.TH "SRC/dlaqz0.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
SRC/dlaqz0.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.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 "
.in -1c
.SH "Function/Subroutine Documentation"
.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, , SIAM J\&. Numer\&. Anal\&., 10(1973),
!>      pp\&. 241--256\&.
!>
!> Ref: B\&. Kagstrom, D\&. Kressner, , SIAM J\&. Numer\&.
!>      Anal\&., 29(2006), pp\&. 199--227\&.
!>
!> Ref: T\&. Steel, D\&. Camps, K\&. Meerbergen, R\&. Vandebril 
!> 
.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 
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.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 
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.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

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Definition at line \fB302\fP of file \fBdlaqz0\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.