table of contents
| hgeqz(3) | LAPACK | hgeqz(3) |
NAME¶
hgeqz - hgeqz: generalized Hessenberg eig
SYNOPSIS¶
Functions¶
subroutine chgeqz (job, compq, compz, n, ilo, ihi, h, ldh,
t, ldt, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, info)
CHGEQZ subroutine dhgeqz (job, compq, compz, n, ilo, ihi, h,
ldh, t, ldt, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, info)
DHGEQZ subroutine shgeqz (job, compq, compz, n, ilo, ihi, h,
ldh, t, ldt, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, info)
SHGEQZ subroutine zhgeqz (job, compq, compz, n, ilo, ihi, h,
ldh, t, ldt, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, info)
ZHGEQZ
Detailed Description¶
Function Documentation¶
subroutine chgeqz (character job, character compq, character compz, integer n, integer ilo, integer ihi, complex, dimension( ldh, * ) h, integer ldh, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer info)¶
CHGEQZ
Purpose:
CHGEQZ 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 & G.W. Stewart, 'An Algorithm for Generalized Matrix
Eigenvalue Problems', SIAM J. Numer. Anal., 10(1973),
pp. 241--256.
Parameters
JOB
JOB is CHARACTER*1
= 'E': Compute eigenvalues only;
= 'S': Computer eigenvalues and the Schur form.
COMPQ
COMPQ is CHARACTER*1
= 'N': Left Schur vectors (Q) are not computed;
= 'I': Q is initialized to the unit matrix and the matrix Q
of left Schur vectors of (H,T) is returned;
= 'V': Q must contain a unitary matrix Q1 on entry and
the product Q1*Q is returned.
COMPZ
COMPZ is CHARACTER*1
= 'N': Right Schur vectors (Z) are not computed;
= 'I': Q is initialized to the unit matrix and the matrix Z
of right Schur vectors of (H,T) is returned;
= 'V': Z must contain a unitary matrix Z1 on entry and
the product Z1*Z is returned.
N
N is INTEGER
The order of the matrices H, T, Q, and Z. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
ILO and IHI mark the rows and columns of H which are in
Hessenberg form. It is assumed that A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N.
If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
H
H is COMPLEX array, dimension (LDH, N)
On entry, the N-by-N upper Hessenberg matrix H.
On exit, if JOB = 'S', H contains the upper triangular
matrix S from the generalized Schur factorization.
If JOB = 'E', the diagonal of H matches that of S, but
the rest of H is unspecified.
LDH
LDH is INTEGER
The leading dimension of the array H. LDH >= max( 1, N ).
T
T is COMPLEX array, dimension (LDT, N)
On entry, the N-by-N upper triangular matrix T.
On exit, if JOB = 'S', T contains the upper triangular
matrix P from the generalized Schur factorization.
If JOB = 'E', the diagonal of T matches that of P, but
the rest of T is unspecified.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max( 1, N ).
ALPHA
ALPHA is COMPLEX array, dimension (N)
The complex scalars alpha that define the eigenvalues of
GNEP. ALPHA(i) = S(i,i) in the generalized Schur
factorization.
BETA
BETA is COMPLEX array, dimension (N)
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
Q is COMPLEX array, dimension (LDQ, N)
On entry, if COMPQ = 'V', the unitary matrix Q1 used in the
reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPQ = 'I', the unitary matrix of left Schur
vectors of (H,T), and if COMPQ = 'V', the unitary matrix of
left Schur vectors of (A,B).
Not referenced if COMPQ = 'N'.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1.
If COMPQ='V' or 'I', then LDQ >= N.
Z
Z is COMPLEX array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the unitary matrix Z1 used in the
reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPZ = 'I', the unitary matrix of right Schur
vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
right Schur vectors of (A,B).
Not referenced if COMPZ = 'N'.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If COMPZ='V' or 'I', then LDZ >= N.
WORK
WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is 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.
RWORK
RWORK is REAL array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1,...,N: the QZ iteration did not converge. (H,T) is not
in Schur form, but ALPHA(i) and BETA(i),
i=INFO+1,...,N should be correct.
= N+1,...,2*N: the shift calculation failed. (H,T) is not
in Schur form, but ALPHA(i) and BETA(i),
i=INFO-N+1,...,N should be correct.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
We assume that complex ABS works as long as its value is less than
overflow.
subroutine dhgeqz (character job, character compq, character compz, integer n, integer ilo, integer ihi, double precision, dimension( ldh, * ) h, integer ldh, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer lwork, integer info)¶
DHGEQZ
Purpose:
DHGEQZ computes the eigenvalues of a real matrix pair (H,T),
where H is an upper Hessenberg matrix and T is upper triangular,
using the double-shift QZ method.
Matrix pairs of this type are produced by the reduction to
generalized upper Hessenberg form of a real matrix pair (A,B):
A = Q1*H*Z1**T, B = Q1*T*Z1**T,
as computed by DGGHRD.
If JOB='S', then the Hessenberg-triangular pair (H,T) is
also 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, i.e., 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.
Optionally, the orthogonal matrix Q from the generalized Schur
factorization may be postmultiplied into an input matrix Q1, and the
orthogonal matrix Z may be postmultiplied into an input matrix Z1.
If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced
the matrix pair (A,B) to generalized upper Hessenberg form, then the
output matrices Q1*Q and Z1*Z are the orthogonal factors from the
generalized Schur factorization of (A,B):
A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T.
To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
complex and beta real.
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.
Real eigenvalues can be read directly from the generalized Schur
form:
alpha = S(i,i), beta = P(i,i).
Ref: C.B. Moler & G.W. Stewart, 'An Algorithm for Generalized Matrix
Eigenvalue Problems', SIAM J. Numer. Anal., 10(1973),
pp. 241--256.
Parameters
JOB
JOB is CHARACTER*1
= 'E': Compute eigenvalues only;
= 'S': Compute eigenvalues and the Schur form.
COMPQ
COMPQ is CHARACTER*1
= 'N': Left Schur vectors (Q) are not computed;
= 'I': Q is initialized to the unit matrix and the matrix Q
of left Schur vectors of (H,T) is returned;
= 'V': Q must contain an orthogonal matrix Q1 on entry and
the product Q1*Q is returned.
COMPZ
COMPZ is CHARACTER*1
= 'N': Right Schur vectors (Z) are not computed;
= 'I': Z is initialized to the unit matrix and the matrix Z
of right Schur vectors of (H,T) is returned;
= 'V': Z must contain an orthogonal matrix Z1 on entry and
the product Z1*Z is returned.
N
N is INTEGER
The order of the matrices H, T, Q, and Z. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
ILO and IHI mark the rows and columns of H which are in
Hessenberg form. It is assumed that A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N.
If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
H
H is DOUBLE PRECISION array, dimension (LDH, N)
On entry, the N-by-N upper Hessenberg matrix H.
On exit, if JOB = 'S', H contains the upper quasi-triangular
matrix S from the generalized Schur factorization.
If JOB = 'E', the diagonal blocks of H match those of S, but
the rest of H is unspecified.
LDH
LDH is INTEGER
The leading dimension of the array H. LDH >= max( 1, N ).
T
T is DOUBLE PRECISION array, dimension (LDT, N)
On entry, the N-by-N upper triangular matrix T.
On exit, if JOB = 'S', T contains the upper triangular
matrix P from the generalized Schur factorization;
2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
are reduced to positive diagonal form, i.e., if H(j+1,j) is
non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
T(j+1,j+1) > 0.
If JOB = 'E', the diagonal blocks of T match those of P, but
the rest of T is unspecified.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max( 1, N ).
ALPHAR
ALPHAR is DOUBLE PRECISION array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of GNEP.
ALPHAI
ALPHAI is DOUBLE PRECISION array, dimension (N)
The imaginary parts 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)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
BETA
BETA is DOUBLE PRECISION array, dimension (N)
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.
Q
Q is DOUBLE PRECISION array, dimension (LDQ, N)
On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
vectors of (H,T), and if COMPQ = 'V', the orthogonal matrix
of left Schur vectors of (A,B).
Not referenced if COMPQ = 'N'.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1.
If COMPQ='V' or 'I', then LDQ >= N.
Z
Z is DOUBLE PRECISION array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPZ = 'I', the orthogonal matrix of
right Schur vectors of (H,T), and if COMPZ = 'V', the
orthogonal matrix of right Schur vectors of (A,B).
Not referenced if COMPZ = 'N'.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If COMPZ='V' or 'I', then LDZ >= N.
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is 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.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1,...,N: the QZ iteration did not converge. (H,T) is not
in Schur form, but ALPHAR(i), ALPHAI(i), and
BETA(i), i=INFO+1,...,N should be correct.
= N+1,...,2*N: the shift calculation failed. (H,T) is not
in Schur form, but ALPHAR(i), ALPHAI(i), and
BETA(i), i=INFO-N+1,...,N should be correct.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
Iteration counters:
JITER -- counts iterations.
IITER -- counts iterations run since ILAST was last
changed. This is therefore reset only when a 1-by-1 or
2-by-2 block deflates off the bottom.
subroutine shgeqz (character job, character compq, character compz, integer n, integer ilo, integer ihi, real, dimension( ldh, * ) h, integer ldh, real, dimension( ldt, * ) t, integer ldt, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( ldq, * ) q, integer ldq, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer info)¶
SHGEQZ
Purpose:
SHGEQZ computes the eigenvalues of a real matrix pair (H,T),
where H is an upper Hessenberg matrix and T is upper triangular,
using the double-shift QZ method.
Matrix pairs of this type are produced by the reduction to
generalized upper Hessenberg form of a real matrix pair (A,B):
A = Q1*H*Z1**T, B = Q1*T*Z1**T,
as computed by SGGHRD.
If JOB='S', then the Hessenberg-triangular pair (H,T) is
also 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, i.e., 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.
Optionally, the orthogonal matrix Q from the generalized Schur
factorization may be postmultiplied into an input matrix Q1, and the
orthogonal matrix Z may be postmultiplied into an input matrix Z1.
If Q1 and Z1 are the orthogonal matrices from SGGHRD that reduced
the matrix pair (A,B) to generalized upper Hessenberg form, then the
output matrices Q1*Q and Z1*Z are the orthogonal factors from the
generalized Schur factorization of (A,B):
A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T.
To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
complex and beta real.
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.
Real eigenvalues can be read directly from the generalized Schur
form:
alpha = S(i,i), beta = P(i,i).
Ref: C.B. Moler & G.W. Stewart, 'An Algorithm for Generalized Matrix
Eigenvalue Problems', SIAM J. Numer. Anal., 10(1973),
pp. 241--256.
Parameters
JOB
JOB is CHARACTER*1
= 'E': Compute eigenvalues only;
= 'S': Compute eigenvalues and the Schur form.
COMPQ
COMPQ is CHARACTER*1
= 'N': Left Schur vectors (Q) are not computed;
= 'I': Q is initialized to the unit matrix and the matrix Q
of left Schur vectors of (H,T) is returned;
= 'V': Q must contain an orthogonal matrix Q1 on entry and
the product Q1*Q is returned.
COMPZ
COMPZ is CHARACTER*1
= 'N': Right Schur vectors (Z) are not computed;
= 'I': Z is initialized to the unit matrix and the matrix Z
of right Schur vectors of (H,T) is returned;
= 'V': Z must contain an orthogonal matrix Z1 on entry and
the product Z1*Z is returned.
N
N is INTEGER
The order of the matrices H, T, Q, and Z. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
ILO and IHI mark the rows and columns of H which are in
Hessenberg form. It is assumed that A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N.
If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
H
H is REAL array, dimension (LDH, N)
On entry, the N-by-N upper Hessenberg matrix H.
On exit, if JOB = 'S', H contains the upper quasi-triangular
matrix S from the generalized Schur factorization.
If JOB = 'E', the diagonal blocks of H match those of S, but
the rest of H is unspecified.
LDH
LDH is INTEGER
The leading dimension of the array H. LDH >= max( 1, N ).
T
T is REAL array, dimension (LDT, N)
On entry, the N-by-N upper triangular matrix T.
On exit, if JOB = 'S', T contains the upper triangular
matrix P from the generalized Schur factorization;
2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
are reduced to positive diagonal form, i.e., if H(j+1,j) is
non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
T(j+1,j+1) > 0.
If JOB = 'E', the diagonal blocks of T match those of P, but
the rest of T is unspecified.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max( 1, N ).
ALPHAR
ALPHAR is REAL array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of GNEP.
ALPHAI
ALPHAI is REAL array, dimension (N)
The imaginary parts 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)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
BETA
BETA is REAL array, dimension (N)
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.
Q
Q is REAL array, dimension (LDQ, N)
On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
vectors of (H,T), and if COMPQ = 'V', the orthogonal matrix
of left Schur vectors of (A,B).
Not referenced if COMPQ = 'N'.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1.
If COMPQ='V' or 'I', then LDQ >= N.
Z
Z is REAL array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
the reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPZ = 'I', the orthogonal matrix of
right Schur vectors of (H,T), and if COMPZ = 'V', the
orthogonal matrix of right Schur vectors of (A,B).
Not referenced if COMPZ = 'N'.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If COMPZ='V' or 'I', then LDZ >= N.
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is 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.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1,...,N: the QZ iteration did not converge. (H,T) is not
in Schur form, but ALPHAR(i), ALPHAI(i), and
BETA(i), i=INFO+1,...,N should be correct.
= N+1,...,2*N: the shift calculation failed. (H,T) is not
in Schur form, but ALPHAR(i), ALPHAI(i), and
BETA(i), i=INFO-N+1,...,N should be correct.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
Iteration counters:
JITER -- counts iterations.
IITER -- counts iterations run since ILAST was last
changed. This is therefore reset only when a 1-by-1 or
2-by-2 block deflates off the bottom.
subroutine zhgeqz (character job, character compq, character compz, integer n, integer ilo, integer ihi, complex*16, dimension( ldh, * ) h, integer ldh, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer info)¶
ZHGEQZ
Purpose:
ZHGEQZ 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 ZGGHRD.
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 ZGGHRD 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 & G.W. Stewart, 'An Algorithm for Generalized Matrix
Eigenvalue Problems', SIAM J. Numer. Anal., 10(1973),
pp. 241--256.
Parameters
JOB
JOB is CHARACTER*1
= 'E': Compute eigenvalues only;
= 'S': Computer eigenvalues and the Schur form.
COMPQ
COMPQ is CHARACTER*1
= 'N': Left Schur vectors (Q) are not computed;
= 'I': Q is initialized to the unit matrix and the matrix Q
of left Schur vectors of (H,T) is returned;
= 'V': Q must contain a unitary matrix Q1 on entry and
the product Q1*Q is returned.
COMPZ
COMPZ is CHARACTER*1
= 'N': Right Schur vectors (Z) are not computed;
= 'I': Q is initialized to the unit matrix and the matrix Z
of right Schur vectors of (H,T) is returned;
= 'V': Z must contain a unitary matrix Z1 on entry and
the product Z1*Z is returned.
N
N is INTEGER
The order of the matrices H, T, Q, and Z. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
ILO and IHI mark the rows and columns of H which are in
Hessenberg form. It is assumed that A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N.
If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
H
H is COMPLEX*16 array, dimension (LDH, N)
On entry, the N-by-N upper Hessenberg matrix H.
On exit, if JOB = 'S', H contains the upper triangular
matrix S from the generalized Schur factorization.
If JOB = 'E', the diagonal of H matches that of S, but
the rest of H is unspecified.
LDH
LDH is INTEGER
The leading dimension of the array H. LDH >= max( 1, N ).
T
T is COMPLEX*16 array, dimension (LDT, N)
On entry, the N-by-N upper triangular matrix T.
On exit, if JOB = 'S', T contains the upper triangular
matrix P from the generalized Schur factorization.
If JOB = 'E', the diagonal of T matches that of P, but
the rest of T is unspecified.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max( 1, N ).
ALPHA
ALPHA is COMPLEX*16 array, dimension (N)
The complex scalars alpha that define the eigenvalues of
GNEP. ALPHA(i) = S(i,i) in the generalized Schur
factorization.
BETA
BETA is COMPLEX*16 array, dimension (N)
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
Q is COMPLEX*16 array, dimension (LDQ, N)
On entry, if COMPQ = 'V', the unitary matrix Q1 used in the
reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPQ = 'I', the unitary matrix of left Schur
vectors of (H,T), and if COMPQ = 'V', the unitary matrix of
left Schur vectors of (A,B).
Not referenced if COMPQ = 'N'.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1.
If COMPQ='V' or 'I', then LDQ >= N.
Z
Z is COMPLEX*16 array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the unitary matrix Z1 used in the
reduction of (A,B) to generalized Hessenberg form.
On exit, if COMPZ = 'I', the unitary matrix of right Schur
vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
right Schur vectors of (A,B).
Not referenced if COMPZ = 'N'.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1.
If COMPZ='V' or 'I', then LDZ >= N.
WORK
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is 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.
RWORK
RWORK is DOUBLE PRECISION array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1,...,N: the QZ iteration did not converge. (H,T) is not
in Schur form, but ALPHA(i) and BETA(i),
i=INFO+1,...,N should be correct.
= N+1,...,2*N: the shift calculation failed. (H,T) is not
in Schur form, but ALPHA(i) and BETA(i),
i=INFO-N+1,...,N should be correct.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
We assume that complex ABS works as long as its value is less than
overflow.
Author¶
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