SRC/zhgeqz.f(3) Library Functions Manual SRC/zhgeqz.f(3) NAME SRC/zhgeqz.f SYNOPSIS Functions/Subroutines subroutine zhgeqz (job, compq, compz, n, ilo, ihi, h, ldh, t, ldt, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, info) ZHGEQZ Function/Subroutine Documentation 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. 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