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NAG Toolbox: nag_lapack_dsbev (f08ha)

Purpose

nag_lapack_dsbev (f08ha) computes all the eigenvalues and, optionally, all the eigenvectors of a real nn by nn symmetric band matrix AA of bandwidth (2kd + 1) (2kd+1) .

Syntax

[ab, w, z, info] = f08ha(jobz, uplo, kd, ab, 'n', n)
[ab, w, z, info] = nag_lapack_dsbev(jobz, uplo, kd, ab, 'n', n)

Description

The symmetric band 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:     kd – int64int32nag_int scalar
If uplo = 'U'uplo='U', the number of superdiagonals, kdkd, of the matrix AA.
If uplo = 'L'uplo='L', the number of subdiagonals, kdkd, of the matrix AA.
Constraint: kd0kd0.
4:     ab(ldab, : :) – double array
The first dimension of the array ab must be at least kd + 1kd+1
The second dimension of the array must be at least max (1,n)max(1,n)
The upper or lower triangle of the nn by nn symmetric band matrix AA.
The matrix is stored in rows 11 to kd + 1kd+1, more precisely,
  • if uplo = 'U'uplo='U', the elements of the upper triangle of AA within the band must be stored with element AijAij in ab(kd + 1 + ij,j)​ for ​max (1,jkd)ijabkd+1+i-jj​ for ​max(1,j-kd)ij;
  • if uplo = 'L'uplo='L', the elements of the lower triangle of AA within the band must be stored with element AijAij in ab(1 + ij,j)​ for ​jimin (n,j + kd).ab1+i-jj​ for ​jimin(n,j+kd).

Optional Input Parameters

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

Input Parameters Omitted from the MATLAB Interface

ldab ldz work

Output Parameters

1:     ab(ldab, : :) – double array
The first dimension of the array ab will be kd + 1kd+1
The second dimension of the array will be max (1,n)max(1,n)
ldabkd + 1ldabkd+1.
ab stores values generated during the reduction to tridiagonal form.
The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix TT are returned in ab using the same storage format as described above.
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: kd, 5: ab, 6: ldab, 7: w, 8: z, 9: ldz, 10: work, 11: 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 if jobz = 'V'jobz='V' and is proportional to kd n2 kd n2  otherwise.
The complex analogue of this function is nag_lapack_zhbev (f08hn).

Example

function nag_lapack_dsbev_example
jobz = 'Vectors';
uplo = 'U';
kd = int64(2);
ab = [0, 0, 3, 4, 5;
     0, 2, 3, 4, 5;
     1, 2, 3, 4, 5];
[abOut, w, z, info] = nag_lapack_dsbev(jobz, uplo, kd, ab)
 

abOut =

         0         0    3.0000    6.9338    1.5841
         0    3.6056    6.9682   -2.3328   -0.2640
    1.0000    5.4615    8.9115    2.8591   -3.2322


w =

   -3.2474
   -2.6633
    1.7511
    4.1599
   14.9997


z =

    0.0394    0.6238    0.5635   -0.5165    0.1582
    0.5721   -0.2575   -0.3896   -0.5955    0.3161
   -0.4372   -0.5900    0.4008   -0.1470    0.5277
   -0.4424    0.4308   -0.5581    0.0470    0.5523
    0.5332    0.1039    0.2421    0.5956    0.5400


info =

                    0


function f08ha_example
jobz = 'Vectors';
uplo = 'U';
kd = int64(2);
ab = [0, 0, 3, 4, 5;
     0, 2, 3, 4, 5;
     1, 2, 3, 4, 5];
[abOut, w, z, info] = f08ha(jobz, uplo, kd, ab)
 

abOut =

         0         0    3.0000    6.9338    1.5841
         0    3.6056    6.9682   -2.3328   -0.2640
    1.0000    5.4615    8.9115    2.8591   -3.2322


w =

   -3.2474
   -2.6633
    1.7511
    4.1599
   14.9997


z =

    0.0394    0.6238    0.5635   -0.5165    0.1582
    0.5721   -0.2575   -0.3896   -0.5955    0.3161
   -0.4372   -0.5900    0.4008   -0.1470    0.5277
   -0.4424    0.4308   -0.5581    0.0470    0.5523
    0.5332    0.1039    0.2421    0.5956    0.5400


info =

                    0



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