.TH "laqz2" 3 "Sat Dec 9 2023 21:42:18" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
laqz2 \- laqz2: step in ggev3, gges3
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "recursive subroutine \fBclaqz2\fP (ilschur, ilq, ilz, n, ilo, ihi, nw, a, lda, b, ldb, q, ldq, z, ldz, ns, nd, alpha, beta, qc, ldqc, zc, ldzc, work, lwork, rwork, rec, info)"
.br
.RI "\fBCLAQZ2\fP "
.ti -1c
.RI "subroutine \fBdlaqz2\fP (ilq, ilz, k, istartm, istopm, ihi, a, lda, b, ldb, nq, qstart, q, ldq, nz, zstart, z, ldz)"
.br
.RI "\fBDLAQZ2\fP "
.ti -1c
.RI "subroutine \fBslaqz2\fP (ilq, ilz, k, istartm, istopm, ihi, a, lda, b, ldb, nq, qstart, q, ldq, nz, zstart, z, ldz)"
.br
.RI "\fBSLAQZ2\fP "
.ti -1c
.RI "recursive subroutine \fBzlaqz2\fP (ilschur, ilq, ilz, n, ilo, ihi, nw, a, lda, b, ldb, q, ldq, z, ldz, ns, nd, alpha, beta, qc, ldqc, zc, ldzc, work, lwork, rwork, rec, info)"
.br
.RI "\fBZLAQZ2\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "recursive subroutine claqz2 (logical, intent(in) ilschur, logical, intent(in) ilq, logical, intent(in) ilz, integer, intent(in) n, integer, intent(in) ilo, integer, intent(in) ihi, integer, intent(in) nw, complex, dimension( lda, * ), intent(inout) a, integer, intent(in) lda, complex, dimension( ldb, * ), intent(inout) b, integer, intent(in) ldb, complex, dimension( ldq, * ), intent(inout) q, integer, intent(in) ldq, complex, dimension( ldz, * ), intent(inout) z, integer, intent(in) ldz, integer, intent(out) ns, integer, intent(out) nd, complex, dimension( * ), intent(inout) alpha, complex, dimension( * ), intent(inout) beta, complex, dimension( ldqc, * ) qc, integer, intent(in) ldqc, complex, dimension( ldzc, * ) zc, integer, intent(in) ldzc, complex, dimension( * ) work, integer, intent(in) lwork, real, dimension( * ) rwork, integer, intent(in) rec, integer, intent(out) info)"

.PP
\fBCLAQZ2\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 CLAQZ2 performs AED
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIILSCHUR\fP 
.PP
.nf
          ILSCHUR is LOGICAL
              Determines whether or not to update the full Schur form
.fi
.PP
.br
\fIILQ\fP 
.PP
.nf
          ILQ is LOGICAL
              Determines whether or not to update the matrix Q
.fi
.PP
.br
\fIILZ\fP 
.PP
.nf
          ILZ is LOGICAL
              Determines whether or not to update the matrix Z
.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,B) which
          are to be normalized
.fi
.PP
.br
\fINW\fP 
.PP
.nf
          NW is INTEGER
          The desired size of the deflation window\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX array, dimension (LDA, N)
.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)
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&.  LDB >= max( 1, N )\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          Q is COMPLEX array, dimension (LDQ, N)
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
          Z is COMPLEX array, dimension (LDZ, N)
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
          LDZ is INTEGER
.fi
.PP
.br
\fINS\fP 
.PP
.nf
          NS is INTEGER
          The number of unconverged eigenvalues available to
          use as shifts\&.
.fi
.PP
.br
\fIND\fP 
.PP
.nf
          ND is INTEGER
          The number of converged eigenvalues found\&.
.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
\fIQC\fP 
.PP
.nf
          QC is COMPLEX array, dimension (LDQC, NW)
.fi
.PP
.br
\fILDQC\fP 
.PP
.nf
          LDQC is INTEGER
.fi
.PP
.br
\fIZC\fP 
.PP
.nf
          ZC is COMPLEX array, dimension (LDZC, NW)
.fi
.PP
.br
\fILDZC\fP 
.PP
.nf
          LDZ is INTEGER
.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
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Thijs Steel, KU Leuven, KU Leuven 
.RE
.PP
\fBDate\fP
.RS 4
May 2020 
.RE
.PP

.SS "subroutine dlaqz2 (logical, intent(in) ilq, logical, intent(in) ilz, integer, intent(in) k, integer, intent(in) istartm, integer, intent(in) istopm, integer, intent(in) ihi, double precision, dimension( lda, * ) a, integer, intent(in) lda, double precision, dimension( ldb, * ) b, integer, intent(in) ldb, integer, intent(in) nq, integer, intent(in) qstart, double precision, dimension( ldq, * ) q, integer, intent(in) ldq, integer, intent(in) nz, integer, intent(in) zstart, double precision, dimension( ldz,                          * ) z, integer, intent(in) ldz)"

.PP
\fBDLAQZ2\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
      DLAQZ2 chases a 2x2 shift bulge in a matrix pencil down a single position
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIILQ\fP 
.PP
.nf
          ILQ is LOGICAL
              Determines whether or not to update the matrix Q
.fi
.PP
.br
\fIILZ\fP 
.PP
.nf
          ILZ is LOGICAL
              Determines whether or not to update the matrix Z
.fi
.PP
.br
\fIK\fP 
.PP
.nf
          K is INTEGER
              Index indicating the position of the bulge\&.
              On entry, the bulge is located in
              (A(k+1:k+2,k:k+1),B(k+1:k+2,k:k+1))\&.
              On exit, the bulge is located in
              (A(k+2:k+3,k+1:k+2),B(k+2:k+3,k+1:k+2))\&.
.fi
.PP
.br
\fIISTARTM\fP 
.PP
.nf
          ISTARTM is INTEGER
.fi
.PP
.br
\fIISTOPM\fP 
.PP
.nf
          ISTOPM is INTEGER
              Updates to (A,B) are restricted to
              (istartm:k+3,k:istopm)\&. It is assumed
              without checking that istartm <= k+1 and
              k+2 <= istopm
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
          IHI is INTEGER
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is DOUBLE PRECISION array, dimension (LDA,N)
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
              The leading dimension of A as declared in
              the calling procedure\&.
.fi
.PP
 
.br
\fIB\fP 
.PP
.nf
          B is DOUBLE PRECISION array, dimension (LDB,N)
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
              The leading dimension of B as declared in
              the calling procedure\&.
.fi
.PP
.br
\fINQ\fP 
.PP
.nf
          NQ is INTEGER
              The order of the matrix Q
.fi
.PP
.br
\fIQSTART\fP 
.PP
.nf
          QSTART is INTEGER
              Start index of the matrix Q\&. Rotations are applied
              To columns k+2-qStart:k+4-qStart of Q\&.
.fi
.PP
 
.br
\fIQ\fP 
.PP
.nf
          Q is DOUBLE PRECISION array, dimension (LDQ,NQ)
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
              The leading dimension of Q as declared in
              the calling procedure\&.
.fi
.PP
.br
\fINZ\fP 
.PP
.nf
          NZ is INTEGER
              The order of the matrix Z
.fi
.PP
.br
\fIZSTART\fP 
.PP
.nf
          ZSTART is INTEGER
              Start index of the matrix Z\&. Rotations are applied
              To columns k+1-qStart:k+3-qStart of Z\&.
.fi
.PP
 
.br
\fIZ\fP 
.PP
.nf
          Z is DOUBLE PRECISION array, dimension (LDZ,NZ)
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
          LDZ is INTEGER
              The leading dimension of Q as declared in
              the calling procedure\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Thijs Steel, KU Leuven 
.RE
.PP
\fBDate\fP
.RS 4
May 2020 
.RE
.PP

.SS "subroutine slaqz2 (logical, intent(in) ilq, logical, intent(in) ilz, integer, intent(in) k, integer, intent(in) istartm, integer, intent(in) istopm, integer, intent(in) ihi, real, dimension( lda, * ) a, integer, intent(in) lda, real, dimension( ldb, * ) b, integer, intent(in) ldb, integer, intent(in) nq, integer, intent(in) qstart, real, dimension( ldq, * ) q, integer, intent(in) ldq, integer, intent(in) nz, integer, intent(in) zstart, real, dimension( ldz, * ) z, integer, intent(in) ldz)"

.PP
\fBSLAQZ2\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
      SLAQZ2 chases a 2x2 shift bulge in a matrix pencil down a single position
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIILQ\fP 
.PP
.nf
          ILQ is LOGICAL
              Determines whether or not to update the matrix Q
.fi
.PP
.br
\fIILZ\fP 
.PP
.nf
          ILZ is LOGICAL
              Determines whether or not to update the matrix Z
.fi
.PP
.br
\fIK\fP 
.PP
.nf
          K is INTEGER
              Index indicating the position of the bulge\&.
              On entry, the bulge is located in
              (A(k+1:k+2,k:k+1),B(k+1:k+2,k:k+1))\&.
              On exit, the bulge is located in
              (A(k+2:k+3,k+1:k+2),B(k+2:k+3,k+1:k+2))\&.
.fi
.PP
.br
\fIISTARTM\fP 
.PP
.nf
          ISTARTM is INTEGER
.fi
.PP
.br
\fIISTOPM\fP 
.PP
.nf
          ISTOPM is INTEGER
              Updates to (A,B) are restricted to
              (istartm:k+3,k:istopm)\&. It is assumed
              without checking that istartm <= k+1 and
              k+2 <= istopm
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
          IHI is INTEGER
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is REAL array, dimension (LDA,N)
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
              The leading dimension of A as declared in
              the calling procedure\&.
.fi
.PP
 
.br
\fIB\fP 
.PP
.nf
          B is REAL array, dimension (LDB,N)
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
              The leading dimension of B as declared in
              the calling procedure\&.
.fi
.PP
.br
\fINQ\fP 
.PP
.nf
          NQ is INTEGER
              The order of the matrix Q
.fi
.PP
.br
\fIQSTART\fP 
.PP
.nf
          QSTART is INTEGER
              Start index of the matrix Q\&. Rotations are applied
              To columns k+2-qStart:k+4-qStart of Q\&.
.fi
.PP
 
.br
\fIQ\fP 
.PP
.nf
          Q is REAL array, dimension (LDQ,NQ)
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
              The leading dimension of Q as declared in
              the calling procedure\&.
.fi
.PP
.br
\fINZ\fP 
.PP
.nf
          NZ is INTEGER
              The order of the matrix Z
.fi
.PP
.br
\fIZSTART\fP 
.PP
.nf
          ZSTART is INTEGER
              Start index of the matrix Z\&. Rotations are applied
              To columns k+1-qStart:k+3-qStart of Z\&.
.fi
.PP
 
.br
\fIZ\fP 
.PP
.nf
          Z is REAL array, dimension (LDZ,NZ)
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
          LDZ is INTEGER
              The leading dimension of Q as declared in
              the calling procedure\&.
.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 zlaqz2 (logical, intent(in) ilschur, logical, intent(in) ilq, logical, intent(in) ilz, integer, intent(in) n, integer, intent(in) ilo, integer, intent(in) ihi, integer, intent(in) nw, 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( ldq,         * ), intent(inout) q, integer, intent(in) ldq, complex*16, dimension( ldz, * ), intent(inout) z, integer, intent(in) ldz, integer, intent(out) ns, integer, intent(out) nd, complex*16, dimension( * ), intent(inout) alpha, complex*16, dimension( * ), intent(inout) beta, complex*16, dimension( ldqc, * ) qc, integer, intent(in) ldqc, complex*16, dimension( ldzc, * ) zc, integer, intent(in) ldzc, complex*16, dimension( * ) work, integer, intent(in) lwork, double precision, dimension( * ) rwork, integer, intent(in) rec, integer, intent(out) info)"

.PP
\fBZLAQZ2\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 ZLAQZ2 performs AED
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIILSCHUR\fP 
.PP
.nf
          ILSCHUR is LOGICAL
              Determines whether or not to update the full Schur form
.fi
.PP
 
.br
\fIILQ\fP 
.PP
.nf
          ILQ is LOGICAL
              Determines whether or not to update the matrix Q
.fi
.PP
.br
\fIILZ\fP 
.PP
.nf
          ILZ is LOGICAL
              Determines whether or not to update the matrix Z
.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,B) which
          are to be normalized
.fi
.PP
.br
\fINW\fP 
.PP
.nf
          NW is INTEGER
          The desired size of the deflation window\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX*16 array, dimension (LDA, N)
.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)
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&.  LDB >= max( 1, N )\&.
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          Q is COMPLEX*16 array, dimension (LDQ, N)
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
          Z is COMPLEX*16 array, dimension (LDZ, N)
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
          LDZ is INTEGER
.fi
.PP
.br
\fINS\fP 
.PP
.nf
          NS is INTEGER
          The number of unconverged eigenvalues available to
          use as shifts\&.
.fi
.PP
.br
\fIND\fP 
.PP
.nf
          ND is INTEGER
          The number of converged eigenvalues found\&.
.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
\fIQC\fP 
.PP
.nf
          QC is COMPLEX*16 array, dimension (LDQC, NW)
.fi
.PP
.br
\fILDQC\fP 
.PP
.nf
          LDQC is INTEGER
.fi
.PP
.br
\fIZC\fP 
.PP
.nf
          ZC is COMPLEX*16 array, dimension (LDZC, NW)
.fi
.PP
.br
\fILDZC\fP 
.PP
.nf
          LDZ is INTEGER
.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
\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 DOUBLE PRECISION 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
.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\&.