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NAG Toolbox: nag_lapack_dspgv (f08ta)

Purpose

nag_lapack_dspgv (f08ta) computes all the eigenvalues and, optionally, all the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form
Az = λBz ,   ABz = λz   or   BAz = λz ,
Az=λBz ,   ABz=λz   or   BAz=λz ,
where AA and BB are symmetric, stored in packed format, and BB is also positive definite.

Syntax

[ap, bp, w, z, info] = f08ta(itype, jobz, uplo, n, ap, bp)
[ap, bp, w, z, info] = nag_lapack_dspgv(itype, jobz, uplo, n, ap, bp)

Description

nag_lapack_dspgv (f08ta) first performs a Cholesky factorization of the matrix BB as B = UTU B=UTU , when uplo = 'U'uplo='U' or B = LLT B=LLT , when uplo = 'L'uplo='L'. The generalized problem is then reduced to a standard symmetric eigenvalue problem
Cx = λx ,
Cx=λx ,
which is solved for the eigenvalues and, optionally, the eigenvectors; the eigenvectors are then backtransformed to give the eigenvectors of the original problem.
For the problem Az = λBz Az=λBz , the eigenvectors are normalized so that the matrix of eigenvectors, ZZ, satisfies
ZT A Z = Λ   and   ZT B Z = I ,
ZT A Z = Λ   and   ZT B Z = I ,
where Λ Λ  is the diagonal matrix whose diagonal elements are the eigenvalues. For the problem A B z = λ z A B z = λ z  we correspondingly have
Z1 A ZT = Λ   and   ZT B Z = I ,
Z-1 A Z-T = Λ   and   ZT B Z = I ,
and for B A z = λ z B A z = λ z  we have
ZT A Z = Λ   and   ZT B1 Z = I .
ZT A Z = Λ   and   ZT B-1 Z = I .

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:     itype – int64int32nag_int scalar
Specifies the problem type to be solved.
itype = 1itype=1
Az = λBzAz=λBz.
itype = 2itype=2
ABz = λzABz=λz.
itype = 3itype=3
BAz = λzBAz=λz.
Constraint: itype = 1itype=1, 22 or 33.
2:     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'.
3:     uplo – string (length ≥ 1)
If uplo = 'U'uplo='U', the upper triangles of AA and BB are stored.
If uplo = 'L'uplo='L', the lower triangles of AA and BB are stored.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
4:     n – int64int32nag_int scalar
nn, the order of the matrices AA and BB.
Constraint: n0n0.
5:     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.
6:     bp( : :) – double array
Note: the dimension of the array bp 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 BB, packed by columns.
More precisely,
  • if uplo = 'U'uplo='U', the upper triangle of BB must be stored with element BijBij in bp(i + j(j1) / 2)bpi+j(j-1)/2 for ijij;
  • if uplo = 'L'uplo='L', the lower triangle of BB must be stored with element BijBij in bp(i + (2nj)(j1) / 2)bpi+(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).
The contents of ap are destroyed.
2:     bp( : :) – double array
Note: the dimension of the array bp must be at least max (1,n × (n + 1) / 2)max(1,n×(n+1)/2).
The triangular factor UU or LL from the Cholesky factorization B = UTUB=UTU or B = LLTB=LLT, in the same storage format as BB.
3:     w(n) – double array
The eigenvalues in ascending order.
4:     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 matrix ZZ of eigenvectors. The eigenvectors are normalized as follows:
  • if itype = 1itype=1 or 22, ZTBZ = IZTBZ=I;
  • if itype = 3itype=3, ZTB1Z = IZTB-1Z=I.
If jobz = 'N'jobz='N', z is not referenced.
5:     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: itype, 2: jobz, 3: uplo, 4: n, 5: ap, 6: bp, 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
nag_lapack_dpptrf (f07gd) or nag_lapack_dspev (f08ga) returned an error code:
nn if info = iinfo=i, nag_lapack_dspev (f08ga) failed to converge; ii off-diagonal elements of an intermediate tridiagonal form did not converge to zero;
> n>n if info = n + iinfo=n+i, for 1in1in, then the leading minor of order ii of BB is not positive definite. The factorization of BB could not be completed and no eigenvalues or eigenvectors were computed.

Accuracy

If BB is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of BB differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of BB would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds.
The example program below illustrates the computation of approximate error bounds.

Further Comments

The total number of floating point operations is proportional to n3 n3 .
The complex analogue of this function is nag_lapack_zhpgv (f08tn).

Example

function nag_lapack_dspgv_example
itype = int64(1);
jobz = 'No vectors';
uplo = 'U';
n = int64(4);
ap = [0.24;
     0.39;
     -0.11;
     0.42;
     0.79;
     -0.25;
     -0.16;
     0.63;
     0.48;
     -0.03];
bp = [4.16;
     -3.12;
     5.03;
     0.56;
     -0.83;
     0.76;
     -0.1;
     1.09;
     0.34;
     1.18];
[apOut, bpOut, w, z, info] = nag_lapack_dspgv(itype, jobz, uplo, n, ap, bp)
 

apOut =

    0.0875
    0.4683
    0.5244
    0.4892
   -0.6812
   -0.3775
   -0.4476
   -0.4576
   -0.9487
   -1.6875


bpOut =

    2.0396
   -1.5297
    1.6401
    0.2746
   -0.2500
    0.7887
   -0.0490
    0.6189
    0.6443
    0.6161


w =

   -2.2254
   -0.4548
    0.1001
    1.1270


z =

     0


info =

                    0


function f08ta_example
itype = int64(1);
jobz = 'No vectors';
uplo = 'U';
n = int64(4);
ap = [0.24;
     0.39;
     -0.11;
     0.42;
     0.79;
     -0.25;
     -0.16;
     0.63;
     0.48;
     -0.03];
bp = [4.16;
     -3.12;
     5.03;
     0.56;
     -0.83;
     0.76;
     -0.1;
     1.09;
     0.34;
     1.18];
[apOut, bpOut, w, z, info] = f08ta(itype, jobz, uplo, n, ap, bp)
 

apOut =

    0.0875
    0.4683
    0.5244
    0.4892
   -0.6812
   -0.3775
   -0.4476
   -0.4576
   -0.9487
   -1.6875


bpOut =

    2.0396
   -1.5297
    1.6401
    0.2746
   -0.2500
    0.7887
   -0.0490
    0.6189
    0.6443
    0.6161


w =

   -2.2254
   -0.4548
    0.1001
    1.1270


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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