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_zggev (f08wn)

Purpose

nag_lapack_zggev (f08wn) computes for a pair of nn by nn complex nonsymmetric matrices (A,B)(A,B) the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the QZQZ algorithm.

Syntax

[a, b, alpha, beta, vl, vr, info] = f08wn(jobvl, jobvr, a, b, 'n', n)
[a, b, alpha, beta, vl, vr, info] = nag_lapack_zggev(jobvl, jobvr, a, b, 'n', n)

Description

A generalized eigenvalue for a pair of matrices (A,B)(A,B) is a scalar λλ or a ratio α / β = λα/β=λ, such that AλBA-λB is singular. It is usually represented as the pair (α,β)(α,β), as there is a reasonable interpretation for β = 0β=0, and even for both being zero.
The right generalized eigenvector vjvj corresponding to the generalized eigenvalue λjλj of (A,B)(A,B) satisfies
A vj = λj B vj .
A vj = λj B vj .
The left generalized eigenvector ujuj corresponding to the generalized eigenvalue λjλj of (A,B)(A,B) satisfies
ujH A = λj ujH B ,
ujH A = λj ujH B ,
where ujHujH is the conjugate-transpose of ujuj.
All the eigenvalues and, if required, all the eigenvectors of the complex generalized eigenproblem Ax = λBxAx=λBx, where AA and BB are complex, square matrices, are determined using the QZQZ algorithm. The complex QZQZ algorithm consists of three stages:
  1. AA is reduced to upper Hessenberg form (with real, non-negative subdiagonal elements) and at the same time BB is reduced to upper triangular form.
  2. AA is further reduced to triangular form while the triangular form of BB is maintained and the diagonal elements of BB are made real and non-negative. This is the generalized Schur form of the pair (A,B) (A,B) .
    This function does not actually produce the eigenvalues λjλj, but instead returns αjαj and βjβj such that
    λj = αj / βj,  j = 1,2,,n.
    λj=αj/βj,  j=1,2,,n.
    The division by βjβj becomes your responsibility, since βjβj may be zero, indicating an infinite eigenvalue.
  3. If the eigenvectors are required they are obtained from the triangular matrices and then transformed back into the original coordinate system.

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
Wilkinson J H (1979) Kronecker's canonical form and the QZQZ algorithm Linear Algebra Appl. 28 285–303

Parameters

Compulsory Input Parameters

1:     jobvl – string (length ≥ 1)
If jobvl = 'N'jobvl='N', do not compute the left generalized eigenvectors.
If jobvl = 'V'jobvl='V', compute the left generalized eigenvectors.
Constraint: jobvl = 'N'jobvl='N' or 'V''V'.
2:     jobvr – string (length ≥ 1)
If jobvr = 'N'jobvr='N', do not compute the right generalized eigenvectors.
If jobvr = 'V'jobvr='V', compute the right generalized eigenvectors.
Constraint: jobvr = 'N'jobvr='N' or 'V''V'.
3:     a(lda, : :) – complex array
The first dimension of the array a must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,n)max(1,n)
The matrix AA in the pair (A,B)(A,B).
4:     b(ldb, : :) – complex array
The first dimension of the array b must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,n)max(1,n)
The matrix BB in the pair (A,B)(A,B).

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the arrays a, b and the second dimension of the arrays a, b. (An error is raised if these dimensions are not equal.)
nn, the order of the matrices AA and BB.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

lda ldb ldvl ldvr work lwork rwork

Output Parameters

1:     a(lda, : :) – complex array
The first dimension of the array a will be max (1,n)max(1,n)
The second dimension of the array will be max (1,n)max(1,n)
2:     b(ldb, : :) – complex array
The first dimension of the array b will be max (1,n)max(1,n)
The second dimension of the array will be max (1,n)max(1,n)
ldbmax (1,n)ldbmax(1,n).
b has been overwritten.
3:     alpha(n) – complex array
See the description of beta.
4:     beta(n) – complex array
alpha(j) / beta(j)alphaj/betaj, for j = 1,2,,nj=1,2,,n, will be the generalized eigenvalues.
Note:  the quotients alpha(j) / beta(j)alphaj/betaj may easily overflow or underflow, and beta(j)betaj may even be zero. Thus, you should avoid naively computing the ratio αj / βjαj/βj. However, max|αj|max|αj| will always be less than and usually comparable with a2a2 in magnitude, and max|βj|max|βj| will always be less than and usually comparable with b2b2.
5:     vl(ldvl, : :) – complex array
The first dimension, ldvl, of the array vl will be
  • if jobvl = 'V'jobvl='V', ldvl max (1,n) ldvl max(1,n) ;
  • otherwise ldvl1ldvl1.
The second dimension of the array will be max (1,n)max(1,n) if jobvl = 'V'jobvl='V', and at least 11 otherwise
If jobvl = 'V'jobvl='V', the left generalized eigenvectors ujuj are stored one after another in the columns of vl, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have |real part| + |imag. part| = 1|real part|+|imag. part|=1.
If jobvl = 'N'jobvl='N', vl is not referenced.
6:     vr(ldvr, : :) – complex array
The first dimension, ldvr, of the array vr will be
  • if jobvr = 'V'jobvr='V', ldvr max (1,n) ldvr max(1,n) ;
  • otherwise ldvr1ldvr1.
The second dimension of the array will be max (1,n)max(1,n) if jobvr = 'V'jobvr='V', and at least 11 otherwise
If jobvr = 'V'jobvr='V', the right generalized eigenvectors vjvj are stored one after another in the columns of vr, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have |real part| + |imag. part| = 1|real part|+|imag. part|=1.
If jobvr = 'N'jobvr='N', vr is not referenced.
7:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

  info = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: jobvl, 2: jobvr, 3: n, 4: a, 5: lda, 6: b, 7: ldb, 8: alpha, 9: beta, 10: vl, 11: ldvl, 12: vr, 13: ldvr, 14: work, 15: lwork, 16: rwork, 17: 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.
W INFO = 1tonINFO=1ton
The QZQZ iteration failed. No eigenvectors have been calculated, but alpha(j)alphaj and beta(j)betaj should be correct for j = info + 1,,nj=info+1,,n.
  INFO = N + 1INFO=N+1
Unexpected error returned from nag_lapack_zhgeqz (f08xs).
  INFO = N + 2INFO=N+2
Error returned from nag_lapack_ztgevc (f08yx).

Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrices (A + E)(A+E) and (B + F)(B+F), where
(E,F)F = O(ε) (A,B)F ,
(E,F) F = O(ε) (A,B) F ,
and εε is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details.
Note:  interpretation of results obtained with the QZQZ algorithm often requires a clear understanding of the effects of small changes in the original data. These effects are reviewed in Wilkinson (1979), in relation to the significance of small values of αjαj and βjβj. It should be noted that if αjαj and βjβj are both small for any jj, it may be that no reliance can be placed on any of the computed eigenvalues λi = αi / βiλi=αi/βi. You are recommended to study Wilkinson (1979) and, if in difficulty, to seek expert advice on determining the sensitivity of the eigenvalues to perturbations in the data.

Further Comments

The total number of floating point operations is proportional to n3n3.
The real analogue of this function is nag_lapack_dggev (f08wa).

Example

function nag_lapack_zggev_example
jobvl = 'No left vectors';
jobvr = 'Vectors (right)';
a = [ -21.1 - 22.5i,  53.5 - 50.5i,  -34.5 + 127.5i,  7.5 + 0.5i;
      -0.46 - 7.78i,  -3.5 - 37.5i,  -15.5 + 58.5i,  -10.5 - 1.5i;
      4.3 - 5.5i,  39.7 - 17.1i,  -68.5 + 12.5i,  -7.5 - 3.5i;
      5.5 + 4.4i,  14.4 + 43.3i,  -32.5 - 46i,  -19 - 32.5i];
b = [ 1 - 5i,  1.6 + 1.2i,  -3 + 0i,  0 - 1i;
      0.8 - 0.6i,  3 - 5i,  -4 + 3i,  -2.4 - 3.2i;
      1 + 0i,  2.4 + 1.8i,  -4 - 5i,  0 - 3i;
      0 + 1i,  -1.8 + 2.4i,  0 - 4i,  4 - 5i];
[aOut, bOut, alpha, beta, vl, vr, info] = nag_lapack_zggev(jobvl, jobvr, a, b)
 

aOut =

   1.0e+02 *

   0.1903 - 0.5710i   0.5359 - 0.8982i  -0.8131 - 0.6323i   1.0666 - 0.4479i
   0.0000 + 0.0000i   0.1188 - 0.2970i   0.0356 + 0.2763i  -0.0067 - 0.1642i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.1096 - 0.0365i  -0.2502 - 0.0820i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.2187 - 0.2734i


bOut =

   6.3443 + 0.0000i   3.3986 + 0.7119i  -0.5152 - 2.3820i   6.5818 + 2.4299i
   0.0000 + 0.0000i   5.9409 + 0.0000i  -2.4480 - 0.3427i   5.7385 - 0.7017i
   0.0000 + 0.0000i   0.0000 + 0.0000i   3.6536 + 0.0000i  -1.4096 - 3.9326i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   5.4681 + 0.0000i


alpha =

  19.0329 -57.0986i
  11.8818 -29.7045i
  10.9609 - 3.6536i
  21.8722 -27.3403i


beta =

   6.3443 + 0.0000i
   5.9409 + 0.0000i
   3.6536 + 0.0000i
   5.4681 + 0.0000i


vl =

  6.9226e-310 + 0.0000e+00i


vr =

  -0.8238 - 0.1762i   0.6397 + 0.3603i   0.9775 + 0.0225i  -0.9062 + 0.0938i
  -0.1530 + 0.0707i   0.0042 - 0.0005i   0.1591 - 0.1137i  -0.0074 + 0.0069i
  -0.0707 - 0.1530i   0.0402 + 0.0226i   0.1209 - 0.1537i   0.0302 - 0.0031i
   0.1530 - 0.0707i  -0.0226 + 0.0402i   0.1537 + 0.1209i  -0.0146 - 0.1410i


info =

                    0


function f08wn_example
jobvl = 'No left vectors';
jobvr = 'Vectors (right)';
a = [ -21.1 - 22.5i,  53.5 - 50.5i,  -34.5 + 127.5i,  7.5 + 0.5i;
      -0.46 - 7.78i,  -3.5 - 37.5i,  -15.5 + 58.5i,  -10.5 - 1.5i;
      4.3 - 5.5i,  39.7 - 17.1i,  -68.5 + 12.5i,  -7.5 - 3.5i;
      5.5 + 4.4i,  14.4 + 43.3i,  -32.5 - 46i,  -19 - 32.5i];
b = [ 1 - 5i,  1.6 + 1.2i,  -3 + 0i,  0 - 1i;
      0.8 - 0.6i,  3 - 5i,  -4 + 3i,  -2.4 - 3.2i;
      1 + 0i,  2.4 + 1.8i,  -4 - 5i,  0 - 3i;
      0 + 1i,  -1.8 + 2.4i,  0 - 4i,  4 - 5i];
[aOut, bOut, alpha, beta, vl, vr, info] = f08wn(jobvl, jobvr, a, b)
 

aOut =

   1.0e+02 *

   0.1903 - 0.5710i   0.5359 - 0.8982i  -0.8131 - 0.6323i   1.0666 - 0.4479i
   0.0000 + 0.0000i   0.1188 - 0.2970i   0.0356 + 0.2763i  -0.0067 - 0.1642i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.1096 - 0.0365i  -0.2502 - 0.0820i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.2187 - 0.2734i


bOut =

   6.3443 + 0.0000i   3.3986 + 0.7119i  -0.5152 - 2.3820i   6.5818 + 2.4299i
   0.0000 + 0.0000i   5.9409 + 0.0000i  -2.4480 - 0.3427i   5.7385 - 0.7017i
   0.0000 + 0.0000i   0.0000 + 0.0000i   3.6536 + 0.0000i  -1.4096 - 3.9326i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   5.4681 + 0.0000i


alpha =

  19.0329 -57.0986i
  11.8818 -29.7045i
  10.9609 - 3.6536i
  21.8722 -27.3403i


beta =

   6.3443 + 0.0000i
   5.9409 + 0.0000i
   3.6536 + 0.0000i
   5.4681 + 0.0000i


vl =

   0.0000 + 0.0000i


vr =

  -0.8238 - 0.1762i   0.6397 + 0.3603i   0.9775 + 0.0225i  -0.9062 + 0.0938i
  -0.1530 + 0.0707i   0.0042 - 0.0005i   0.1591 - 0.1137i  -0.0074 + 0.0069i
  -0.0707 - 0.1530i   0.0402 + 0.0226i   0.1209 - 0.1537i   0.0302 - 0.0031i
   0.1530 - 0.0707i  -0.0226 + 0.0402i   0.1537 + 0.1209i  -0.0146 - 0.1410i


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