?hgeqz

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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Date 3/22/2024

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?hgeqz

Implements the QZ method for finding the generalized eigenvalues of the matrix pair (H,T).

Syntax

call shgeqz(job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, info)

call dhgeqz(job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, info)

call chgeqz(job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, info)

call zhgeqz(job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, info)

call hgeqz(h, t [,ilo] [,ihi] [,alphar] [,alphai] [,beta] [,q] [,z] [,job] [,compq] [,compz] [,info])

call hgeqz(h, t [,ilo] [,ihi] [,alpha] [,beta] [,q] [,z] [,job] [,compq] [, compz] [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine computes the eigenvalues of a real/complex matrix pair (H,T), where H is an upper Hessenberg matrix and T is upper triangular, using the double-shift version (for real flavors) or single-shift version (for complex flavors) of the QZ method. Matrix pairs of this type are produced by the reduction to generalized upper Hessenberg form of a real/complex matrix pair (A,B):

A = Q₁*H*Z₁^H, B = Q₁*T*Z₁^H,

as computed by ?gghrd.

For real flavors:

If job = 'S', then the Hessenberg-triangular pair (H,T) is reduced to generalized Schur form,

H = Q*S*Z^T, T = Q*P*Z^T,

where Q and Z are orthogonal matrices, P is an upper triangular matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal blocks. The 1-by-1 blocks correspond to real eigenvalues of the matrix pair (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of eigenvalues.

Additionally, the 2-by-2 upper triangular diagonal blocks of P corresponding to 2-by-2 blocks of S are reduced to positive diagonal form, that is, if S_{j + 1, j} is non-zero, then P_{j + 1, j} = P_{j, j + 1} = 0, P_{j, j} > 0, and P_{j + 1, j + 1} > 0.

For complex flavors:

If job = 'S', then the Hessenberg-triangular pair (H,T) is reduced to generalized Schur form,

H = Q* S*Z^H, T = Q*P*Z^H,

where Q and Z are unitary matrices, and S and P are upper triangular.

For all function flavors:

Optionally, the orthogonal/unitary matrix Q from the generalized Schur factorization may be post-multiplied by an input matrix Q₁, and the orthogonal/unitary matrix Z may be post-multiplied by an input matrix Z₁.

If Q₁ and Z₁ are the orthogonal/unitary matrices from ?gghrd that reduced the matrix pair (A,B) to generalized upper Hessenberg form, then the output matrices Q₁Q and Z₁Z are the orthogonal/unitary factors from the generalized Schur factorization of (A,B):

A = (Q₁Q)*S *(Z₁Z)^H, B = (Q₁Q)*P*(Z₁Z)^H.

To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently, of (A,B)) are computed as a pair of values (alpha,beta). For chgeqz/zhgeqz, alpha and beta are complex, and for shgeqz/dhgeqz, alpha is complex and beta real. If beta is nonzero, λ = alpha/beta is an eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)

A*x = λ*B*x

and if alpha is nonzero, μ = beta/alpha is an eigenvalue of the alternate form of the GNEP

μ*A*y = B*y .

Real eigenvalues (for real flavors) or the values of alpha and beta for the i-th eigenvalue (for complex flavors) can be read directly from the generalized Schur form:

alpha = S_{i, i}, beta = P_{i, i}.

Input Parameters

job

CHARACTER*1. Specifies the operations to be performed. Must be 'E' or 'S'.

If job = 'E', then compute eigenvalues only;

If job = 'S', then compute eigenvalues and the Schur form.

compq

CHARACTER*1. Must be 'N', 'I', or 'V'.

If compq = 'N', left Schur vectors (q) are not computed;

If compq = 'I', q is initialized to the unit matrix and the matrix of left Schur vectors of (H,T) is returned;

If compq = 'V', q must contain an orthogonal/unitary matrix Q₁ on entry and the product Q₁*Q is returned.

compz

CHARACTER*1. Must be 'N', 'I', or 'V'.

If compz = 'N', right Schur vectors (z) are not computed;

If compz = 'I', z is initialized to the unit matrix and the matrix of right Schur vectors of (H,T) is returned;

If compz = 'V', z must contain an orthogonal/unitary matrix Z₁ on entry and the product Z₁*Z is returned.

n

INTEGER. The order of the matrices H, T, Q, and Z

(n≥ 0).

ilo, ihi

INTEGER. ilo and ihi mark the rows and columns of H which are in Hessenberg form. It is assumed that H is already upper triangular in rows and columns 1:ilo-1 and ihi+1:n.

Constraint:

If n > 0, then 1 ≤ilo≤ihi≤n;

if n = 0, then ilo = 1 and ihi = 0.

h, t, q, z, work

REAL for shgeqz

DOUBLE PRECISION for dhgeqz

COMPLEX for chgeqz

DOUBLE COMPLEX for zhgeqz.

Arrays:

On entry, h(ldh,*) contains the n-by-n upper Hessenberg matrix H.

The second dimension of h must be at least max(1, n).

On entry, t(ldt,*) contains the n-by-n upper triangular matrix T.

The second dimension of t must be at least max(1, n).

q(ldq,*) :

On entry, if compq = 'V', this array contains the orthogonal/unitary matrix Q₁ used in the reduction of (A,B) to generalized Hessenberg form.

If compq = 'N', then q is not referenced.

The second dimension of q must be at least max(1, n).

z(ldz,*) :

On entry, if compz = 'V', this array contains the orthogonal/unitary matrix Z₁ used in the reduction of (A,B) to generalized Hessenberg form.

If compz = 'N', then z is not referenced.

The second dimension of z must be at least max(1, n).

work is a workspace array, its dimension max(1, lwork).

ldh

INTEGER. The leading dimension of h; at least max(1, n).

ldt

INTEGER. The leading dimension of t; at least max(1, n).

ldq

INTEGER. The leading dimension of q;

If compq = 'N', then ldq≥ 1.

If compq = 'I'or 'V', then ldq≥ max(1, n).

ldz

INTEGER. The leading dimension of z;

If compq = 'N', then ldz≥ 1.

If compq = 'I'or 'V', then ldz≥ max(1, n).

lwork

INTEGER. The dimension of the array work.

lwork≥ max(1, n).

If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla. See Application Notes for details.

rwork

REAL for chgeqz

DOUBLE PRECISION for zhgeqz.

Workspace array, size at least max(1, n). Used in complex flavors only.

Output Parameters

h

For real flavors:

If job = 'S', then on exit h contains the upper quasi-triangular matrix S from the generalized Schur factorization.

If job = 'E', then on exit the diagonal blocks of h match those of S, but the rest of h is unspecified.

For complex flavors:

If job = 'S', then, on exit, h contains the upper triangular matrix S from the generalized Schur factorization.

If job = 'E', then on exit the diagonal of h matches that of S, but the rest of h is unspecified.

t

If job = 'S', then, on exit, t contains the upper triangular matrix P from the generalized Schur factorization.

For real flavors:

2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S are reduced to positive diagonal form, that is, if h(j+1,j) is non-zero, then t(j+1,j)=t(j,j+1)=0 and t(j,j) and t(j+1,j+1) will be positive.

If job = 'E', then on exit the diagonal blocks of t match those of P, but the rest of t is unspecified.

For complex flavors:

if job = 'E', then on exit the diagonal of t matches that of P, but the rest of t is unspecified.

alphar, alphai

REAL for shgeqz;

DOUBLE PRECISION for dhgeqz.

Arrays, size at least max(1, n). The real and imaginary parts, respectively, of each scalar alpha defining an eigenvalue of GNEP.

If alphai(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-th eigenvalues are a complex conjugate pair, with

alphai(j+1) = -alphai(j).

alpha

COMPLEX for chgeqz;

DOUBLE COMPLEX for zhgeqz.

Array, size at least max(1, n).

The complex scalars alpha that define the eigenvalues of GNEP. alphai(i) = S_{i, i} in the generalized Schur factorization.

beta

REAL for shgeqz

DOUBLE PRECISION for dhgeqz

COMPLEX for chgeqz

DOUBLE COMPLEX for zhgeqz.

Array, size at least max(1, n).

For real flavors:

The scalars beta that define the eigenvalues of GNEP.

Together, the quantities alpha = (alphar(j), alphai(j)) and beta = beta(j) represent the j-th eigenvalue of the matrix pair (A,B), in one of the forms lambda = alpha/beta or mu = beta/alpha. Since either lambda or mu may overflow, they should not, in general, be computed.

For complex flavors:

The real non-negative scalars beta that define the eigenvalues of GNEP.

beta(i) = P_{i, i} in the generalized Schur factorization. Together, the quantities alpha = alpha(j) and beta = beta(j) represent the j-th eigenvalue of the matrix pair (A,B), in one of the forms lambda = alpha/beta or mu = beta/alpha. Since either lambda or mu may overflow, they should not, in general, be computed.

q

On exit, if compq = 'I', q is overwritten by the orthogonal/unitary matrix of left Schur vectors of the pair (H,T), and if compq = 'V', q is overwritten by the orthogonal/unitary matrix of left Schur vectors of (A,B).

z

On exit, if compz = 'I', z is overwritten by the orthogonal/unitary matrix of right Schur vectors of the pair (H,T), and if compz = 'V', z is overwritten by the orthogonal/unitary matrix of right Schur vectors of (A,B).

work(1)

If info≥ 0, on exit, work(1) contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.

info

INTEGER.

If info = 0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

If info = 1,..., n, the QZ iteration did not converge.

(H,T) is not in Schur form, but alphar(i), alphai(i) (for real flavors), alpha(i) (for complex flavors), and beta(i), i=info+1,..., n should be correct.

If info = n+1,...,2n, the shift calculation failed.

(H,T) is not in Schur form, but alphar(i), alphai(i) (for real flavors), alpha(i) (for complex flavors), and beta(i), i =info-n+1,..., n should be correct.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine hgeqz interface are the following:

h

Holds the matrix H of size (n,n).

t

Holds the matrix T of size (n,n).

alphar

Holds the vector of length n. Used in real flavors only.

alphai

Holds the vector of length n. Used in real flavors only.

alpha

Holds the vector of length n. Used in complex flavors only.

beta

Holds the vector of length n.

q

Holds the matrix Q of size (n,n).

z

Holds the matrix Z of size (n,n).

ilo

Default value for this argument is ilo = 1.

ihi

Default value for this argument is ihi = n.

job

Must be 'E' or 'S'. The default value is 'E'.

compq

If omitted, this argument is restored based on the presence of argument q as follows:

compq = 'I', if q is present,

compq = 'N', if q is omitted.

If present, compq must be equal to 'I' or 'V' and the argument q must also be present.

Note that there will be an error condition if compq is present and q omitted.

compz

If omitted, this argument is restored based on the presence of argument z as follows:

compz = 'I', if z is present,

compz = 'N', if z is omitted.

If present, compz must be equal to 'I' or 'V' and the argument z must also be present.

Note an error condition if compz is present and z is omitted.

Application Notes

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set lwork = -1.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (work(1)) for subsequent runs.

If you set lwork = -1, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

Parent topic: Generalized Nonsymmetric Eigenvalue Problems: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?hgeqz

Syntax

Include Files

Description

Input Parameters

Output Parameters

LAPACK 95 Interface Notes

Application Notes