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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zhgeqz (f08xs)

Purpose

nag_lapack_zhgeqz (f08xs) implements the QZQZ method for finding generalized eigenvalues of the complex matrix pair (A,B)(A,B) of order nn, which is in the generalized upper Hessenberg form.

Syntax

[a, b, alpha, beta, q, z, info] = f08xs(job, compq, compz, ilo, ihi, a, b, q, z, 'n', n)
[a, b, alpha, beta, q, z, info] = nag_lapack_zhgeqz(job, compq, compz, ilo, ihi, a, b, q, z, 'n', n)

Description

nag_lapack_zhgeqz (f08xs) implements a single-shift version of the QZQZ method for finding the generalized eigenvalues of the complex matrix pair (A,B)(A,B) which is in the generalized upper Hessenberg form. If the matrix pair (A,B)(A,B) is not in the generalized upper Hessenberg form, then the function nag_lapack_zgghrd (f08ws) should be called before invoking nag_lapack_zhgeqz (f08xs).
This problem is mathematically equivalent to solving the matrix equation
det(AλB) = 0.
det(A-λB)=0.
Note that, to avoid underflow, overflow and other arithmetic problems, the generalized eigenvalues λjλj are never computed explicitly by this function but defined as ratios between two computed values, αjαj and βjβj:
λj = αj / βj.
λj=αj/βj.
The parameters αjαj, in general, are finite complex values and βjβj are finite real non-negative values.
If desired, the matrix pair (A,B)(A,B) may be reduced to generalized Schur form. That is, the transformed matrices AA and BB are upper triangular and the diagonal values of AA and BB provide αα and ββ.
The parameter job specifies two options. If job = 'S'job='S' then the matrix pair (A,B)(A,B) is simultaneously reduced to Schur form by applying one unitary transformation (usually called QQ) on the left and another (usually called ZZ) on the right. That is,
AQHAZ
BQHBZ
AQHAZ BQHBZ
If job = 'E'job='E', then at each iteration the same transformations are computed but they are only applied to those parts of AA and BB which are needed to compute αα and ββ. This option could be used if generalized eigenvalues are required but not generalized eigenvectors.
If job = 'S'job='S' and compq = 'V'compq='V' or 'I''I', and compz = 'V'compz='V' or 'I''I', then the unitary transformations used to reduce the pair (A,B)(A,B) are accumulated into the input arrays q and z. If generalized eigenvectors are required then job must be set to job = 'S'job='S' and if left (right) generalized eigenvectors are to be computed then compq (compz) must be set to compq = 'V'compq='V' or 'I''I' rather than compq = 'N'compq='N'.
If compq = 'I'compq='I', then eigenvectors are accumulated on the identity matrix and on exit the array q contains the left eigenvector matrix QQ. However, if compq = 'V'compq='V' then the transformations are accumulated in the user-supplied matrix Q0Q0 in array q on entry and thus on exit q contains the matrix product QQ0QQ0. A similar convention is used for compz.

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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems SIAM J. Numer. Anal. 10 241–256
Stewart G W and Sun J-G (1990) Matrix Perturbation Theory Academic Press, London

Parameters

Compulsory Input Parameters

1:     job – string (length ≥ 1)
Specifies the operations to be performed on (A,B)(A,B).
job = 'E'job='E'
The matrix pair (A,B)(A,B) on exit might not be in the generalized Schur form.
job = 'S'job='S'
The matrix pair (A,B)(A,B) on exit will be in the generalized Schur form.
Constraint: job = 'E'job='E' or 'S''S'.
2:     compq – string (length ≥ 1)
Specifies the operations to be performed on QQ:
compq = 'N'compq='N'
The array q is unchanged.
compq = 'V'compq='V'
The left transformation QQ is accumulated on the array q.
compq = 'I'compq='I'
The array q is initialized to the identity matrix before the left transformation QQ is accumulated in q.
Constraint: compq = 'N'compq='N', 'V''V' or 'I''I'.
3:     compz – string (length ≥ 1)
Specifies the operations to be performed on ZZ.
compz = 'N'compz='N'
The array z is unchanged.
compz = 'V'compz='V'
The right transformation ZZ is accumulated on the array z.
compz = 'I'compz='I'
The array z is initialized to the identity matrix before the right transformation ZZ is accumulated in z.
Constraint: compz = 'N'compz='N', 'V''V' or 'I''I'.
4:     ilo – int64int32nag_int scalar
5:     ihi – int64int32nag_int scalar
The indices iloilo and ihiihi, respectively which define the upper triangular parts of AA. The submatrices A(1 : ilo1,1 : ilo1)A(1:ilo-1,1:ilo-1) and A(ihi + 1 : n,ihi + 1 : n)A(ihi+1:n,ihi+1:n) are then upper triangular. These parameters are provided by nag_lapack_zggbal (f08wv) if the matrix pair was previously balanced; otherwise, ilo = 1ilo=1 and ihi = nihi=n.
Constraints:
  • if n > 0n>0, 1 ilo ihi n 1 ilo ihi n ;
  • if n = 0n=0, ilo = 1ilo=1 and ihi = 0ihi=0.
6:     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 nn by nn upper Hessenberg matrix AA. The elements below the first subdiagonal must be set to zero.
7:     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 nn by nn upper triangular matrix BB. The elements below the diagonal must be zero.
8:     q(ldq, : :) – complex array
The first dimension, ldq, of the array q must satisfy
  • if compq = 'V'compq='V' or 'I''I', ldqnldqn;
  • if compq = 'N'compq='N', ldq1ldq1.
The second dimension of the array must be at least max (1,n)max(1,n) if compq = 'V'compq='V' or 'I''I' and at least 11 if compq = 'N'compq='N'
If compq = 'V'compq='V', the matrix Q0Q0. The matrix Q0Q0 is usually the matrix QQ returned by nag_lapack_zgghrd (f08ws).
If compq = 'N'compq='N', q is not referenced.
9:     z(ldz, : :) – complex array
The first dimension, ldz, of the array z must satisfy
  • if compz = 'V'compz='V' or 'I''I', ldznldzn;
  • if compz = 'N'compz='N', ldz1ldz1.
The second dimension of the array must be at least max (1,n)max(1,n) if compz = 'V'compz='V' or 'I''I' and at least 11 if compz = 'N'compz='N'
If compz = 'V'compz='V', the matrix Z0Z0. The matrix Z0Z0 is usually the matrix ZZ returned by nag_lapack_zgghrd (f08ws).
If compz = 'N'compz='N', z is not referenced.

Optional Input Parameters

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

Input Parameters Omitted from the MATLAB Interface

lda ldb ldq ldz 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)
ldamax (1,n)ldamax(1,n).
If job = 'S'job='S', the matrix pair (A,B)(A,B) will be simultaneously reduced to generalized Schur form.
If job = 'E'job='E', the 11 by 11 and 22 by 22 diagonal blocks of the matrix pair (A,B)(A,B) will give generalized eigenvalues but the remaining elements will be irrelevant.
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).
If job = 'S'job='S', the matrix pair (A,B)(A,B) will be simultaneously reduced to generalized Schur form.
If job = 'E'job='E', the 11 by 11 and 22 by 22 diagonal blocks of the matrix pair (A,B)(A,B) will give generalized eigenvalues but the remaining elements will be irrelevant.
3:     alpha(n) – complex array
αjαj, for j = 1,2,,nj=1,2,,n.
4:     beta(n) – complex array
βjβj, for j = 1,2,,nj=1,2,,n.
5:     q(ldq, : :) – complex array
The first dimension, ldq, of the array q will be
  • if compq = 'V'compq='V' or 'I''I', ldqnldqn;
  • if compq = 'N'compq='N', ldq1ldq1.
The second dimension of the array will be max (1,n)max(1,n) if compq = 'V'compq='V' or 'I''I' and at least 11 if compq = 'N'compq='N'
If compq = 'V'compq='V', q contains the matrix product QQ0QQ0.
If compq = 'I'compq='I', q contains the transformation matrix QQ.
6:     z(ldz, : :) – complex array
The first dimension, ldz, of the array z will be
  • if compz = 'V'compz='V' or 'I''I', ldznldzn;
  • if compz = 'N'compz='N', ldz1ldz1.
The second dimension of the array will be max (1,n)max(1,n) if compz = 'V'compz='V' or 'I''I' and at least 11 if compz = 'N'compz='N'
If compz = 'V'compz='V', z contains the matrix product ZZ0ZZ0.
If compz = 'I'compz='I', z contains the transformation matrix ZZ.
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: job, 2: compq, 3: compz, 4: n, 5: ilo, 6: ihi, 7: a, 8: lda, 9: b, 10: ldb, 11: alpha, 12: beta, 13: q, 14: ldq, 15: z, 16: ldz, 17: work, 18: lwork, 19: rwork, 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
If 1infon1infon, the QZQZ iteration did not converge and the matrix pair (A,B)(A,B) is not in the generalized Schur form at exit. However, if info < ninfo<n, then the computed αiαi and βiβi should be correct for i = info + 1,,ni=info+1,,n.
If n + 1info2 × nn+1info2×n, the computation of shifts failed and the matrix pair (A,B)(A,B) is not in the generalized Schur form at exit. However, if info < 2 × ninfo<2×n, then the computed αiαi and βiβi should be correct for i = infon + 1,,ni=info-n+1,,n.
If info > 2 × ninfo>2×n, then an unexpected Library error has occurred. Please contact NAG with details of your program.

Accuracy

Please consult Section 4.11 of the LAPACK Users' Guide (see Anderson et al. (1999)) and Chapter 6 of Stewart and Sun (1990), for more information.

Further Comments

nag_lapack_zhgeqz (f08xs) is the fifth step in the solution of the complex generalized eigenvalue problem and is called after nag_lapack_zgghrd (f08ws).
The number of floating point operations taken by this function is proportional to n3n3.
The real analogue of this function is nag_lapack_dhgeqz (f08xe).

Example

function nag_lapack_zhgeqz_example
job = 'E';
compq = 'N';
compz = 'N';
ilo = int64(1);
ihi = int64(4);
a = [ -2.867890104378767 - 1.594524360587801i, ...
    -0.809337098604556 - 0.3276607283529636i,  -4.900373446187322 - 0.9865105961392747i, ...
      -0.04834623303205134 + 1.162636735910666i;
      -2.671939461604618 + 0.5945064559939077i, ...
    -0.7895240421486864 + 0.04903482075256903i,  -4.954929775736532 - 0.1634387045312732i, ...
      -0.4386325532444865 - 0.5739313215365673i;
      0 + 0i,  -0.09825782595897961 - 0.01149417965898412i, ...
     -1.167669110453865 - 0.1365936851015428i,  -1.756232676852781 - 0.2054437263770907i;
      0 + 0i,  0 + 0i,  0.08729329881919053 + 0.003819531014388017i, ...
      0.03170217359735389 + 0.001387133226909417i];
b = [-1.774823934929885,  -0.7210490086119171 + 0.04290055077107185i, ...
     -5.020721861715561 + 1.189845102979979i,  -0.1450254390211963 + 0.7257437885879333i;
      0 + 0i,  -0.2176281526219798 + 0.03516041416104906i, ...
     -2.541102900685018 - 0.1458063680541886i,  -0.8228500482725508 - 0.4184333588843852i;
      0 + 0i,  0 + 0i,  -1.395782135347712 - 0.1632782984144689i, ...
      -1.747484189780436 - 0.2044203302132494i;
      0 + 0i,  0 + 0i,  0 + 0i,  -0.09963146114886638 - 0.004359389105629694i];
q = [complex(0)];
z = [complex(0)];
[aOut, bOut, alpha, beta, qOut, zOut, info] = ...
    nag_lapack_zhgeqz(job, compq, compz, ilo, ihi, a, b, q, z)
 

aOut =

  -0.1438 + 0.3741i  -1.7000 + 0.1473i  -1.4170 - 1.0916i  -6.6330 - 0.8784i
   0.0000 + 0.0000i   0.2824 + 0.5209i  -1.5812 + 0.5647i   1.9829 - 3.4956i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.1522 - 0.2800i   1.1701 - 1.7439i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.8591 - 0.0636i


bOut =

   0.2263 + 0.0000i   0.2959 + 0.6808i  -0.9706 + 0.3572i  -3.4142 + 1.1586i
   0.0000 + 0.0000i   0.5723 + 0.0000i  -1.7815 + 0.5004i   1.7329 - 3.4482i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.3323 + 0.0000i   1.7834 - 2.2524i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   1.2738 + 0.0000i


alpha =

  -0.1438 + 0.3741i
   0.2824 + 0.5209i
   0.1522 - 0.2800i
   0.8591 - 0.0636i


beta =

   0.2263 + 0.0000i
   0.5723 + 0.0000i
   0.3323 + 0.0000i
   1.2738 + 0.0000i


qOut =

   0.0000 + 0.0000i


zOut =

   0.0000 + 0.0000i


info =

                    0


function f08xs_example
job = 'E';
compq = 'N';
compz = 'N';
ilo = int64(1);
ihi = int64(4);
a = [ -2.867890104378767 - 1.594524360587801i, ...
    -0.809337098604556 - 0.3276607283529636i,  -4.900373446187322 - 0.9865105961392747i, ...
      -0.04834623303205134 + 1.162636735910666i;
      -2.671939461604618 + 0.5945064559939077i, ...
    -0.7895240421486864 + 0.04903482075256903i,  -4.954929775736532 - 0.1634387045312732i, ...
      -0.4386325532444865 - 0.5739313215365673i;
      0 + 0i,  -0.09825782595897961 - 0.01149417965898412i, ...
     -1.167669110453865 - 0.1365936851015428i,  -1.756232676852781 - 0.2054437263770907i;
      0 + 0i,  0 + 0i,  0.08729329881919053 + 0.003819531014388017i, ...
      0.03170217359735389 + 0.001387133226909417i];
b = [-1.774823934929885,  -0.7210490086119171 + 0.04290055077107185i, ...
     -5.020721861715561 + 1.189845102979979i,  -0.1450254390211963 + 0.7257437885879333i;
      0 + 0i,  -0.2176281526219798 + 0.03516041416104906i, ...
     -2.541102900685018 - 0.1458063680541886i,  -0.8228500482725508 - 0.4184333588843852i;
      0 + 0i,  0 + 0i,  -1.395782135347712 - 0.1632782984144689i, ...
      -1.747484189780436 - 0.2044203302132494i;
      0 + 0i,  0 + 0i,  0 + 0i,  -0.09963146114886638 - 0.004359389105629694i];
q = [complex(0)];
z = [complex(0)];
[aOut, bOut, alpha, beta, qOut, zOut, info] = ...
    f08xs(job, compq, compz, ilo, ihi, a, b, q, z)
 

aOut =

  -0.1438 + 0.3741i  -1.7000 + 0.1473i  -1.4170 - 1.0916i  -6.6330 - 0.8784i
   0.0000 + 0.0000i   0.2824 + 0.5209i  -1.5812 + 0.5647i   1.9829 - 3.4956i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.1522 - 0.2800i   1.1701 - 1.7439i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.8591 - 0.0636i


bOut =

   0.2263 + 0.0000i   0.2959 + 0.6808i  -0.9706 + 0.3572i  -3.4142 + 1.1586i
   0.0000 + 0.0000i   0.5723 + 0.0000i  -1.7815 + 0.5004i   1.7329 - 3.4482i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.3323 + 0.0000i   1.7834 - 2.2524i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   1.2738 + 0.0000i


alpha =

  -0.1438 + 0.3741i
   0.2824 + 0.5209i
   0.1522 - 0.2800i
   0.8591 - 0.0636i


beta =

   0.2263 + 0.0000i
   0.5723 + 0.0000i
   0.3323 + 0.0000i
   1.2738 + 0.0000i


qOut =

   0.0000 + 0.0000i


zOut =

   0.0000 + 0.0000i


info =

                    0



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