LAPACK

NAME¶

laqz3 - laqz3: step in ggev3, gges3

SYNOPSIS¶

Functions¶

subroutine claqz3 (ilschur, ilq, ilz, n, ilo, ihi, nshifts, nblock_desired, alpha, beta, a, lda, b, ldb, q, ldq, z, ldz, qc, ldqc, zc, ldzc, work, lwork, info)
CLAQZ3 recursive subroutine dlaqz3 (ilschur, ilq, ilz, n, ilo, ihi, nw, a, lda, b, ldb, q, ldq, z, ldz, ns, nd, alphar, alphai, beta, qc, ldqc, zc, ldzc, work, lwork, rec, info)
DLAQZ3 recursive subroutine slaqz3 (ilschur, ilq, ilz, n, ilo, ihi, nw, a, lda, b, ldb, q, ldq, z, ldz, ns, nd, alphar, alphai, beta, qc, ldqc, zc, ldzc, work, lwork, rec, info)
SLAQZ3 subroutine zlaqz3 (ilschur, ilq, ilz, n, ilo, ihi, nshifts, nblock_desired, alpha, beta, a, lda, b, ldb, q, ldq, z, ldz, qc, ldqc, zc, ldzc, work, lwork, info)
ZLAQZ3

Detailed Description¶

Function Documentation¶

subroutine claqz3 (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) nshifts, integer, intent(in) nblock_desired, complex, dimension( * ), intent(inout) alpha, complex, dimension( * ), intent(inout) beta, 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, complex, dimension( ldqc, * ), intent(inout) qc, integer, intent(in) ldqc, complex, dimension( ldzc, * ), intent(inout) zc, integer, intent(in) ldzc, complex, dimension( * ), intent(inout) work, integer, intent(in) lwork, integer, intent(out) info)¶

CLAQZ3

Purpose:



 CLAQZ3 Executes a single multishift QZ sweep

Parameters

ILSCHUR



          ILSCHUR is LOGICAL


              Determines whether or not to update the full Schur form

ILQ



          ILQ is LOGICAL


              Determines whether or not to update the matrix Q

ILZ



          ILZ is LOGICAL


              Determines whether or not to update the matrix Z



          N is INTEGER


          The order of the matrices A, B, Q, and Z.  N >= 0.

ILO



          ILO is INTEGER

IHI



          IHI is INTEGER

NSHIFTS



          NSHIFTS is INTEGER


          The desired number of shifts to use

NBLOCK_DESIRED



          NBLOCK_DESIRED is INTEGER


          The desired size of the computational windows

ALPHA



          ALPHA is COMPLEX array. SR contains


          the alpha parts of the shifts to use.

BETA



          BETA is COMPLEX array. SS contains


          the scale of the shifts to use.



          A is COMPLEX array, dimension (LDA, N)

LDA



          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max( 1, N ).



          B is COMPLEX array, dimension (LDB, N)

LDB



          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max( 1, N ).



          Q is COMPLEX array, dimension (LDQ, N)

LDQ



          LDQ is INTEGER



          Z is COMPLEX array, dimension (LDZ, N)

LDZ



          LDZ is INTEGER



          QC is COMPLEX array, dimension (LDQC, NBLOCK_DESIRED)

LDQC



          LDQC is INTEGER



          ZC is COMPLEX array, dimension (LDZC, NBLOCK_DESIRED)

LDZC



          LDZ is INTEGER

WORK



          WORK is COMPLEX 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.

INFO



          INFO is INTEGER


          = 0: successful exit


          < 0: if INFO = -i, the i-th argument had an illegal value

Author

Thijs Steel, KU Leuven

Date

May 2020

recursive subroutine dlaqz3 (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, 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( ldq, * ), intent(inout) q, integer, intent(in) ldq, double precision, dimension( ldz, * ), intent(inout) z, integer, intent(in) ldz, integer, intent(out) ns, integer, intent(out) nd, double precision, dimension( * ), intent(inout) alphar, double precision, dimension( * ), intent(inout) alphai, double precision, dimension( * ), intent(inout) beta, double precision, dimension( ldqc, * ) qc, integer, intent(in) ldqc, double precision, dimension( ldzc, * ) zc, integer, intent(in) ldzc, double precision, dimension( * ) work, integer, intent(in) lwork, integer, intent(in) rec, integer, intent(out) info)¶

DLAQZ3

Purpose:



 DLAQZ3 performs AED

Parameters

ILSCHUR



          ILSCHUR is LOGICAL


              Determines whether or not to update the full Schur form

ILQ



          ILQ is LOGICAL


              Determines whether or not to update the matrix Q

ILZ



          ILZ is LOGICAL


              Determines whether or not to update the matrix Z



          N is INTEGER


          The order of the matrices A, B, Q, and Z.  N >= 0.

ILO



          ILO is INTEGER

IHI



          IHI is INTEGER


          ILO and IHI mark the rows and columns of (A,B) which


          are to be normalized



          NW is INTEGER


          The desired size of the deflation window.



          A is DOUBLE PRECISION array, dimension (LDA, N)

LDA



          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max( 1, N ).



          B is DOUBLE PRECISION array, dimension (LDB, N)

LDB



          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max( 1, N ).



          Q is DOUBLE PRECISION array, dimension (LDQ, N)

LDQ



          LDQ is INTEGER



          Z is DOUBLE PRECISION array, dimension (LDZ, N)

LDZ



          LDZ is INTEGER



          NS is INTEGER


          The number of unconverged eigenvalues available to


          use as shifts.



          ND is INTEGER


          The number of converged eigenvalues found.

ALPHAR



          ALPHAR is DOUBLE PRECISION array, dimension (N)


          The real parts of each scalar alpha defining an eigenvalue


          of GNEP.

ALPHAI



          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).

BETA



          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.



          QC is DOUBLE PRECISION array, dimension (LDQC, NW)

LDQC



          LDQC is INTEGER



          ZC is DOUBLE PRECISION array, dimension (LDZC, NW)

LDZC



          LDZ is INTEGER

WORK



          WORK is DOUBLE PRECISION 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.

REC



          REC is INTEGER


             REC indicates the current recursion level. Should be set


             to 0 on first call.

INFO



          INFO is INTEGER


          = 0: successful exit


          < 0: if INFO = -i, the i-th argument had an illegal value

Author

Thijs Steel, KU Leuven

Date

May 2020

recursive subroutine slaqz3 (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, real, dimension( lda, * ), intent(inout) a, integer, intent(in) lda, real, dimension( ldb, * ), intent(inout) b, integer, intent(in) ldb, real, dimension( ldq, * ), intent(inout) q, integer, intent(in) ldq, real, dimension( ldz, * ), intent(inout) z, integer, intent(in) ldz, integer, intent(out) ns, integer, intent(out) nd, real, dimension( * ), intent(inout) alphar, real, dimension( * ), intent(inout) alphai, real, dimension( * ), intent(inout) beta, real, dimension( ldqc, * ) qc, integer, intent(in) ldqc, real, dimension( ldzc, * ) zc, integer, intent(in) ldzc, real, dimension( * ) work, integer, intent(in) lwork, integer, intent(in) rec, integer, intent(out) info)¶

SLAQZ3

Purpose:



 SLAQZ3 performs AED

Parameters

ILSCHUR



          ILSCHUR is LOGICAL


              Determines whether or not to update the full Schur form

ILQ



          ILQ is LOGICAL


              Determines whether or not to update the matrix Q

ILZ



          ILZ is LOGICAL


              Determines whether or not to update the matrix Z



          N is INTEGER


          The order of the matrices A, B, Q, and Z.  N >= 0.

ILO



          ILO is INTEGER

IHI



          IHI is INTEGER


          ILO and IHI mark the rows and columns of (A,B) which


          are to be normalized



          NW is INTEGER


          The desired size of the deflation window.



          A is REAL array, dimension (LDA, N)

LDA



          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max( 1, N ).



          B is REAL array, dimension (LDB, N)

LDB



          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max( 1, N ).



          Q is REAL array, dimension (LDQ, N)

LDQ



          LDQ is INTEGER



          Z is REAL array, dimension (LDZ, N)

LDZ



          LDZ is INTEGER



          NS is INTEGER


          The number of unconverged eigenvalues available to


          use as shifts.



          ND is INTEGER


          The number of converged eigenvalues found.

ALPHAR



          ALPHAR is REAL array, dimension (N)


          The real parts of each scalar alpha defining an eigenvalue


          of GNEP.

ALPHAI



          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).

BETA



          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.



          QC is REAL array, dimension (LDQC, NW)

LDQC



          LDQC is INTEGER



          ZC is REAL array, dimension (LDZC, NW)

LDZC



          LDZ is INTEGER

WORK



          WORK is REAL 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.

REC



          REC is INTEGER


             REC indicates the current recursion level. Should be set


             to 0 on first call.

INFO



          INFO is INTEGER


          = 0: successful exit


          < 0: if INFO = -i, the i-th argument had an illegal value

Author

Thijs Steel, KU Leuven

Date

May 2020

subroutine zlaqz3 (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) nshifts, integer, intent(in) nblock_desired, complex16, dimension( ), intent(inout) alpha, complex16, dimension( ), intent(inout) beta, complex16, dimension( lda, ), intent(inout) a, integer, intent(in) lda, complex16, dimension( ldb, ), intent(inout) b, integer, intent(in) ldb, complex16, dimension( ldq, ), intent(inout) q, integer, intent(in) ldq, complex16, dimension( ldz, ), intent(inout) z, integer, intent(in) ldz, complex16, dimension( ldqc, ), intent(inout) qc, integer, intent(in) ldqc, complex16, dimension( ldzc, ), intent(inout) zc, integer, intent(in) ldzc, complex16, dimension( ), intent(inout) work, integer, intent(in) lwork, integer, intent(out) info)¶

ZLAQZ3

Purpose:



 ZLAQZ3 Executes a single multishift QZ sweep

Parameters

ILSCHUR



          ILSCHUR is LOGICAL


              Determines whether or not to update the full Schur form

ILQ



          ILQ is LOGICAL


              Determines whether or not to update the matrix Q

ILZ



          ILZ is LOGICAL


              Determines whether or not to update the matrix Z



          N is INTEGER


          The order of the matrices A, B, Q, and Z.  N >= 0.

ILO



          ILO is INTEGER

IHI



          IHI is INTEGER

NSHIFTS



          NSHIFTS is INTEGER


          The desired number of shifts to use

NBLOCK_DESIRED



          NBLOCK_DESIRED is INTEGER


          The desired size of the computational windows

ALPHA



          ALPHA is COMPLEX*16 array. SR contains


          the alpha parts of the shifts to use.

BETA



          BETA is COMPLEX*16 array. SS contains


          the scale of the shifts to use.



          A is COMPLEX*16 array, dimension (LDA, N)

LDA



          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max( 1, N ).



          B is COMPLEX*16 array, dimension (LDB, N)

LDB



          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max( 1, N ).



          Q is COMPLEX*16 array, dimension (LDQ, N)

LDQ



          LDQ is INTEGER



          Z is COMPLEX*16 array, dimension (LDZ, N)

LDZ



          LDZ is INTEGER



          QC is COMPLEX*16 array, dimension (LDQC, NBLOCK_DESIRED)

LDQC



          LDQC is INTEGER



          ZC is COMPLEX*16 array, dimension (LDZC, NBLOCK_DESIRED)

LDZC



          LDZ is INTEGER

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.

INFO



          INFO is INTEGER


          = 0: successful exit


          < 0: if INFO = -i, the i-th argument had an illegal value

Author

Thijs Steel, KU Leuven

Date

May 2020

Author¶

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