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---
:name: chgeqz
:md5sum: 8994134138840bb33bb79c034bde7992
:category: :subroutine
:arguments:
- job:
:type: char
:intent: input
- compq:
:type: char
:intent: input
- compz:
:type: char
:intent: input
- n:
:type: integer
:intent: input
- ilo:
:type: integer
:intent: input
- ihi:
:type: integer
:intent: input
- h:
:type: complex
:intent: input/output
:dims:
- ldh
- n
- ldh:
:type: integer
:intent: input
- t:
:type: complex
:intent: input/output
:dims:
- ldt
- n
- ldt:
:type: integer
:intent: input
- alpha:
:type: complex
:intent: output
:dims:
- n
- beta:
:type: complex
:intent: output
:dims:
- n
- q:
:type: complex
:intent: input/output
:dims:
- ldq
- n
- ldq:
:type: integer
:intent: input
- z:
:type: complex
:intent: input/output
:dims:
- ldz
- n
- ldz:
:type: integer
:intent: input
- work:
:type: complex
:intent: output
:dims:
- MAX(1,lwork)
- lwork:
:type: integer
:intent: input
:option: true
:default: n
- rwork:
:type: real
:intent: workspace
:dims:
- n
- info:
:type: integer
:intent: output
:substitutions: {}
:fortran_help: " SUBROUTINE CHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* CHGEQZ computes the eigenvalues of a complex matrix pair (H,T),\n\
* where H is an upper Hessenberg matrix and T is upper triangular,\n\
* using the single-shift QZ method.\n\
* Matrix pairs of this type are produced by the reduction to\n\
* generalized upper Hessenberg form of a complex matrix pair (A,B):\n\
* \n\
* A = Q1*H*Z1**H, B = Q1*T*Z1**H,\n\
* \n\
* as computed by CGGHRD.\n\
* \n\
* If JOB='S', then the Hessenberg-triangular pair (H,T) is\n\
* also reduced to generalized Schur form,\n\
* \n\
* H = Q*S*Z**H, T = Q*P*Z**H,\n\
* \n\
* where Q and Z are unitary matrices and S and P are upper triangular.\n\
* \n\
* Optionally, the unitary matrix Q from the generalized Schur\n\
* factorization may be postmultiplied into an input matrix Q1, and the\n\
* unitary matrix Z may be postmultiplied into an input matrix Z1.\n\
* If Q1 and Z1 are the unitary matrices from CGGHRD that reduced\n\
* the matrix pair (A,B) to generalized Hessenberg form, then the output\n\
* matrices Q1*Q and Z1*Z are the unitary factors from the generalized\n\
* Schur factorization of (A,B):\n\
* \n\
* A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H.\n\
* \n\
* To avoid overflow, eigenvalues of the matrix pair (H,T)\n\
* (equivalently, of (A,B)) are computed as a pair of complex values\n\
* (alpha,beta). If beta is nonzero, lambda = alpha / beta is an\n\
* eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)\n\
* A*x = lambda*B*x\n\
* and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the\n\
* alternate form of the GNEP\n\
* mu*A*y = B*y.\n\
* The values of alpha and beta for the i-th eigenvalue can be read\n\
* directly from the generalized Schur form: alpha = S(i,i),\n\
* beta = P(i,i).\n\
*\n\
* Ref: C.B. Moler & G.W. Stewart, \"An Algorithm for Generalized Matrix\n\
* Eigenvalue Problems\", SIAM J. Numer. Anal., 10(1973),\n\
* pp. 241--256.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* JOB (input) CHARACTER*1\n\
* = 'E': Compute eigenvalues only;\n\
* = 'S': Computer eigenvalues and the Schur form.\n\
*\n\
* COMPQ (input) CHARACTER*1\n\
* = 'N': Left Schur vectors (Q) are not computed;\n\
* = 'I': Q is initialized to the unit matrix and the matrix Q\n\
* of left Schur vectors of (H,T) is returned;\n\
* = 'V': Q must contain a unitary matrix Q1 on entry and\n\
* the product Q1*Q is returned.\n\
*\n\
* COMPZ (input) CHARACTER*1\n\
* = 'N': Right Schur vectors (Z) are not computed;\n\
* = 'I': Q is initialized to the unit matrix and the matrix Z\n\
* of right Schur vectors of (H,T) is returned;\n\
* = 'V': Z must contain a unitary matrix Z1 on entry and\n\
* the product Z1*Z is returned.\n\
*\n\
* N (input) INTEGER\n\
* The order of the matrices H, T, Q, and Z. N >= 0.\n\
*\n\
* ILO (input) INTEGER\n\
* IHI (input) INTEGER\n\
* ILO and IHI mark the rows and columns of H which are in\n\
* Hessenberg form. It is assumed that A is already upper\n\
* triangular in rows and columns 1:ILO-1 and IHI+1:N.\n\
* If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.\n\
*\n\
* H (input/output) COMPLEX array, dimension (LDH, N)\n\
* On entry, the N-by-N upper Hessenberg matrix H.\n\
* On exit, if JOB = 'S', H contains the upper triangular\n\
* matrix S from the generalized Schur factorization.\n\
* If JOB = 'E', the diagonal of H matches that of S, but\n\
* the rest of H is unspecified.\n\
*\n\
* LDH (input) INTEGER\n\
* The leading dimension of the array H. LDH >= max( 1, N ).\n\
*\n\
* T (input/output) COMPLEX array, dimension (LDT, N)\n\
* On entry, the N-by-N upper triangular matrix T.\n\
* On exit, if JOB = 'S', T contains the upper triangular\n\
* matrix P from the generalized Schur factorization.\n\
* If JOB = 'E', the diagonal of T matches that of P, but\n\
* the rest of T is unspecified.\n\
*\n\
* LDT (input) INTEGER\n\
* The leading dimension of the array T. LDT >= max( 1, N ).\n\
*\n\
* ALPHA (output) COMPLEX array, dimension (N)\n\
* The complex scalars alpha that define the eigenvalues of\n\
* GNEP. ALPHA(i) = S(i,i) in the generalized Schur\n\
* factorization.\n\
*\n\
* BETA (output) COMPLEX array, dimension (N)\n\
* The real non-negative scalars beta that define the\n\
* eigenvalues of GNEP. BETA(i) = P(i,i) in the generalized\n\
* Schur factorization.\n\
*\n\
* Together, the quantities alpha = ALPHA(j) and beta = BETA(j)\n\
* represent the j-th eigenvalue of the matrix pair (A,B), in\n\
* one of the forms lambda = alpha/beta or mu = beta/alpha.\n\
* Since either lambda or mu may overflow, they should not,\n\
* in general, be computed.\n\
*\n\
* Q (input/output) COMPLEX array, dimension (LDQ, N)\n\
* On entry, if COMPZ = 'V', the unitary matrix Q1 used in the\n\
* reduction of (A,B) to generalized Hessenberg form.\n\
* On exit, if COMPZ = 'I', the unitary matrix of left Schur\n\
* vectors of (H,T), and if COMPZ = 'V', the unitary matrix of\n\
* left Schur vectors of (A,B).\n\
* Not referenced if COMPZ = 'N'.\n\
*\n\
* LDQ (input) INTEGER\n\
* The leading dimension of the array Q. LDQ >= 1.\n\
* If COMPQ='V' or 'I', then LDQ >= N.\n\
*\n\
* Z (input/output) COMPLEX array, dimension (LDZ, N)\n\
* On entry, if COMPZ = 'V', the unitary matrix Z1 used in the\n\
* reduction of (A,B) to generalized Hessenberg form.\n\
* On exit, if COMPZ = 'I', the unitary matrix of right Schur\n\
* vectors of (H,T), and if COMPZ = 'V', the unitary matrix of\n\
* right Schur vectors of (A,B).\n\
* Not referenced if COMPZ = 'N'.\n\
*\n\
* LDZ (input) INTEGER\n\
* The leading dimension of the array Z. LDZ >= 1.\n\
* If COMPZ='V' or 'I', then LDZ >= N.\n\
*\n\
* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))\n\
* On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.\n\
*\n\
* LWORK (input) INTEGER\n\
* The dimension of the array WORK. LWORK >= max(1,N).\n\
*\n\
* If LWORK = -1, then a workspace query is assumed; the routine\n\
* only calculates the optimal size of the WORK array, returns\n\
* this value as the first entry of the WORK array, and no error\n\
* message related to LWORK is issued by XERBLA.\n\
*\n\
* RWORK (workspace) REAL array, dimension (N)\n\
*\n\
* INFO (output) INTEGER\n\
* = 0: successful exit\n\
* < 0: if INFO = -i, the i-th argument had an illegal value\n\
* = 1,...,N: the QZ iteration did not converge. (H,T) is not\n\
* in Schur form, but ALPHA(i) and BETA(i),\n\
* i=INFO+1,...,N should be correct.\n\
* = N+1,...,2*N: the shift calculation failed. (H,T) is not\n\
* in Schur form, but ALPHA(i) and BETA(i),\n\
* i=INFO-N+1,...,N should be correct.\n\
*\n\n\
* Further Details\n\
* ===============\n\
*\n\
* We assume that complex ABS works as long as its value is less than\n\
* overflow.\n\
*\n\
* =====================================================================\n\
*\n"
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