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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dggevx (f08wb)

Purpose

nag_lapack_dggevx (f08wb) computes for a pair of nn by nn real nonsymmetric matrices (A,B)(A,B) the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the QZQZ 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)(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 eigenvector vjvj corresponding to the eigenvalue λjλj of (A,B)(A,B) satisfies
A vj = λj B vj .
A vj = λj B vj .
The left eigenvector ujuj corresponding to the 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 generalized eigenproblem Ax = λBxAx=λBx, where AA and BB are real, square matrices, are determined using the QZQZ algorithm. The QZQZ algorithm consists of four stages:
  1. AA is reduced to upper Hessenberg form and at the same time BB is reduced to upper triangular form.
  2. AA is further reduced to quasi-triangular form while the triangular form of BB is maintained. This is the real generalized Schur form of the pair (A,B) (A,B) .
  3. The quasi-triangular form of AA is reduced to triangular form and the eigenvalues extracted. 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. Pairs of complex eigenvalues occur with αj / βjαj/βj and αj + 1 / βj + 1αj+1/βj+1 complex conjugates, even though αjαj and αj + 1αj+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 Section [Description] in (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 QZQZ 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'balanc='N'
Do not diagonally scale or permute.
balanc = 'P'balanc='P'
Permute only.
balanc = 'S'balanc='S'
Scale only.
balanc = 'B'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'balanc='B' is recommended.
Constraint: balanc = 'N'balanc='N', 'P''P', 'S''S' or 'B''B'.
2:     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'.
3:     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'.
4:     sense – string (length ≥ 1)
Determines which reciprocal condition numbers are computed.
sense = 'N'sense='N'
None are computed.
sense = 'E'sense='E'
Computed for eigenvalues only.
sense = 'V'sense='V'
Computed for eigenvectors only.
sense = 'B'sense='B'
Computed for eigenvalues and eigenvectors.
Constraint: sense = 'N'sense='N', 'E''E', 'V''V' or 'B''B'.
5:     a(lda, : :) – double 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).
6:     b(ldb, : :) – double 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 iwork bwork

Output Parameters

1:     a(lda, : :) – double 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)
ldamax (1,n)ldamax(1,n).
a has been overwritten. If jobvl = 'V'jobvl='V' or jobvr = 'V'jobvr='V' or both, then AA contains the first part of the real Schur form of the ‘balanced’ versions of the input AA and BB.
2:     b(ldb, : :) – double 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:     alphar(n) – double array
The element alphar(j)alpharj contains the real part of αjαj.
4:     alphai(n) – double array
The element alphai(j)alphaij contains the imaginary part of αjαj.
5:     beta(n) – double array
(alphar(j) + alphai(j) × i) / beta(j)(alpharj+alphaij×i)/betaj, for j = 1,2,,nj=1,2,,n, will be the generalized eigenvalues.
If alphai(j)alphaij is zero, then the jjth eigenvalue is real; if positive, then the jjth and (j + 1)(j+1)st eigenvalues are a complex conjugate pair, with alphai(j + 1)alphaij+1 negative.
Note:  the quotients alphar(j) / beta(j)alpharj/betaj and alphai(j) / beta(j)alphaij/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.
6:     vl(ldvl, : :) – double 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 eigenvectors ujuj are stored one after another in the columns of vl, in the same order as the corresponding eigenvalues.
If the jjth eigenvalue is real, then uj = vl( : ,j)uj=vl:,j, the jjth column of vlvl.
If the jjth and (j + 1)(j+1)th eigenvalues form a complex conjugate pair, then uj = vl( : ,j) + i × vl( : ,j + 1)uj=vl:,j+i×vl:,j+1 and u(j + 1) = vl( : ,j)i × vl( : ,j + 1)u(j+1)=vl:,j-i×vl:,j+1. Each eigenvector will be scaled so the largest component has |real part| + |imag. part| = 1|real part|+|imag. part|=1.
If jobvl = 'N'jobvl='N', vl is not referenced.
7:     vr(ldvr, : :) – double 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 eigenvectors vjvj are stored one after another in the columns of vr, in the same order as their eigenvalues.
If the jjth eigenvalue is real, then v(j) = vr( : ,j)v(j)=vr:,j, the jjth column of VRVR.
If the jjth and (j + 1)(j+1)th eigenvalues form a complex conjugate pair, then vj = vr( : ,j) + i × vr( : ,j + 1)vj=vr:,j+i×vr:,j+1 and vj + 1 = vr( : ,j) i × vr( : ,j + 1) vj+1 = vr:,j -i×vr:,j+1.
Each eigenvector will be scaled so the largest component has |real part| + |imag. part| = 1|real part|+|imag. part|=1.
If jobvr = 'N'jobvr='N', vr is not referenced.
8:     ilo – int64int32nag_int scalar
9:     ihi – int64int32nag_int scalar
ilo and ihi are integer values such that a(i,j) = 0aij=0 and b(i,j) = 0bij=0 if i > ji>j and j = 1,2,,ilo1j=1,2,,ilo-1 or i = ihi + 1,,ni=ihi+1,,n.
If balanc = 'N'balanc='N' or 'S''S', ilo = 1ilo=1 and ihi = nihi=n.
10:   lscale(n) – double array
Details of the permutations and scaling factors applied to the left side of AA and BB.
If plj plj is the index of the row interchanged with row jj, and dlj dlj is the scaling factor applied to row jj, then:
  • lscale(j) = plj lscalej = plj , for j = 1,2,,ilo1j=1,2,,ilo-1;
  • lscale = dlj lscale = dlj , for j = ilo,,ihij=ilo,,ihi;
  • lscale = plj lscale = plj , for j = ihi + 1,,nj=ihi+1,,n.
The order in which the interchanges are made is n to ihi + 1ihi+1, then 11 to ilo1ilo-1.
11:   rscale(n) – double array
Details of the permutations and scaling factors applied to the right side of AA and BB.
If prjprj is the index of the column interchanged with column jj, and drjdrj is the scaling factor applied to column jj, then:
  • rscale(j) = prjrscalej=prj, for j = 1,2,,ilo1j=1,2,,ilo-1;
  • if rscale = drjrscale=drj, for j = ilo,,ihij=ilo,,ihi;
  • if rscale = prjrscale=prj, for j = ihi + 1,,nj=ihi+1,,n.
The order in which the interchanges are made is n to ihi + 1ihi+1, then 11 to ilo1ilo-1.
12:   abnrm – double scalar
The 11-norm of the balanced matrix AA.
13:   bbnrm – double scalar
The 11-norm of the balanced matrix BB.
14:   rconde( : :) – double array
Note: the dimension of the array rconde must be at least max (1,n)max(1,n).
If sense = 'E'sense='E' or 'B''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 rconde(j)rcondej, rcondv(j)rcondvj, and the jjth columns of vl and vr all correspond to the jjth eigenpair.
If sense = 'V'sense='V', rconde is not referenced.
15:   rcondv( : :) – double array
Note: the dimension of the array rcondv must be at least max (1,n)max(1,n).
If sense = 'V'sense='V' or 'B''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'sense='E', rcondv is not referenced.
16:   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: 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 = 1tonINFO=1ton
The QZQZ iteration failed. No eigenvectors have been calculated, but alphar(j)alpharj, alphai(j)alphaij, 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_dhgeqz (f08xe).
  INFO = N + 2INFO=N+2
Error returned from nag_lapack_dtgevc (f08yk).

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.
An approximate error bound on the chordal distance between the iith computed generalized eigenvalue ww and the corresponding exact eigenvalue λ λ  is
ε × abnrm,bbnrm2 / rconde(i) .
ε × abnrm,bbnrm2 / rcondei .
An approximate error bound for the angle between the iith computed eigenvector vl(i) vli  or vr(i) vri  is given by
ε × abnrm,bbnrm2 / rcondv(i) .
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 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 complex analogue of this function is nag_lapack_zggevx (f08wp).

Example

function nag_lapack_dggevx_example
balanc = 'Balance';
jobvl = 'No vectors (left)';
jobvr = 'Vectors (right)';
sense = 'Both reciprocal condition numbers';
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];
[aOut, bOut, alphar, alphai, beta, vl, vr, ilo, ihi, lscale, rscale, ...
 abnrm, bbnrm, rconde, rcondv, info] = ...
    nag_lapack_dggevx(balanc, jobvl, jobvr, sense, a, b);

epsilon = nag_machine_precision
small = nag_machine_real_safe;
absnrm = sqrt(abnrm^2+bbnrm^2);
tol = epsilon*absnrm;

for j=1:4

  % Print out information on the jth eigenvalue
  if (abs(alphar(j)) + abs(alphai(j)))*small >= abs(beta(j))
    fprintf('\nEigenvalue(%d) is numerically infinite or undetermined\n');
    fprintf('alphar(%d) = %11.4e, alphai(%d) = %11.4e, beta(%d) = %11.4e\n', ...
            j, alphar(j), j, alphai(j), j, beta(j));
  else
    fprintf('\nEigenvalue(%d) = %s\n\n', j, num2str(complex(alphar(j), alphai(j))/beta(j)));
  end
  fprintf('Reciprocal condition number = %8.1e\n', rconde(j));
  if rconde(j) > 0
    fprintf('Error bound                 = %8.1e\n', tol/rconde(j));
  else
    fprintf('Error bound is infinite\n');
  end

  % Print out information on the jth eigenvector
  fprintf('\nEigenvector(%d)\n', j);
  if alphai(j) == 0
    disp(vr(:, j));
  elseif alphai(j) > 0
    disp(complex(vr(:, j), vr(:, j+1)));
  else
    disp(complex(vr(:, j-1), vr(:, j)));
  end
  fprintf('Reciprocal condition number = %8.1e\n', rcondv(j));
  if rcondv(j) > 0
    fprintf('Error bound                 = %8.1e\n', tol/rcondv(j));
  else
    fprintf('Error bound is infinite\n');
  end
end

for i=1:4
end
 

epsilon =

   1.1102e-16


Eigenvalue(1) = 2

Reciprocal condition number =  9.5e-02
Error bound                 =  2.5e-14

Eigenvector(1)
   -1.0000
   -0.0057
   -0.0629
   -0.0629

Reciprocal condition number =  1.3e-01
Error bound                 =  1.9e-14

Eigenvalue(2) = 3+4i

Reciprocal condition number =  1.7e-01
Error bound                 =  1.4e-14

Eigenvector(2)
  -0.4255 - 0.5745i
  -0.0851 - 0.1149i
  -0.1430 - 0.0009i
  -0.1430 - 0.0009i

Reciprocal condition number =  3.8e-02
Error bound                 =  6.2e-14

Eigenvalue(3) = 3-4i

Reciprocal condition number =  1.7e-01
Error bound                 =  1.4e-14

Eigenvector(3)
  -0.4255 - 0.5745i
  -0.0851 - 0.1149i
  -0.1430 - 0.0009i
  -0.1430 - 0.0009i

Reciprocal condition number =  3.8e-02
Error bound                 =  6.2e-14

Eigenvalue(4) = 4

Reciprocal condition number =  5.1e-01
Error bound                 =  4.6e-15

Eigenvector(4)
   -1.0000
   -0.0111
    0.0333
   -0.1556

Reciprocal condition number =  7.1e-02
Error bound                 =  3.3e-14

function f08wb_example
balanc = 'Balance';
jobvl = 'No vectors (left)';
jobvr = 'Vectors (right)';
sense = 'Both reciprocal condition numbers';
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];
[aOut, bOut, alphar, alphai, beta, vl, vr, ilo, ihi, lscale, rscale, ...
 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=1:4

  % Print out information on the jth eigenvalue
  if (abs(alphar(j)) + abs(alphai(j)))*small >= abs(beta(j))
    fprintf('\nEigenvalue(%d) is numerically infinite or undetermined\n');
    fprintf('alphar(%d) = %11.4e, alphai(%d) = %11.4e, beta(%d) = %11.4e\n', ...
            j, alphar(j), j, alphai(j), j, beta(j));
  else
    fprintf('\nEigenvalue(%d) = %s\n\n', j, num2str(complex(alphar(j), alphai(j))/beta(j)));
  end
  fprintf('Reciprocal condition number = %8.1e\n', rconde(j));
  if rconde(j) > 0
    fprintf('Error bound                 = %8.1e\n', tol/rconde(j));
  else
    fprintf('Error bound is infinite\n');
  end

  % Print out information on the jth eigenvector
  fprintf('\nEigenvector(%d)\n', j);
  if alphai(j) == 0
    disp(vr(:, j));
  elseif alphai(j) > 0
    disp(complex(vr(:, j), vr(:, j+1)));
  else
    disp(complex(vr(:, j-1), vr(:, j)));
  end
  fprintf('Reciprocal condition number = %8.1e\n', rcondv(j));
  if rcondv(j) > 0
    fprintf('Error bound                 = %8.1e\n', tol/rcondv(j));
  else
    fprintf('Error bound is infinite\n');
  end
end

for i=1:4
end
 

epsilon =

   1.1102e-16


Eigenvalue(1) = 2

Reciprocal condition number =  9.5e-02
Error bound                 =  2.5e-14

Eigenvector(1)
   -1.0000
   -0.0057
   -0.0629
   -0.0629

Reciprocal condition number =  1.3e-01
Error bound                 =  1.9e-14

Eigenvalue(2) = 3+4i

Reciprocal condition number =  1.7e-01
Error bound                 =  1.4e-14

Eigenvector(2)
  -0.4255 - 0.5745i
  -0.0851 - 0.1149i
  -0.1430 - 0.0009i
  -0.1430 - 0.0009i

Reciprocal condition number =  3.8e-02
Error bound                 =  6.2e-14

Eigenvalue(3) = 3-4i

Reciprocal condition number =  1.7e-01
Error bound                 =  1.4e-14

Eigenvector(3)
  -0.4255 - 0.5745i
  -0.0851 - 0.1149i
  -0.1430 - 0.0009i
  -0.1430 - 0.0009i

Reciprocal condition number =  3.8e-02
Error bound                 =  6.2e-14

Eigenvalue(4) = 4

Reciprocal condition number =  5.1e-01
Error bound                 =  4.6e-15

Eigenvector(4)
   -1.0000
   -0.0111
    0.0333
   -0.1556

Reciprocal condition number =  7.1e-02
Error bound                 =  3.3e-14


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