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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dggevx (f08wb)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dggevx (f08wb) computes for a pair of n by n real nonsymmetric matrices A,B the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the QZ algorithm.
Optionally it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors, reciprocal condition numbers for the eigenvalues, and reciprocal condition numbers for the right eigenvectors.

Syntax

[a, b, alphar, alphai, beta, vl, vr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, info] = f08wb(balanc, jobvl, jobvr, sense, a, b, 'n', n)
[a, b, alphar, alphai, beta, vl, vr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, info] = nag_lapack_dggevx(balanc, jobvl, jobvr, sense, a, b, 'n', n)

Description

A generalized eigenvalue for a pair of matrices A,B is a scalar λ or a ratio α/β=λ, such that A-λB is singular. It is usually represented as the pair α,β, as there is a reasonable interpretation for β=0, and even for both being zero.
The right eigenvector vj corresponding to the eigenvalue λj of A,B satisfies
A vj = λj B vj .  
The left eigenvector uj corresponding to the eigenvalue λj of A,B satisfies
ujH A = λj ujH B ,  
where ujH is the conjugate-transpose of uj.
All the eigenvalues and, if required, all the eigenvectors of the generalized eigenproblem Ax=λBx, where A and B are real, square matrices, are determined using the QZ algorithm. The QZ algorithm consists of four stages:
1. A is reduced to upper Hessenberg form and at the same time B is reduced to upper triangular form.
2. A is further reduced to quasi-triangular form while the triangular form of B is maintained. This is the real generalized Schur form of the pair A,B .
3. The quasi-triangular form of A is reduced to triangular form and the eigenvalues extracted. This function does not actually produce the eigenvalues λj, but instead returns αj and βj such that
λj=αj/βj,  j=1,2,,n.  
The division by βj becomes your responsibility, since βj may be zero, indicating an infinite eigenvalue. Pairs of complex eigenvalues occur with αj/βj and αj+1/βj+1 complex conjugates, even though αj and αj+1 are not conjugate.
4. If the eigenvectors are required they are obtained from the triangular matrices and then transformed back into the original coordinate system.
For details of the balancing option, see Description in nag_lapack_dggbal (f08wh).

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 QZ algorithm Linear Algebra Appl. 28 285–303

Parameters

Compulsory Input Parameters

1:     balanc – string (length ≥ 1)
Specifies the balance option to be performed.
balanc='N'
Do not diagonally scale or permute.
balanc='P'
Permute only.
balanc='S'
Scale only.
balanc='B'
Both permute and scale.
Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. Permuting does not change condition numbers (in exact arithmetic), but balancing does. In the absence of other information, balanc='B' is recommended.
Constraint: balanc='N', 'P', 'S' or 'B'.
2:     jobvl – string (length ≥ 1)
If jobvl='N', do not compute the left generalized eigenvectors.
If jobvl='V', compute the left generalized eigenvectors.
Constraint: jobvl='N' or 'V'.
3:     jobvr – string (length ≥ 1)
If jobvr='N', do not compute the right generalized eigenvectors.
If jobvr='V', compute the right generalized eigenvectors.
Constraint: jobvr='N' or 'V'.
4:     sense – string (length ≥ 1)
Determines which reciprocal condition numbers are computed.
sense='N'
None are computed.
sense='E'
Computed for eigenvalues only.
sense='V'
Computed for eigenvectors only.
sense='B'
Computed for eigenvalues and eigenvectors.
Constraint: sense='N', 'E', 'V' or 'B'.
5:     alda: – double array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
The matrix A in the pair A,B.
6:     bldb: – double array
The first dimension of the array b must be at least max1,n.
The second dimension of the array b must be at least max1,n.
The matrix B in the pair 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.)
n, the order of the matrices A and B.
Constraint: n0.

Output Parameters

1:     alda: – double array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,n.
a has been overwritten. If jobvl='V' or jobvr='V' or both, then A contains the first part of the real Schur form of the ‘balanced’ versions of the input A and B.
2:     bldb: – double array
The first dimension of the array b will be max1,n.
The second dimension of the array b will be max1,n.
b has been overwritten.
3:     alpharn – double array
The element alpharj contains the real part of αj.
4:     alphain – double array
The element alphaij contains the imaginary part of αj.
5:     betan – double array
alpharj+alphaij×i/betaj, for j=1,2,,n, will be the generalized eigenvalues.
If alphaij is zero, then the jth eigenvalue is real; if positive, then the jth and j+1st eigenvalues are a complex conjugate pair, with alphaij+1 negative.
Note:  the quotients alpharj/betaj and alphaij/betaj may easily overflow or underflow, and betaj may even be zero. Thus, you should avoid naively computing the ratio αj/βj. However, maxαj will always be less than and usually comparable with a2 in magnitude, and maxβj will always be less than and usually comparable with b2.
6:     vlldvl: – double array
The first dimension, ldvl, of the array vl will be
  • if jobvl='V', ldvl= max1,n ;
  • otherwise ldvl=1.
The second dimension of the array vl will be max1,n if jobvl='V' and 1 otherwise.
If jobvl='V', the left generalized eigenvectors uj 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.
If jobvl='N', vl is not referenced.
7:     vrldvr: – double array
The first dimension, ldvr, of the array vr will be
  • if jobvr='V', ldvr= max1,n ;
  • otherwise ldvr=1.
The second dimension of the array vr will be max1,n if jobvr='V' and 1 otherwise.
If jobvr='V', the right generalized eigenvectors vj 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.
If jobvr='N', vr is not referenced.
8:     ilo int64int32nag_int scalar
9:     ihi int64int32nag_int scalar
ilo and ihi are integer values such that aij=0 and bij=0 if i>j and j=1,2,,ilo-1 or i=ihi+1,,n.
If balanc='N' or 'S', ilo=1 and ihi=n.
10:   lscalen – double array
Details of the permutations and scaling factors applied to the left side of A and B.
If plj is the index of the row interchanged with row j, and dlj is the scaling factor applied to row j, then:
  • lscalej = plj , for j=1,2,,ilo-1;
  • lscale = dlj , for j=ilo,,ihi;
  • lscale = plj , for j=ihi+1,,n.
The order in which the interchanges are made is n to ihi+1, then 1 to ilo-1.
11:   rscalen – double array
Details of the permutations and scaling factors applied to the right side of A and B.
If prj is the index of the column interchanged with column j, and drj is the scaling factor applied to column j, then:
  • rscalej=prj, for j=1,2,,ilo-1;
  • if rscale=drj, for j=ilo,,ihi;
  • if rscale=prj, for j=ihi+1,,n.
The order in which the interchanges are made is n to ihi+1, then 1 to ilo-1.
12:   abnrm – double scalar
The 1-norm of the balanced matrix A.
13:   bbnrm – double scalar
The 1-norm of the balanced matrix B.
14:   rconde: – double array
The dimension of the array rconde will be max1,n
If sense='E' or 'B', the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of rconde are set to the same value. Thus rcondej, rcondvj, and the jth columns of vl and vr all correspond to the jth eigenpair.
If sense='V', rconde is not referenced.
15:   rcondv: – double array
The dimension of the array rcondv will be max1,n
If sense='V' or 'B', the estimated reciprocal condition numbers of the eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of rcondv are set to the same value.
If sense='E', rcondv is not referenced.
16:   info int64int32nag_int scalar
info=0 unless the function detects an error (see 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=-i
If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: balanc, 2: jobvl, 3: jobvr, 4: sense, 5: n, 6: a, 7: lda, 8: b, 9: ldb, 10: alphar, 11: alphai, 12: beta, 13: vl, 14: ldvl, 15: vr, 16: ldvr, 17: ilo, 18: ihi, 19: lscale, 20: rscale, 21: abnrm, 22: bbnrm, 23: rconde, 24: rcondv, 25: work, 26: lwork, 27: iwork, 28: bwork, 29: 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=1ton
The QZ iteration failed. No eigenvectors have been calculated, but alpharj, alphaij, and betaj should be correct for j=info+1,,n.
   info=n+1
Unexpected error returned from nag_lapack_dhgeqz (f08xe).
   info=n+2
Error returned from nag_lapack_dtgevc (f08yk).

Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrices A+E and B+F, where
E,F F = Oε A,B F ,  
and ε is the machine precision.
An approximate error bound on the chordal distance between the ith computed generalized eigenvalue w and the corresponding exact eigenvalue λ  is
ε × abnrm,bbnrm2 / rcondei .  
An approximate error bound for the angle between the ith computed eigenvector uj  or vj  is given by
ε × abnrm,bbnrm2 / rcondvi .  
For further explanation of the reciprocal condition numbers rconde and rcondv, see Section 4.11 of Anderson et al. (1999).
Note:  interpretation of results obtained with the QZ 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 αj and βj. It should be noted that if αj and βj are both small for any j, it may be that no reliance can be placed on any of the computed eigenvalues λi=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 n3.
The complex analogue of this function is nag_lapack_zggevx (f08wp).

Example

This example finds all the eigenvalues and right eigenvectors of the matrix pair A,B, where
A = 3.9 12.5 -34.5 -0.5 4.3 21.5 -47.5 7.5 4.3 21.5 -43.5 3.5 4.4 26.0 -46.0 6.0   and   B= 1.0 2.0 -3.0 1.0 1.0 3.0 -5.0 4.0 1.0 3.0 -4.0 3.0 1.0 3.0 -4.0 4.0 ,  
together with estimates of the condition number and forward error bounds for each eigenvalue and eigenvector. The option to balance the matrix pair is used.
Note that the block size (NB) of 64 assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
function f08wb_example


fprintf('f08wb example results\n\n');

% Generalized eigenvalues and right eigenvectors of (A, B):
n = 4;
a = [3.9, 12.5, -34.5, -0.5;
     4.3, 21.5, -47.5,  7.5;
     4.3, 21.5, -43.5,  3.5;
     4.4, 26,   -46,    6];
b = [1,    2,    -3,    1;
     1,    3,    -5,    4;
     1,    3,    -4,    3;
     1,    3,    -4,    4];

balanc = 'Balance';
jobvl = 'No vectors (left)';
jobvr = 'Vectors (right)';
sense = 'Both reciprocal condition numbers';
[~, ~, alphar, alphai, beta, ~, VR, ~, ~, ~, ~, abnrm, bbnrm, ...
	     rconde, rcondv, info] = ...
  f08wb( ...
	 balanc, jobvl, jobvr, sense, a, b);

epsilon = x02aj;
small   = x02am;
absnrm  = sqrt(abnrm^2+bbnrm^2);
tol     = epsilon*absnrm;

for j=1:n

  % display information on the jth eigenvalue
  if (abs(alphar(j)) + abs(alphai(j)))*small >= abs(beta(j))
    fprintf('\n%4d: Eigenvalue is numerically infinite or undetermined\n',j);
    fprintf('%4d: alphar = %11.4e, alphai = %11.4e, beta = %11.4e\n', ...
	    j, alphar(j), alphai(j), beta(j));
  else
    fprintf('\n%4d: Eigenvalue =  ', j);
    if alphai(j)==0
      fprintf('%10.4e\n',alphar(j)/beta(j));
    elseif alphai(j)>=0
      fprintf('%10.4e + %10.4ei\n',alphar(j)/beta(j), alphai(j)/beta(j));
    else
      fprintf('%10.4e - %10.4ei\n',alphar(j)/beta(j), -alphai(j)/beta(j));
    end
  end

  if rconde(j) > 0
    fprintf('\n      Condition number            = %8.1e\n', 1/rconde(j));
    fprintf('      Error bound                 = %8.1e\n', tol/rconde(j));
  else
    fprintf('\n      Reciprocal condition number = %8.1e\n', rconde(j));
    fprintf('      Error bound is infinite\n');
  end

  fprintf('\n      Eigenvector:\n');
  if alphai(j) == 0
    fprintf('%30.4e\n',VR(:, j));
  else
    if alphai(j) > 0
      k = j;
    else   
      k = j-1;
    end
    for l = 1:n
      if VR(l,k+1)>=0
        fprintf('%30.4e + %10.4ei\n',VR(l, k), VR(l, k+1));
      else
        fprintf('%30.4e - %10.4ei\n',VR(l, k), -VR(l, k+1));
      end
    end
  end

  if rcondv(j) > 0
    fprintf('\n      Condition number            = %8.1e\n', 1/rcondv(j));
    fprintf('      Error bound                 = %8.1e\n', tol/rcondv(j));
  else
    fprintf('\n      Reciprocal condition number = %8.1e\n', rcondv(j));
    fprintf('      Error bound is infinite\n');
  end
end


f08wb example results


   1: Eigenvalue =  2.0000e+00

      Condition number            =  1.1e+01
      Error bound                 =  2.5e-14

      Eigenvector:
                   -1.0000e+00
                   -5.7143e-03
                   -6.2857e-02
                   -6.2857e-02

      Condition number            =  8.0e+00
      Error bound                 =  1.9e-14

   2: Eigenvalue =  3.0000e+00 + 4.0000e+00i

      Condition number            =  6.1e+00
      Error bound                 =  1.4e-14

      Eigenvector:
                   -4.2550e-01 - 5.7450e-01i
                   -8.5099e-02 - 1.1490e-01i
                   -1.4298e-01 - 8.6125e-04i
                   -1.4298e-01 - 8.6125e-04i

      Condition number            =  2.6e+01
      Error bound                 =  6.2e-14

   3: Eigenvalue =  3.0000e+00 - 4.0000e+00i

      Condition number            =  6.1e+00
      Error bound                 =  1.4e-14

      Eigenvector:
                   -4.2550e-01 - 5.7450e-01i
                   -8.5099e-02 - 1.1490e-01i
                   -1.4298e-01 - 8.6125e-04i
                   -1.4298e-01 - 8.6125e-04i

      Condition number            =  2.6e+01
      Error bound                 =  6.2e-14

   4: Eigenvalue =  4.0000e+00

      Condition number            =  1.9e+00
      Error bound                 =  4.6e-15

      Eigenvector:
                   -1.0000e+00
                   -1.1111e-02
                    3.3333e-02
                   -1.5556e-01

      Condition number            =  1.4e+01
      Error bound                 =  3.3e-14

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