gghrd(3)

NAME
gghrd - gghrd: reduction to Hessenberg

SYNOPSIS

  Functions

subroutine cgghrd (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)
CGGHRD
subroutine dgghrd (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)
DGGHRD
subroutine sgghrd (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)
SGGHRD
subroutine zgghrd (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)
ZGGHRD

Detailed Description

Function Documentation

  subroutine cgghrd (character compq, character compz, integer n, integer ilo, integer ihi, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( ldz, * ) z, integer ldz, integer info)

CGGHRD

Purpose:

    !>
    !> CGGHRD reduces a pair of complex matrices (A,B) to generalized upper
    !> Hessenberg form using unitary transformations, where A is a
    !> general matrix and B is upper triangular.  The form of the generalized
    !> eigenvalue problem is
    !>    A*x = lambda*B*x,
    !> and B is typically made upper triangular by computing its QR
    !> factorization and moving the unitary matrix Q to the left side
    !> of the equation.
    !>
    !> This subroutine simultaneously reduces A to a Hessenberg matrix H:
    !>    Q**H*A*Z = H
    !> and transforms B to another upper triangular matrix T:
    !>    Q**H*B*Z = T
    !> in order to reduce the problem to its standard form
    !>    H*y = lambda*T*y
    !> where y = Z**H*x.
    !>
    !> The unitary matrices Q and Z are determined as products of Givens
    !> rotations.  They may either be formed explicitly, or they may be
    !> postmultiplied into input matrices Q1 and Z1, so that
    !>      Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H
    !>      Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H
    !> If Q1 is the unitary matrix from the QR factorization of B in the
    !> original equation A*x = lambda*B*x, then CGGHRD reduces the original
    !> problem to generalized Hessenberg form.
    !>

Parameters
    COMPQ

    !>          COMPQ is CHARACTER*1
    !>          = 'N': do not compute Q;
    !>          = 'I': Q is initialized to the unit matrix, and the
    !>                 unitary matrix Q is returned;
    !>          = 'V': Q must contain a unitary matrix Q1 on entry,
    !>                 and the product Q1*Q is returned.
    !>

    COMPZ

    !>          COMPZ is CHARACTER*1
    !>          = 'N': do not compute Z;
    !>          = 'I': Z is initialized to the unit matrix, and the
    !>                 unitary matrix Z is returned;
    !>          = 'V': Z must contain a unitary matrix Z1 on entry,
    !>                 and the product Z1*Z is returned.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices A and B.  N >= 0.
    !>

    ILO

    !>          ILO is INTEGER
    !>

    IHI

    !>          IHI is INTEGER
    !>
    !>          ILO and IHI mark the rows and columns of A which are to be
    !>          reduced.  It is assumed that A is already upper triangular
    !>          in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are
    !>          normally set by a previous call to CGGBAL; otherwise they
    !>          should be set to 1 and N respectively.
    !>          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
    !>

    A

    !>          A is COMPLEX array, dimension (LDA, N)
    !>          On entry, the N-by-N general matrix to be reduced.
    !>          On exit, the upper triangle and the first subdiagonal of A
    !>          are overwritten with the upper Hessenberg matrix H, and the
    !>          rest is set to zero.
    !>

    LDA

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

    B

    !>          B is COMPLEX array, dimension (LDB, N)
    !>          On entry, the N-by-N upper triangular matrix B.
    !>          On exit, the upper triangular matrix T = Q**H B Z.  The
    !>          elements below the diagonal are set to zero.
    !>

    LDB

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

    Q

    !>          Q is COMPLEX array, dimension (LDQ, N)
    !>          On entry, if COMPQ = 'V', the unitary matrix Q1, typically
    !>          from the QR factorization of B.
    !>          On exit, if COMPQ='I', the unitary matrix Q, and if
    !>          COMPQ = 'V', the product Q1*Q.
    !>          Not referenced if COMPQ='N'.
    !>

    LDQ

    !>          LDQ is INTEGER
    !>          The leading dimension of the array Q.
    !>          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
    !>

    Z

    !>          Z is COMPLEX array, dimension (LDZ, N)
    !>          On entry, if COMPZ = 'V', the unitary matrix Z1.
    !>          On exit, if COMPZ='I', the unitary matrix Z, and if
    !>          COMPZ = 'V', the product Z1*Z.
    !>          Not referenced if COMPZ='N'.
    !>

    LDZ

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z.
    !>          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  This routine reduces A to Hessenberg and B to triangular form by
    !>  an unblocked reduction, as described in _Matrix_Computations_,
    !>  by Golub and van Loan (Johns Hopkins Press).
    !>

  subroutine dgghrd (character compq, character compz, integer n, integer ilo, integer ihi, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( ldz, * ) z, integer ldz, integer info)

DGGHRD

Purpose:

    !>
    !> DGGHRD reduces a pair of real matrices (A,B) to generalized upper
    !> Hessenberg form using orthogonal transformations, where A is a
    !> general matrix and B is upper triangular.  The form of the
    !> generalized eigenvalue problem is
    !>    A*x = lambda*B*x,
    !> and B is typically made upper triangular by computing its QR
    !> factorization and moving the orthogonal matrix Q to the left side
    !> of the equation.
    !>
    !> This subroutine simultaneously reduces A to a Hessenberg matrix H:
    !>    Q**T*A*Z = H
    !> and transforms B to another upper triangular matrix T:
    !>    Q**T*B*Z = T
    !> in order to reduce the problem to its standard form
    !>    H*y = lambda*T*y
    !> where y = Z**T*x.
    !>
    !> The orthogonal matrices Q and Z are determined as products of Givens
    !> rotations.  They may either be formed explicitly, or they may be
    !> postmultiplied into input matrices Q1 and Z1, so that
    !>
    !>      Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
    !>
    !>      Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
    !>
    !> If Q1 is the orthogonal matrix from the QR factorization of B in the
    !> original equation A*x = lambda*B*x, then DGGHRD reduces the original
    !> problem to generalized Hessenberg form.
    !>

Parameters
    COMPQ

    !>          COMPQ is CHARACTER*1
    !>          = 'N': do not compute Q;
    !>          = 'I': Q is initialized to the unit matrix, and the
    !>                 orthogonal matrix Q is returned;
    !>          = 'V': Q must contain an orthogonal matrix Q1 on entry,
    !>                 and the product Q1*Q is returned.
    !>

    COMPZ

    !>          COMPZ is CHARACTER*1
    !>          = 'N': do not compute Z;
    !>          = 'I': Z is initialized to the unit matrix, and the
    !>                 orthogonal matrix Z is returned;
    !>          = 'V': Z must contain an orthogonal matrix Z1 on entry,
    !>                 and the product Z1*Z is returned.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices A and B.  N >= 0.
    !>

    ILO

    !>          ILO is INTEGER
    !>

    IHI

    !>          IHI is INTEGER
    !>
    !>          ILO and IHI mark the rows and columns of A which are to be
    !>          reduced.  It is assumed that A is already upper triangular
    !>          in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are
    !>          normally set by a previous call to DGGBAL; otherwise they
    !>          should be set to 1 and N respectively.
    !>          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
    !>

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA, N)
    !>          On entry, the N-by-N general matrix to be reduced.
    !>          On exit, the upper triangle and the first subdiagonal of A
    !>          are overwritten with the upper Hessenberg matrix H, and the
    !>          rest is set to zero.
    !>

    LDA

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

    B

    !>          B is DOUBLE PRECISION array, dimension (LDB, N)
    !>          On entry, the N-by-N upper triangular matrix B.
    !>          On exit, the upper triangular matrix T = Q**T B Z.  The
    !>          elements below the diagonal are set to zero.
    !>

    LDB

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

    Q

    !>          Q is DOUBLE PRECISION array, dimension (LDQ, N)
    !>          On entry, if COMPQ = 'V', the orthogonal matrix Q1,
    !>          typically from the QR factorization of B.
    !>          On exit, if COMPQ='I', the orthogonal matrix Q, and if
    !>          COMPQ = 'V', the product Q1*Q.
    !>          Not referenced if COMPQ='N'.
    !>

    LDQ

    !>          LDQ is INTEGER
    !>          The leading dimension of the array Q.
    !>          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
    !>

    Z

    !>          Z is DOUBLE PRECISION array, dimension (LDZ, N)
    !>          On entry, if COMPZ = 'V', the orthogonal matrix Z1.
    !>          On exit, if COMPZ='I', the orthogonal matrix Z, and if
    !>          COMPZ = 'V', the product Z1*Z.
    !>          Not referenced if COMPZ='N'.
    !>

    LDZ

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z.
    !>          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  This routine reduces A to Hessenberg and B to triangular form by
    !>  an unblocked reduction, as described in _Matrix_Computations_,
    !>  by Golub and Van Loan (Johns Hopkins Press.)
    !>

  subroutine sgghrd (character compq, character compz, integer n, integer ilo, integer ihi, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldq, * ) q, integer ldq, real, dimension( ldz, * ) z, integer ldz, integer info)

SGGHRD

Purpose:

    !>
    !> SGGHRD reduces a pair of real matrices (A,B) to generalized upper
    !> Hessenberg form using orthogonal transformations, where A is a
    !> general matrix and B is upper triangular.  The form of the
    !> generalized eigenvalue problem is
    !>    A*x = lambda*B*x,
    !> and B is typically made upper triangular by computing its QR
    !> factorization and moving the orthogonal matrix Q to the left side
    !> of the equation.
    !>
    !> This subroutine simultaneously reduces A to a Hessenberg matrix H:
    !>    Q**T*A*Z = H
    !> and transforms B to another upper triangular matrix T:
    !>    Q**T*B*Z = T
    !> in order to reduce the problem to its standard form
    !>    H*y = lambda*T*y
    !> where y = Z**T*x.
    !>
    !> The orthogonal matrices Q and Z are determined as products of Givens
    !> rotations.  They may either be formed explicitly, or they may be
    !> postmultiplied into input matrices Q1 and Z1, so that
    !>
    !>      Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
    !>
    !>      Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
    !>
    !> If Q1 is the orthogonal matrix from the QR factorization of B in the
    !> original equation A*x = lambda*B*x, then SGGHRD reduces the original
    !> problem to generalized Hessenberg form.
    !>

Parameters
    COMPQ

    !>          COMPQ is CHARACTER*1
    !>          = 'N': do not compute Q;
    !>          = 'I': Q is initialized to the unit matrix, and the
    !>                 orthogonal matrix Q is returned;
    !>          = 'V': Q must contain an orthogonal matrix Q1 on entry,
    !>                 and the product Q1*Q is returned.
    !>

    COMPZ

    !>          COMPZ is CHARACTER*1
    !>          = 'N': do not compute Z;
    !>          = 'I': Z is initialized to the unit matrix, and the
    !>                 orthogonal matrix Z is returned;
    !>          = 'V': Z must contain an orthogonal matrix Z1 on entry,
    !>                 and the product Z1*Z is returned.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices A and B.  N >= 0.
    !>

    ILO

    !>          ILO is INTEGER
    !>

    IHI

    !>          IHI is INTEGER
    !>
    !>          ILO and IHI mark the rows and columns of A which are to be
    !>          reduced.  It is assumed that A is already upper triangular
    !>          in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are
    !>          normally set by a previous call to SGGBAL; otherwise they
    !>          should be set to 1 and N respectively.
    !>          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
    !>

    A

    !>          A is REAL array, dimension (LDA, N)
    !>          On entry, the N-by-N general matrix to be reduced.
    !>          On exit, the upper triangle and the first subdiagonal of A
    !>          are overwritten with the upper Hessenberg matrix H, and the
    !>          rest is set to zero.
    !>

    LDA

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

    B

    !>          B is REAL array, dimension (LDB, N)
    !>          On entry, the N-by-N upper triangular matrix B.
    !>          On exit, the upper triangular matrix T = Q**T B Z.  The
    !>          elements below the diagonal are set to zero.
    !>

    LDB

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

    Q

    !>          Q is REAL array, dimension (LDQ, N)
    !>          On entry, if COMPQ = 'V', the orthogonal matrix Q1,
    !>          typically from the QR factorization of B.
    !>          On exit, if COMPQ='I', the orthogonal matrix Q, and if
    !>          COMPQ = 'V', the product Q1*Q.
    !>          Not referenced if COMPQ='N'.
    !>

    LDQ

    !>          LDQ is INTEGER
    !>          The leading dimension of the array Q.
    !>          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
    !>

    Z

    !>          Z is REAL array, dimension (LDZ, N)
    !>          On entry, if COMPZ = 'V', the orthogonal matrix Z1.
    !>          On exit, if COMPZ='I', the orthogonal matrix Z, and if
    !>          COMPZ = 'V', the product Z1*Z.
    !>          Not referenced if COMPZ='N'.
    !>

    LDZ

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z.
    !>          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  This routine reduces A to Hessenberg and B to triangular form by
    !>  an unblocked reduction, as described in _Matrix_Computations_,
    !>  by Golub and Van Loan (Johns Hopkins Press.)
    !>

  subroutine zgghrd (character compq, character compz, integer n, integer ilo, integer ihi, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldz, * ) z, integer ldz, integer info)

ZGGHRD

Purpose:

    !>
    !> ZGGHRD reduces a pair of complex matrices (A,B) to generalized upper
    !> Hessenberg form using unitary transformations, where A is a
    !> general matrix and B is upper triangular.  The form of the
    !> generalized eigenvalue problem is
    !>    A*x = lambda*B*x,
    !> and B is typically made upper triangular by computing its QR
    !> factorization and moving the unitary matrix Q to the left side
    !> of the equation.
    !>
    !> This subroutine simultaneously reduces A to a Hessenberg matrix H:
    !>    Q**H*A*Z = H
    !> and transforms B to another upper triangular matrix T:
    !>    Q**H*B*Z = T
    !> in order to reduce the problem to its standard form
    !>    H*y = lambda*T*y
    !> where y = Z**H*x.
    !>
    !> The unitary matrices Q and Z are determined as products of Givens
    !> rotations.  They may either be formed explicitly, or they may be
    !> postmultiplied into input matrices Q1 and Z1, so that
    !>      Q1 * A * Z1**H = (Q1*Q) * H * (Z1*Z)**H
    !>      Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H
    !> If Q1 is the unitary matrix from the QR factorization of B in the
    !> original equation A*x = lambda*B*x, then ZGGHRD reduces the original
    !> problem to generalized Hessenberg form.
    !>

Parameters
    COMPQ

    !>          COMPQ is CHARACTER*1
    !>          = 'N': do not compute Q;
    !>          = 'I': Q is initialized to the unit matrix, and the
    !>                 unitary matrix Q is returned;
    !>          = 'V': Q must contain a unitary matrix Q1 on entry,
    !>                 and the product Q1*Q is returned.
    !>

    COMPZ

    !>          COMPZ is CHARACTER*1
    !>          = 'N': do not compute Z;
    !>          = 'I': Z is initialized to the unit matrix, and the
    !>                 unitary matrix Z is returned;
    !>          = 'V': Z must contain a unitary matrix Z1 on entry,
    !>                 and the product Z1*Z is returned.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices A and B.  N >= 0.
    !>

    ILO

    !>          ILO is INTEGER
    !>

    IHI

    !>          IHI is INTEGER
    !>
    !>          ILO and IHI mark the rows and columns of A which are to be
    !>          reduced.  It is assumed that A is already upper triangular
    !>          in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are
    !>          normally set by a previous call to ZGGBAL; otherwise they
    !>          should be set to 1 and N respectively.
    !>          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
    !>

    A

    !>          A is COMPLEX*16 array, dimension (LDA, N)
    !>          On entry, the N-by-N general matrix to be reduced.
    !>          On exit, the upper triangle and the first subdiagonal of A
    !>          are overwritten with the upper Hessenberg matrix H, and the
    !>          rest is set to zero.
    !>

    LDA

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

    B

    !>          B is COMPLEX*16 array, dimension (LDB, N)
    !>          On entry, the N-by-N upper triangular matrix B.
    !>          On exit, the upper triangular matrix T = Q**H B Z.  The
    !>          elements below the diagonal are set to zero.
    !>

    LDB

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

    Q

    !>          Q is COMPLEX*16 array, dimension (LDQ, N)
    !>          On entry, if COMPQ = 'V', the unitary matrix Q1, typically
    !>          from the QR factorization of B.
    !>          On exit, if COMPQ='I', the unitary matrix Q, and if
    !>          COMPQ = 'V', the product Q1*Q.
    !>          Not referenced if COMPQ='N'.
    !>

    LDQ

    !>          LDQ is INTEGER
    !>          The leading dimension of the array Q.
    !>          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
    !>

    Z

    !>          Z is COMPLEX*16 array, dimension (LDZ, N)
    !>          On entry, if COMPZ = 'V', the unitary matrix Z1.
    !>          On exit, if COMPZ='I', the unitary matrix Z, and if
    !>          COMPZ = 'V', the product Z1*Z.
    !>          Not referenced if COMPZ='N'.
    !>

    LDZ

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z.
    !>          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  This routine reduces A to Hessenberg and B to triangular form by
    !>  an unblocked reduction, as described in _Matrix_Computations_,
    !>  by Golub and van Loan (Johns Hopkins Press).
    !>

Author

Generated automatically by Doxygen for LAPACK from the source code.