HGEQZ: computes the eigenvalues of a complex matrix pair (H,T), where H is an upper Hessenberg matrix and T is upper triangular, using the single-shift QZ method. Matrix pairs of this type are produced by the reduction to generalized upper Hessenberg form of a complex matrix pair (A,B): A = Q1*H*Z1**H, B = Q1*T*Z1**H, as computed by CGGHRD. If JOB='S', then the Hessenberg-triangular pair (H,T) is also 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. Optionally, the unitary matrix Q from the generalized Schur factorization may be postmultiplied into an input matrix Q1, and the unitary matrix Z may be postmultiplied into an input matrix Z1. If Q1 and Z1 are the unitary matrices from CGGHRD that reduced the matrix pair (A,B) to generalized Hessenberg form, then the output matrices Q1*Q and Z1*Z are the unitary factors from the generalized Schur factorization of (A,B): A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H. To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently, of (A,B)) are computed as a pair of complex values (alpha,beta). If beta is nonzero, lambda = alpha / beta is an eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP) A*x = lambda*B*x and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the alternate form of the GNEP mu*A*y = B*y. The values of alpha and beta for the i-th eigenvalue can be read directly from the generalized Schur form: alpha = S(i,i), beta = P(i,i). Ref: C.B. Moler Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973), pp. 241–256. More...

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

	la_chgeqz

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

	la_dhgeqz

	la_qhgeqz

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

	la_shgeqz

	la_whgeqz

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

	la_zhgeqz

Detailed Description

HGEQZ: computes the eigenvalues of a complex matrix pair (H,T), where H is an upper Hessenberg matrix and T is upper triangular, using the single-shift QZ method. Matrix pairs of this type are produced by the reduction to generalized upper Hessenberg form of a complex matrix pair (A,B): A = Q1*H*Z1**H, B = Q1*T*Z1**H, as computed by CGGHRD. If JOB='S', then the Hessenberg-triangular pair (H,T) is also 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. Optionally, the unitary matrix Q from the generalized Schur factorization may be postmultiplied into an input matrix Q1, and the unitary matrix Z may be postmultiplied into an input matrix Z1. If Q1 and Z1 are the unitary matrices from CGGHRD that reduced the matrix pair (A,B) to generalized Hessenberg form, then the output matrices Q1*Q and Z1*Z are the unitary factors from the generalized Schur factorization of (A,B): A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H. To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently, of (A,B)) are computed as a pair of complex values (alpha,beta). If beta is nonzero, lambda = alpha / beta is an eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP) A*x = lambda*B*x and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the alternate form of the GNEP mu*A*y = B*y. The values of alpha and beta for the i-th eigenvalue can be read directly from the generalized Schur form: alpha = S(i,i), beta = P(i,i). Ref: C.B. Moler Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973), pp. 241–256.

Member Function/Subroutine Documentation

◆ chgeqz()

subroutine la_lapack::hgeqz::chgeqz	(	character, intent(in)	job,
		character, intent(in)	compq,
		character, intent(in)	compz,
		integer(ilp), intent(in)	n,
		integer(ilp), intent(in)	ilo,
		integer(ilp), intent(in)	ihi,
		complex(sp), dimension(ldh,*), intent(inout)	h,
		integer(ilp), intent(in)	ldh,
		complex(sp), dimension(ldt,*), intent(inout)	t,
		integer(ilp), intent(in)	ldt,
		complex(sp), dimension(*), intent(out)	alpha,
		complex(sp), dimension(*), intent(out)	beta,
		complex(sp), dimension(ldq,*), intent(inout)	q,
		integer(ilp), intent(in)	ldq,
		complex(sp), dimension(ldz,*), intent(inout)	z,
		integer(ilp), intent(in)	ldz,
		complex(sp), dimension(*), intent(out)	work,
		integer(ilp), intent(in)	lwork,
		real(sp), dimension(*), intent(out)	rwork,
		integer(ilp), intent(out)	info
	)

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◆ dhgeqz()

subroutine la_lapack::hgeqz::dhgeqz	(	character, intent(in)	job,
		character, intent(in)	compq,
		character, intent(in)	compz,
		integer(ilp), intent(in)	n,
		integer(ilp), intent(in)	ilo,
		integer(ilp), intent(in)	ihi,
		real(dp), dimension(ldh,*), intent(inout)	h,
		integer(ilp), intent(in)	ldh,
		real(dp), dimension(ldt,*), intent(inout)	t,
		integer(ilp), intent(in)	ldt,
		real(dp), dimension(*), intent(out)	alphar,
		real(dp), dimension(*), intent(out)	alphai,
		real(dp), dimension(*), intent(out)	beta,
		real(dp), dimension(ldq,*), intent(inout)	q,
		integer(ilp), intent(in)	ldq,
		real(dp), dimension(ldz,*), intent(inout)	z,
		integer(ilp), intent(in)	ldz,
		real(dp), dimension(*), intent(out)	work,
		integer(ilp), intent(in)	lwork,
		integer(ilp), intent(out)	info
	)

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◆ la_chgeqz()

la_lapack::hgeqz::la_chgeqz

◆ la_dhgeqz()

la_lapack::hgeqz::la_dhgeqz

◆ la_qhgeqz()

la_lapack::hgeqz::la_qhgeqz

◆ la_shgeqz()

la_lapack::hgeqz::la_shgeqz

◆ la_whgeqz()

la_lapack::hgeqz::la_whgeqz

◆ la_zhgeqz()

la_lapack::hgeqz::la_zhgeqz

◆ shgeqz()

subroutine la_lapack::hgeqz::shgeqz	(	character, intent(in)	job,
		character, intent(in)	compq,
		character, intent(in)	compz,
		integer(ilp), intent(in)	n,
		integer(ilp), intent(in)	ilo,
		integer(ilp), intent(in)	ihi,
		real(sp), dimension(ldh,*), intent(inout)	h,
		integer(ilp), intent(in)	ldh,
		real(sp), dimension(ldt,*), intent(inout)	t,
		integer(ilp), intent(in)	ldt,
		real(sp), dimension(*), intent(out)	alphar,
		real(sp), dimension(*), intent(out)	alphai,
		real(sp), dimension(*), intent(out)	beta,
		real(sp), dimension(ldq,*), intent(inout)	q,
		integer(ilp), intent(in)	ldq,
		real(sp), dimension(ldz,*), intent(inout)	z,
		integer(ilp), intent(in)	ldz,
		real(sp), dimension(*), intent(out)	work,
		integer(ilp), intent(in)	lwork,
		integer(ilp), intent(out)	info
	)

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◆ zhgeqz()

subroutine la_lapack::hgeqz::zhgeqz	(	character, intent(in)	job,
		character, intent(in)	compq,
		character, intent(in)	compz,
		integer(ilp), intent(in)	n,
		integer(ilp), intent(in)	ilo,
		integer(ilp), intent(in)	ihi,
		complex(dp), dimension(ldh,*), intent(inout)	h,
		integer(ilp), intent(in)	ldh,
		complex(dp), dimension(ldt,*), intent(inout)	t,
		integer(ilp), intent(in)	ldt,
		complex(dp), dimension(*), intent(out)	alpha,
		complex(dp), dimension(*), intent(out)	beta,
		complex(dp), dimension(ldq,*), intent(inout)	q,
		integer(ilp), intent(in)	ldq,
		complex(dp), dimension(ldz,*), intent(inout)	z,
		integer(ilp), intent(in)	ldz,
		complex(dp), dimension(*), intent(out)	work,
		integer(ilp), intent(in)	lwork,
		real(dp), dimension(*), intent(out)	rwork,
		integer(ilp), intent(out)	info
	)

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The documentation for this interface was generated from the following file:

la_lapack.f90

Public Member Functions

Detailed Description

Member Function/Subroutine Documentation

◆ chgeqz()

◆ dhgeqz()

◆ la_chgeqz()

◆ la_dhgeqz()

◆ la_qhgeqz()

◆ la_shgeqz()

◆ la_whgeqz()

◆ la_zhgeqz()

◆ shgeqz()

◆ zhgeqz()