gghrd - Man Page

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

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).

Definition at line 202 of file cgghrd.f.

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.)

Definition at line 205 of file dgghrd.f.

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.)

Definition at line 205 of file sgghrd.f.

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).

Definition at line 202 of file zgghrd.f.