hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

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

NAG Toolbox: nag_lapack_dstevr (f08jd)

Purpose

nag_lapack_dstevr (f08jd) computes selected eigenvalues and, optionally, eigenvectors of a real nn by nn symmetric tridiagonal matrix TT. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Syntax

[d, e, m, w, z, isuppz, info] = f08jd(jobz, range, d, e, vl, vu, il, iu, abstol, 'n', n)
[d, e, m, w, z, isuppz, info] = nag_lapack_dstevr(jobz, range, d, e, vl, vu, il, iu, abstol, 'n', n)

Description

Whenever possible nag_lapack_dstevr (f08jd) computes the eigenspectrum using Relatively Robust Representations. nag_lapack_dstevr (f08jd) computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various ‘good’ LDLTLDLT representations (also known as Relatively Robust Representations). Gram–Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the iith unreduced block of TT:
(a) compute T σi I = Li Di LiT T - σi I = Li Di LiT , such that Li Di LiT Li Di LiT  is a relatively robust representation,
(b) compute the eigenvalues, λjλj, of Li Di LiT Li Di LiT  to high relative accuracy by the dqds algorithm,
(c) if there is a cluster of close eigenvalues, ‘choose’ σiσi close to the cluster, and go to (a),
(d) given the approximate eigenvalue λjλj of Li Di LiT Li Di LiT , compute the corresponding eigenvector by forming a rank-revealing twisted factorization.
The desired accuracy of the output can be specified by the parameter abstol. For more details, see Dhillon (1997) and Parlett and Dhillon (2000).

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
Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Dhillon I (1997) A new O(n2)O(n2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem Computer Science Division Technical Report No. UCB//CSD-97-971 UC Berkeley
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Parlett B N and Dhillon I S (2000) Relatively robust representations of symmetric tridiagonals Linear Algebra Appl. 309 121–151

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:     range – string (length ≥ 1)
If range = 'A'range='A', all eigenvalues will be found.
If range = 'V'range='V', all eigenvalues in the half-open interval (vl,vu](vl,vu] will be found.
If range = 'I'range='I', the ilth to iuth eigenvalues will be found.
Constraint: range = 'A'range='A', 'V''V' or 'I''I'.
3:     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 TT.
4:     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 TT.
5:     vl – double scalar
6:     vu – double scalar
If range = 'V'range='V', the lower and upper bounds of the interval to be searched for eigenvalues.
If range = 'A'range='A' or 'I''I', vl and vu are not referenced.
Constraint: if range = 'V'range='V', vl < vuvl<vu.
7:     il – int64int32nag_int scalar
8:     iu – int64int32nag_int scalar
If range = 'I'range='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If range = 'A'range='A' or 'V''V', il and iu are not referenced.
Constraints:
  • if range = 'I'range='I' and n = 0n=0, il = 1il=1 and iu = 0iu=0;
  • if range = 'I'range='I' and n > 0n>0, 1 il iu n 1 il iu n .
9:     abstol – double scalar
The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] [a,b]  of width less than or equal to
abstol + ε max (|a|,|b|) ,
abstol+ε max(|a|,|b|) ,
where ε ε  is the machine precision. If abstol is less than or equal to zero, then ε T1 ε T1  will be used in its place. See Demmel and Kahan (1990).
If high relative accuracy is important, set abstol to x02am(   ) x02am( ) , although doing so does not currently guarantee that eigenvalues are computed to high relative accuracy. See Barlow and Demmel (1990) for a discussion of which matrices can define their eigenvalues to high relative accuracy.

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 lwork iwork liwork

Output Parameters

1:     d( : :) – double array
Note: the dimension of the array d must be at least max (1,n)max(1,n).
May be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
2:     e( : :) – double array
Note: the dimension of the array e must be at least max (1,n1)max(1,n-1).
May be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
3:     m – int64int32nag_int scalar
The total number of eigenvalues found. 0mn0mn.
If range = 'A'range='A', m = nm=n.
If range = 'I'range='I', m = iuil + 1m=iu-il+1.
4:     w( : :) – double array
Note: the dimension of the array w must be at least max (1,n)max(1,n).
The first m elements contain the selected eigenvalues in ascending order.
5:     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,m)max(1,m) if jobz = 'V'jobz='V', and at least 11 otherwise
If jobz = 'V'jobz='V', the first m columns of ZZ contain the orthonormal eigenvectors of the matrix AA corresponding to the selected eigenvalues, with the iith column of ZZ holding the eigenvector associated with w(i)wi.
If jobz = 'N'jobz='N', z is not referenced.
6:     isuppz( : :) – int64int32nag_int array
Note: the dimension of the array isuppz must be at least max (1,2 × m)max(1,2×m).
The support of the eigenvectors in z, i.e., the indices indicating the nonzero elements in z. The iith eigenvector is nonzero only in elements isuppz(2 × i1)isuppz2×i-1 through isuppz(2 × i)isuppz2×i. Implemented only for range = 'A'range='A' or 'I''I' and iuil = n1iu-il=n-1.
7:     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: range, 3: n, 4: d, 5: e, 6: vl, 7: vu, 8: il, 9: iu, 10: abstol, 11: m, 12: w, 13: z, 14: ldz, 15: isuppz, 16: work, 17: lwork, 18: iwork, 19: liwork, 20: 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
An internal error has occurred in this function. Please refer to info in nag_lapack_dstebz (f08jj).

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' and range = 'A'range='A', otherwise the number of floating point operations will depend upon the number of computed eigenvectors.

Example

function nag_lapack_dstevr_example
jobz = 'Vectors';
range = 'Indices';
d = [1;
     4;
     9;
     16];
e = [1;
     2;
     3];
vl = 0;
vu = 0;
il = int64(2);
iu = int64(3);
abstol = 0;
[dOut, eOut, m, w, z, isuppz, info] = nag_lapack_dstevr(jobz, range, d, e, vl, vu, il, iu, abstol)
 

dOut =

     1
     4
     9
    16


eOut =

     1
     2
     3


m =

                    2


w =

    3.5470
    8.6578
         0
         0


z =

    0.3388    0.0494
    0.8628    0.3781
   -0.3648    0.8558
    0.0879   -0.3497


isuppz =

                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0


info =

                    0


function f08jd_example
jobz = 'Vectors';
range = 'Indices';
d = [1;
     4;
     9;
     16];
e = [1;
     2;
     3];
vl = 0;
vu = 0;
il = int64(2);
iu = int64(3);
abstol = 0;
[dOut, eOut, m, w, z, isuppz, info] = f08jd(jobz, range, d, e, vl, vu, il, iu, abstol)
 

dOut =

     1
     4
     9
    16


eOut =

     1
     2
     3


m =

                    2


w =

    3.5470
    8.6578
         0
         0


z =

    0.3388    0.0494
    0.8628    0.3781
   -0.3648    0.8558
    0.0879   -0.3497


isuppz =

                    0
                    0
                    0
                    0
                    0
                    0
                    0
                    0


info =

                    0



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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013