.TH "laqz0" 3 "Sat Dec 9 2023 21:42:18" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
laqz0 \- laqz0: step in ggev3, gges3
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "recursive subroutine \fBclaqz0\fP (wants, wantq, wantz, n, ilo, ihi, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, rec, info)"
.br
.RI "\fBCLAQZ0\fP "
.ti -1c
.RI "recursive subroutine \fBdlaqz0\fP (wants, wantq, wantz, n, ilo, ihi, a, lda, b, ldb, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, rec, info)"
.br
.RI "\fBDLAQZ0\fP "
.ti -1c
.RI "recursive subroutine \fBslaqz0\fP (wants, wantq, wantz, n, ilo, ihi, a, lda, b, ldb, alphar, alphai, beta, q, ldq, z, ldz, work, lwork, rec, info)"
.br
.RI "\fBSLAQZ0\fP "
.ti -1c
.RI "recursive subroutine \fBzlaqz0\fP (wants, wantq, wantz, n, ilo, ihi, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, rec, info)"
.br
.RI "\fBZLAQZ0\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "recursive subroutine claqz0 (character, intent(in) wants, character, intent(in) wantq, character, intent(in) wantz, integer, intent(in) n, integer, intent(in) ilo, integer, intent(in) ihi, complex, dimension( lda, * ), intent(inout) a, integer, intent(in) lda, complex, dimension( ldb, * ), intent(inout) b, integer, intent(in) ldb, complex, dimension( * ), intent(inout) alpha, complex, dimension( * ), intent(inout) beta, complex, dimension( ldq, * ), intent(inout) q, integer, intent(in) ldq, complex, dimension( ldz, * ), intent(inout) z, integer, intent(in) ldz, complex, dimension( * ), intent(inout) work, integer, intent(in) lwork, real, dimension( * ), intent(out) rwork, integer, intent(in) rec, integer, intent(out) info)"

.PP
\fBCLAQZ0\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 CLAQZ0 computes the eigenvalues of a 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 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, P and S are an upper triangular
 matrices\&.

 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 upper 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 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\&.
 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\&.

 Ref: B\&. Kagstrom, D\&. Kressner, 'Multishift Variants of the QZ
      Algorithm with Aggressive Early Deflation', SIAM J\&. Numer\&.
      Anal\&., 29(2006), pp\&. 199--227\&.

 Ref: T\&. Steel, D\&. Camps, K\&. Meerbergen, R\&. Vandebril 'A multishift,
      multipole rational QZ method with aggressive early deflation'
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIWANTS\fP 
.PP
.nf
          WANTS is CHARACTER*1
          = 'E': Compute eigenvalues only;
          = 'S': Compute eigenvalues and the Schur form\&.
.fi
.PP
.br
\fIWANTQ\fP 
.PP
.nf
          WANTQ 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 (A,B) is returned;
          = 'V': Q must contain an unitary matrix Q1 on entry and
                 the product Q1*Q is returned\&.
.fi
.PP
.br
\fIWANTZ\fP 
.PP
.nf
          WANTZ 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 (A,B) is returned;
          = 'V': Z must contain an unitary matrix Z1 on entry and
                 the product Z1*Z is returned\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A, B, Q, and Z\&.  N >= 0\&.
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
          ILO is INTEGER
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
          IHI is INTEGER
          ILO and IHI mark the rows and columns of A 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\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX array, dimension (LDA, N)
          On entry, the N-by-N upper Hessenberg matrix A\&.
          On exit, if JOB = 'S', A contains the upper triangular
          matrix S from the generalized Schur factorization\&.
          If JOB = 'E', the diagonal of A matches that of S, but
          the rest of A is unspecified\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max( 1, N )\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is COMPLEX array, dimension (LDB, N)
          On entry, the N-by-N upper triangular matrix B\&.
          On exit, if JOB = 'S', B contains the upper triangular
          matrix P from the generalized Schur factorization\&.
          If JOB = 'E', the diagonal of B matches that of P, but
          the rest of B is unspecified\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&.  LDB >= max( 1, N )\&.
.fi
.PP
.br
\fIALPHA\fP 
.PP
.nf
          ALPHA is COMPLEX array, dimension (N)
          Each scalar alpha defining an eigenvalue
          of GNEP\&.
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          BETA is COMPLEX array, dimension (N)
          The scalars beta that define the eigenvalues of GNEP\&.
          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\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          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 (A,B), and if COMPQ = 'V', the unitary matrix
          of left Schur vectors of (A,B)\&.
          Not referenced if COMPQ = 'N'\&.
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
          The leading dimension of the array Q\&.  LDQ >= 1\&.
          If COMPQ='V' or 'I', then LDQ >= N\&.
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
          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'\&.
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
          LDZ is INTEGER
          The leading dimension of the array Z\&.  LDZ >= 1\&.
          If COMPZ='V' or 'I', then LDZ >= N\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is REAL array, dimension (N)
.fi
.PP
.br
\fIREC\fP 
.PP
.nf
          REC is INTEGER
             REC indicates the current recursion level\&. Should be set
             to 0 on first call\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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\&.  (A,B) is not
                     in Schur form, but ALPHA(i) and
                     BETA(i), i=INFO+1,\&.\&.\&.,N should be correct\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Thijs Steel, KU Leuven 
.RE
.PP
\fBDate\fP
.RS 4
May 2020 
.RE
.PP

.SS "recursive subroutine dlaqz0 (character, intent(in) wants, character, intent(in) wantq, character, intent(in) wantz, integer, intent(in) n, integer, intent(in) ilo, integer, intent(in) ihi, double precision, dimension( lda, * ), intent(inout) a, integer, intent(in) lda, double precision, dimension( ldb, * ), intent(inout) b, integer, intent(in) ldb, double precision, dimension( * ), intent(inout) alphar, double precision, dimension( * ), intent(inout) alphai, double precision, dimension( * ), intent(inout) beta, double precision, dimension( ldq, * ), intent(inout) q, integer, intent(in) ldq, double precision, dimension( ldz, * ), intent(inout) z, integer, intent(in) ldz, double precision, dimension( * ), intent(inout) work, integer, intent(in) lwork, integer, intent(in) rec, integer, intent(out) info)"

.PP
\fBDLAQZ0\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DLAQZ0 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\&.

 Ref: B\&. Kagstrom, D\&. Kressner, 'Multishift Variants of the QZ
      Algorithm with Aggressive Early Deflation', SIAM J\&. Numer\&.
      Anal\&., 29(2006), pp\&. 199--227\&.

 Ref: T\&. Steel, D\&. Camps, K\&. Meerbergen, R\&. Vandebril 'A multishift,
      multipole rational QZ method with aggressive early deflation'
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIWANTS\fP 
.PP
.nf
          WANTS is CHARACTER*1
          = 'E': Compute eigenvalues only;
          = 'S': Compute eigenvalues and the Schur form\&.
.fi
.PP
.br
\fIWANTQ\fP 
.PP
.nf
          WANTQ 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 (A,B) is returned;
          = 'V': Q must contain an orthogonal matrix Q1 on entry and
                 the product Q1*Q is returned\&.
.fi
.PP
.br
\fIWANTZ\fP 
.PP
.nf
          WANTZ 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 (A,B) is returned;
          = 'V': Z must contain an orthogonal matrix Z1 on entry and
                 the product Z1*Z is returned\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A, B, Q, and Z\&.  N >= 0\&.
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
          ILO is INTEGER
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
          IHI is INTEGER
          ILO and IHI mark the rows and columns of A 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\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the N-by-N upper Hessenberg matrix A\&.
          On exit, if JOB = 'S', A contains the upper quasi-triangular
          matrix S from the generalized Schur factorization\&.
          If JOB = 'E', the diagonal blocks of A match those of S, but
          the rest of A is unspecified\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max( 1, N )\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is DOUBLE PRECISION array, dimension (LDB, N)
          On entry, the N-by-N upper triangular matrix B\&.
          On exit, if JOB = 'S', B 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 A(j+1,j) is
          non-zero, then B(j+1,j) = B(j,j+1) = 0, B(j,j) > 0, and
          B(j+1,j+1) > 0\&.
          If JOB = 'E', the diagonal blocks of B match those of P, but
          the rest of B is unspecified\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&.  LDB >= max( 1, N )\&.
.fi
.PP
.br
\fIALPHAR\fP 
.PP
.nf
          ALPHAR is DOUBLE PRECISION array, dimension (N)
          The real parts of each scalar alpha defining an eigenvalue
          of GNEP\&.
.fi
.PP
.br
\fIALPHAI\fP 
.PP
.nf
          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)\&.
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          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 (A,B), and if COMPQ = 'V', the orthogonal matrix
          of left Schur vectors of (A,B)\&.
          Not referenced if COMPQ = 'N'\&.
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
          The leading dimension of the array Q\&.  LDQ >= 1\&.
          If COMPQ='V' or 'I', then LDQ >= N\&.
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
          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'\&.
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
          LDZ is INTEGER
          The leading dimension of the array Z\&.  LDZ >= 1\&.
          If COMPZ='V' or 'I', then LDZ >= N\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIREC\fP 
.PP
.nf
          REC is INTEGER
             REC indicates the current recursion level\&. Should be set
             to 0 on first call\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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\&.  (A,B) is not
                     in Schur form, but ALPHAR(i), ALPHAI(i), and
                     BETA(i), i=INFO+1,\&.\&.\&.,N should be correct\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Thijs Steel, KU Leuven 
.RE
.PP
\fBDate\fP
.RS 4
May 2020 
.RE
.PP

.SS "recursive subroutine slaqz0 (character, intent(in) wants, character, intent(in) wantq, character, intent(in) wantz, integer, intent(in) n, integer, intent(in) ilo, integer, intent(in) ihi, real, dimension( lda, * ), intent(inout) a, integer, intent(in) lda, real, dimension( ldb, * ), intent(inout) b, integer, intent(in) ldb, real, dimension( * ), intent(inout) alphar, real, dimension( * ), intent(inout) alphai, real, dimension( * ), intent(inout) beta, real, dimension( ldq, * ), intent(inout) q, integer, intent(in) ldq, real, dimension( ldz, * ), intent(inout) z, integer, intent(in) ldz, real, dimension( * ), intent(inout) work, integer, intent(in) lwork, integer, intent(in) rec, integer, intent(out) info)"

.PP
\fBSLAQZ0\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 SLAQZ0 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\&.

 Ref: B\&. Kagstrom, D\&. Kressner, 'Multishift Variants of the QZ
      Algorithm with Aggressive Early Deflation', SIAM J\&. Numer\&.
      Anal\&., 29(2006), pp\&. 199--227\&.

 Ref: T\&. Steel, D\&. Camps, K\&. Meerbergen, R\&. Vandebril 'A multishift,
      multipole rational QZ method with aggressive early deflation'
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIWANTS\fP 
.PP
.nf
          WANTS is CHARACTER*1
          = 'E': Compute eigenvalues only;
          = 'S': Compute eigenvalues and the Schur form\&.
.fi
.PP
.br
\fIWANTQ\fP 
.PP
.nf
          WANTQ 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 (A,B) is returned;
          = 'V': Q must contain an orthogonal matrix Q1 on entry and
                 the product Q1*Q is returned\&.
.fi
.PP
.br
\fIWANTZ\fP 
.PP
.nf
          WANTZ 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 (A,B) is returned;
          = 'V': Z must contain an orthogonal matrix Z1 on entry and
                 the product Z1*Z is returned\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A, B, Q, and Z\&.  N >= 0\&.
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
          ILO is INTEGER
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
          IHI is INTEGER
          ILO and IHI mark the rows and columns of A 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\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is REAL array, dimension (LDA, N)
          On entry, the N-by-N upper Hessenberg matrix A\&.
          On exit, if JOB = 'S', A contains the upper quasi-triangular
          matrix S from the generalized Schur factorization\&.
          If JOB = 'E', the diagonal blocks of A match those of S, but
          the rest of A is unspecified\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max( 1, N )\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is REAL array, dimension (LDB, N)
          On entry, the N-by-N upper triangular matrix B\&.
          On exit, if JOB = 'S', B 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 A(j+1,j) is
          non-zero, then B(j+1,j) = B(j,j+1) = 0, B(j,j) > 0, and
          B(j+1,j+1) > 0\&.
          If JOB = 'E', the diagonal blocks of B match those of P, but
          the rest of B is unspecified\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&.  LDB >= max( 1, N )\&.
.fi
.PP
.br
\fIALPHAR\fP 
.PP
.nf
          ALPHAR is REAL array, dimension (N)
          The real parts of each scalar alpha defining an eigenvalue
          of GNEP\&.
.fi
.PP
.br
\fIALPHAI\fP 
.PP
.nf
          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)\&.
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          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 (A,B), and if COMPQ = 'V', the orthogonal matrix
          of left Schur vectors of (A,B)\&.
          Not referenced if COMPQ = 'N'\&.
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
          The leading dimension of the array Q\&.  LDQ >= 1\&.
          If COMPQ='V' or 'I', then LDQ >= N\&.
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
          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'\&.
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
          LDZ is INTEGER
          The leading dimension of the array Z\&.  LDZ >= 1\&.
          If COMPZ='V' or 'I', then LDZ >= N\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIREC\fP 
.PP
.nf
          REC is INTEGER
             REC indicates the current recursion level\&. Should be set
             to 0 on first call\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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\&.  (A,B) is not
                     in Schur form, but ALPHAR(i), ALPHAI(i), and
                     BETA(i), i=INFO+1,\&.\&.\&.,N should be correct\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Thijs Steel, KU Leuven 
.RE
.PP
\fBDate\fP
.RS 4
May 2020 
.RE
.PP

.SS "recursive subroutine zlaqz0 (character, intent(in) wants, character, intent(in) wantq, character, intent(in) wantz, integer, intent(in) n, integer, intent(in) ilo, integer, intent(in) ihi, complex*16, dimension( lda, * ), intent(inout) a, integer, intent(in) lda, complex*16, dimension( ldb, * ), intent(inout) b, integer, intent(in) ldb, complex*16, dimension( * ), intent(inout) alpha, complex*16, dimension( * ), intent(inout) beta, complex*16, dimension( ldq,         * ), intent(inout) q, integer, intent(in) ldq, complex*16, dimension( ldz, * ), intent(inout) z, integer, intent(in) ldz, complex*16, dimension( * ), intent(inout) work, integer, intent(in) lwork, double precision, dimension( * ), intent(out) rwork, integer, intent(in) rec, integer, intent(out) info)"

.PP
\fBZLAQZ0\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 ZLAQZ0 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**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, P and S are an upper triangular
 matrices\&.

 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 upper 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 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\&.
 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\&.

 Ref: B\&. Kagstrom, D\&. Kressner, 'Multishift Variants of the QZ
      Algorithm with Aggressive Early Deflation', SIAM J\&. Numer\&.
      Anal\&., 29(2006), pp\&. 199--227\&.

 Ref: T\&. Steel, D\&. Camps, K\&. Meerbergen, R\&. Vandebril 'A multishift,
      multipole rational QZ method with aggressive early deflation'
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIWANTS\fP 
.PP
.nf
          WANTS is CHARACTER*1
          = 'E': Compute eigenvalues only;
          = 'S': Compute eigenvalues and the Schur form\&.
.fi
.PP
.br
\fIWANTQ\fP 
.PP
.nf
          WANTQ 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 (A,B) is returned;
          = 'V': Q must contain an unitary matrix Q1 on entry and
                 the product Q1*Q is returned\&.
.fi
.PP
.br
\fIWANTZ\fP 
.PP
.nf
          WANTZ 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 (A,B) is returned;
          = 'V': Z must contain an unitary matrix Z1 on entry and
                 the product Z1*Z is returned\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A, B, Q, and Z\&.  N >= 0\&.
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
          ILO is INTEGER
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
          IHI is INTEGER
          ILO and IHI mark the rows and columns of A 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\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX*16 array, dimension (LDA, N)
          On entry, the N-by-N upper Hessenberg matrix A\&.
          On exit, if JOB = 'S', A contains the upper triangular
          matrix S from the generalized Schur factorization\&.
          If JOB = 'E', the diagonal blocks of A match those of S, but
          the rest of A is unspecified\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max( 1, N )\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is COMPLEX*16 array, dimension (LDB, N)
          On entry, the N-by-N upper triangular matrix B\&.
          On exit, if JOB = 'S', B contains the upper triangular
          matrix P from the generalized Schur factorization;
          If JOB = 'E', the diagonal blocks of B match those of P, but
          the rest of B is unspecified\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&.  LDB >= max( 1, N )\&.
.fi
.PP
.br
\fIALPHA\fP 
.PP
.nf
          ALPHA is COMPLEX*16 array, dimension (N)
          Each scalar alpha defining an eigenvalue
          of GNEP\&.
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          BETA is COMPLEX*16 array, dimension (N)
          The scalars beta that define the eigenvalues of GNEP\&.
          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\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          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 (A,B), and if COMPQ = 'V', the unitary matrix
          of left Schur vectors of (A,B)\&.
          Not referenced if COMPQ = 'N'\&.
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
          The leading dimension of the array Q\&.  LDQ >= 1\&.
          If COMPQ='V' or 'I', then LDQ >= N\&.
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
          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'\&.
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
          LDZ is INTEGER
          The leading dimension of the array Z\&.  LDZ >= 1\&.
          If COMPZ='V' or 'I', then LDZ >= N\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK\&.
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is DOUBLE PRECISION array, dimension (N)
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIREC\fP 
.PP
.nf
          REC is INTEGER
             REC indicates the current recursion level\&. Should be set
             to 0 on first call\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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\&.  (A,B) is not
                     in Schur form, but ALPHA(i) and
                     BETA(i), i=INFO+1,\&.\&.\&.,N should be correct\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Thijs Steel, KU Leuven 
.RE
.PP
\fBDate\fP
.RS 4
May 2020 
.RE
.PP

.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.