--- name: "hgeqz" section: "3" platform: "linux" package: "lapack-doc" version: "3.12.1-2" release: "Arch Linux 2026-11-01" platforms: ["linux"] url: "https://new.carta.tech/hgeqz" --- # hgeqz(3) hgeqz: generalized Hessenberg eig ## 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, , 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. !> ``` Definition at line **281** of file **chgeqz.f**. ### 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, , 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. !> ``` Definition at line **301** of file **dhgeqz.f**. ### 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, , 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. !> ``` Definition at line **301** of file **shgeqz.f**. ### 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, , 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. !> ``` Definition at line **281** of file **zhgeqz.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.