gghd3(3)

NAME
gghd3 - gghd3: reduction to Hessenberg, level 3

SYNOPSIS

  Functions
subroutine cgghd3 (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, work, lwork, info)
CGGHD3
subroutine dgghd3 (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, work, lwork, info)
DGGHD3
subroutine sgghd3 (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, work, lwork, info)
SGGHD3
subroutine zgghd3 (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, work, lwork, info)
ZGGHD3

Detailed Description

Function Documentation

  subroutine cgghd3 (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, complex, dimension( * ) work, integer lwork, integer info)

CGGHD3

Purpose:

     CGGHD3 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 CGGHD3 reduces the original
     problem to generalized Hessenberg form.
     This is a blocked variant of CGGHRD, using matrix-matrix
     multiplications for parts of the computation to enhance performance.

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.

    WORK

              WORK is COMPLEX array, dimension (LWORK)
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of the array WORK.  LWORK >= 1.
              For optimum performance LWORK >= 6*N*NB, where NB is the
              optimal blocksize.
              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
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      This routine reduces A to Hessenberg form and maintains B in triangular form
      using a blocked variant of Moler and Stewart's original algorithm,
      as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti
      (BIT 2008).

Definition at line 229 of file cgghd3.f.

  subroutine dgghd3 (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, double precision, dimension( * ) work, integer lwork, integer info)

DGGHD3

Purpose:

     DGGHD3 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 DGGHD3 reduces the original
     problem to generalized Hessenberg form.
     This is a blocked variant of DGGHRD, using matrix-matrix
     multiplications for parts of the computation to enhance performance.

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.

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of the array WORK.  LWORK >= 1.
              For optimum performance LWORK >= 6*N*NB, where NB is the
              optimal blocksize.
              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
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      This routine reduces A to Hessenberg form and maintains B in triangular form
      using a blocked variant of Moler and Stewart's original algorithm,
      as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti
      (BIT 2008).

Definition at line 228 of file dgghd3.f.

  subroutine sgghd3 (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, real, dimension( * ) work, integer lwork, integer info)

SGGHD3

Purpose:

     SGGHD3 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 SGGHD3 reduces the original
     problem to generalized Hessenberg form.
     This is a blocked variant of SGGHRD, using matrix-matrix
     multiplications for parts of the computation to enhance performance.

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.

    WORK

              WORK is REAL array, dimension (LWORK)
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of the array WORK.  LWORK >= 1.
              For optimum performance LWORK >= 6*N*NB, where NB is the
              optimal blocksize.
              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
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      This routine reduces A to Hessenberg form and maintains B in triangular form
      using a blocked variant of Moler and Stewart's original algorithm,
      as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti
      (BIT 2008).

Definition at line 228 of file sgghd3.f.

  subroutine zgghd3 (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, complex*16, dimension( * ) work, integer lwork, integer info)

ZGGHD3

Purpose:

     ZGGHD3 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 ZGGHD3 reduces the original
     problem to generalized Hessenberg form.
     This is a blocked variant of CGGHRD, using matrix-matrix
     multiplications for parts of the computation to enhance performance.

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.

    WORK

              WORK is COMPLEX*16 array, dimension (LWORK)
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The length of the array WORK.  LWORK >= 1.
              For optimum performance LWORK >= 6*N*NB, where NB is the
              optimal blocksize.
              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
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      This routine reduces A to Hessenberg form and maintains B in triangular form
      using a blocked variant of Moler and Stewart's original algorithm,
      as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti
      (BIT 2008).

Definition at line 225 of file zgghd3.f.

Author

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