liwork, 12: 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 and job = 'N'job='N', the algorithm failed to converge; ii elements of an intermediate tridiagonal form did not converge to zero; if info = iinfo=i and job = 'V'job='V', then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column i / (n + 1)i/(n+1) through i  mod  (n + 1)i mod (n+1).

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 complex analogue of this function is nag_lapack_zhpevd (f08gq).

Example

function nag_lapack_dspevd_example
job = 'V';
uplo = 'L';
n = int64(4);
ap = [1;
     2;
     3;
     4;
     2;
     3;
     4;
     3;
     4;
     4];
[apOut, w, z, info] = nag_lapack_dspevd(job, uplo, n, ap)
 

apOut =

    1.0000
   -5.3852
    0.4062
    0.5416
   10.1724
   -1.8302
    0.7276
   -0.8160
    0.1223
   -0.3564


w =

   -2.0531
   -0.5146
   -0.2943
   12.8621


z =

   -0.7003   -0.5144   -0.2767   -0.4103
   -0.3592    0.4851    0.6634   -0.4422
    0.1569    0.5420   -0.6504   -0.5085
    0.5965   -0.4543    0.2457   -0.6144


info =

                    0


function f08gc_example
job = 'V';
uplo = 'L';
n = int64(4);
ap = [1;
     2;
     3;
     4;
     2;
     3;
     4;
     3;
     4;
     4];
[apOut, w, z, info] = f08gc(job, uplo, n, ap)
 

apOut =

    1.0000
   -5.3852
    0.4062
    0.5416
   10.1724
   -1.8302
    0.7276
   -0.8160
    0.1223
   -0.3564


w =

   -2.0531
   -0.5146
   -0.2943
   12.8621


z =

   -0.7003   -0.5144   -0.2767   -0.4103
   -0.3592    0.4851    0.6634   -0.4422
    0.1569    0.5420   -0.6504   -0.5085
    0.5965   -0.4543    0.2457   -0.6144


info =

                    0



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