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NAG Toolbox: nag_lapack_dsbevd (f08hc)

Purpose

nag_lapack_dsbevd (f08hc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric band 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 QLQL or QRQR algorithm.

Syntax

[ab, w, z, info] = f08hc(job, uplo, kd, ab, 'n', n)
[ab, w, z, info] = nag_lapack_dsbevd(job, uplo, kd, ab, 'n', n)

Description

nag_lapack_dsbevd (f08hc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric band matrix AA. 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:     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 first dimension of the array ab and the second dimension of the array ab. (An error is raised if these dimensions are not equal.)
nn, the order of the matrix AA.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

ldab ldz work lwork iwork liwork

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( : :) – 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. The iith column of ZZ contains the eigenvector which corresponds to the eigenvalue w(i)wi.
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: kd, 5: ab, 6: ldab, 7: w, 8: z, 9: ldz, 10: work, 11: lwork, 12: iwork, 13: liwork, 14: 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_zhbevd (f08hq).

Example

function nag_lapack_dsbevd_example
job = 'V';
uplo = 'L';
kd = int64(2);
ab = [1, 2, 3, 4, 5;
     2, 3, 4, 5, -4.792575277983584e-39;
     3, 4, 5, 0, -0.1064760684967041];
[abOut, w, z, info] = nag_lapack_dsbevd(job, uplo, kd, ab)
 

abOut =

    1.0000    5.4615    8.9115    2.8591   -3.2322
    3.6056    6.9682   -2.3328   -0.2640   -0.0000
    3.0000    6.9338    1.5841         0   -0.1065


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 f08hc_example
job = 'V';
uplo = 'L';
kd = int64(2);
ab = [1, 2, 3, 4, 5;
     2, 3, 4, 5, -4.792575277983584e-39;
     3, 4, 5, 0, -0.1064760684967041];
[abOut, w, z, info] = f08hc(job, uplo, kd, ab)
 

abOut =

    1.0000    5.4615    8.9115    2.8591   -3.2322
    3.6056    6.9682   -2.3328   -0.2640   -0.0000
    3.0000    6.9338    1.5841         0   -0.1065


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



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Chapter Introduction
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