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gehd2(3) Library Functions Manual gehd2(3)

NAME

gehd2 - gehd2: reduction to Hessenberg, level 2

SYNOPSIS

Functions


subroutine CGEHD2 (n, ilo, ihi, a, lda, tau, work, info)
CGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm. subroutine DGEHD2 (n, ilo, ihi, a, lda, tau, work, info)
DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm. subroutine SGEHD2 (n, ilo, ihi, a, lda, tau, work, info)
SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm. subroutine ZGEHD2 (n, ilo, ihi, a, lda, tau, work, info)
ZGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Detailed Description

Function Documentation

subroutine CGEHD2 (integer n, integer ilo, integer ihi, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) tau, complex, dimension( * ) work, integer info)

CGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Purpose:

!>
!> CGEHD2 reduces a complex general matrix A to upper Hessenberg form H
!> by a unitary similarity transformation:  Q**H * A * Q = H .
!>

Parameters

N 

!>          N is INTEGER
!>          The order of the matrix A.  N >= 0.
!>

ILO

!>          ILO is INTEGER
!>

IHI

!>          IHI is INTEGER
!>
!>          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 CGEBAL; otherwise they should be
!>          set to 1 and N respectively. See Further Details.
!>          1 <= ILO <= IHI <= max(1,N).
!>

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
!>          elements below the first subdiagonal, with the array TAU,
!>          represent the unitary matrix Q as a product of elementary
!>          reflectors. See Further Details.
!>

LDA

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

TAU

!>          TAU is COMPLEX array, dimension (N-1)
!>          The scalar factors of the elementary reflectors (see Further
!>          Details).
!>

WORK

!>          WORK is COMPLEX array, dimension (N)
!>

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:

!>
!>  The matrix Q is represented as a product of (ihi-ilo) elementary
!>  reflectors
!>
!>     Q = H(ilo) H(ilo+1) . . . H(ihi-1).
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - tau * v * v**H
!>
!>  where tau is a complex scalar, and v is a complex vector with
!>  v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
!>  exit in A(i+2:ihi,i), and tau in TAU(i).
!>
!>  The contents of A are illustrated by the following example, with
!>  n = 7, ilo = 2 and ihi = 6:
!>
!>  on entry,                        on exit,
!>
!>  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
!>  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
!>  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
!>  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
!>  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
!>  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
!>  (                         a )    (                          a )
!>
!>  where a denotes an element of the original matrix A, h denotes a
!>  modified element of the upper Hessenberg matrix H, and vi denotes an
!>  element of the vector defining H(i).
!>

Definition at line 148 of file cgehd2.f.

subroutine DGEHD2 (integer n, integer ilo, integer ihi, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer info)

DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Purpose:

!>
!> DGEHD2 reduces a real general matrix A to upper Hessenberg form H by
!> an orthogonal similarity transformation:  Q**T * A * Q = H .
!>

Parameters

N 

!>          N is INTEGER
!>          The order of the matrix A.  N >= 0.
!>

ILO

!>          ILO is INTEGER
!>

IHI

!>          IHI is INTEGER
!>
!>          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 DGEBAL; otherwise they should be
!>          set to 1 and N respectively. See Further Details.
!>          1 <= ILO <= IHI <= max(1,N).
!>

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
!>          elements below the first subdiagonal, with the array TAU,
!>          represent the orthogonal matrix Q as a product of elementary
!>          reflectors. See Further Details.
!>

LDA

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

TAU

!>          TAU is DOUBLE PRECISION array, dimension (N-1)
!>          The scalar factors of the elementary reflectors (see Further
!>          Details).
!>

WORK

!>          WORK is DOUBLE PRECISION array, dimension (N)
!>

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:

!>
!>  The matrix Q is represented as a product of (ihi-ilo) elementary
!>  reflectors
!>
!>     Q = H(ilo) H(ilo+1) . . . H(ihi-1).
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - tau * v * v**T
!>
!>  where tau is a real scalar, and v is a real vector with
!>  v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
!>  exit in A(i+2:ihi,i), and tau in TAU(i).
!>
!>  The contents of A are illustrated by the following example, with
!>  n = 7, ilo = 2 and ihi = 6:
!>
!>  on entry,                        on exit,
!>
!>  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
!>  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
!>  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
!>  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
!>  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
!>  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
!>  (                         a )    (                          a )
!>
!>  where a denotes an element of the original matrix A, h denotes a
!>  modified element of the upper Hessenberg matrix H, and vi denotes an
!>  element of the vector defining H(i).
!>

Definition at line 148 of file dgehd2.f.

subroutine SGEHD2 (integer n, integer ilo, integer ihi, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work, integer info)

SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Purpose:

!>
!> SGEHD2 reduces a real general matrix A to upper Hessenberg form H by
!> an orthogonal similarity transformation:  Q**T * A * Q = H .
!>

Parameters

N 

!>          N is INTEGER
!>          The order of the matrix A.  N >= 0.
!>

ILO

!>          ILO is INTEGER
!>

IHI

!>          IHI is INTEGER
!>
!>          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 SGEBAL; otherwise they should be
!>          set to 1 and N respectively. See Further Details.
!>          1 <= ILO <= IHI <= max(1,N).
!>

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
!>          elements below the first subdiagonal, with the array TAU,
!>          represent the orthogonal matrix Q as a product of elementary
!>          reflectors. See Further Details.
!>

LDA

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

TAU

!>          TAU is REAL array, dimension (N-1)
!>          The scalar factors of the elementary reflectors (see Further
!>          Details).
!>

WORK

!>          WORK is REAL array, dimension (N)
!>

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:

!>
!>  The matrix Q is represented as a product of (ihi-ilo) elementary
!>  reflectors
!>
!>     Q = H(ilo) H(ilo+1) . . . H(ihi-1).
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - tau * v * v**T
!>
!>  where tau is a real scalar, and v is a real vector with
!>  v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
!>  exit in A(i+2:ihi,i), and tau in TAU(i).
!>
!>  The contents of A are illustrated by the following example, with
!>  n = 7, ilo = 2 and ihi = 6:
!>
!>  on entry,                        on exit,
!>
!>  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
!>  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
!>  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
!>  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
!>  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
!>  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
!>  (                         a )    (                          a )
!>
!>  where a denotes an element of the original matrix A, h denotes a
!>  modified element of the upper Hessenberg matrix H, and vi denotes an
!>  element of the vector defining H(i).
!>

Definition at line 148 of file sgehd2.f.

subroutine ZGEHD2 (integer n, integer ilo, integer ihi, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer info)

ZGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Purpose:

!>
!> ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H
!> by a unitary similarity transformation:  Q**H * A * Q = H .
!>

Parameters

N 

!>          N is INTEGER
!>          The order of the matrix A.  N >= 0.
!>

ILO

!>          ILO is INTEGER
!>

IHI

!>          IHI is INTEGER
!>
!>          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 ZGEBAL; otherwise they should be
!>          set to 1 and N respectively. See Further Details.
!>          1 <= ILO <= IHI <= max(1,N).
!>

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
!>          elements below the first subdiagonal, with the array TAU,
!>          represent the unitary matrix Q as a product of elementary
!>          reflectors. See Further Details.
!>

LDA

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

TAU

!>          TAU is COMPLEX*16 array, dimension (N-1)
!>          The scalar factors of the elementary reflectors (see Further
!>          Details).
!>

WORK

!>          WORK is COMPLEX*16 array, dimension (N)
!>

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:

!>
!>  The matrix Q is represented as a product of (ihi-ilo) elementary
!>  reflectors
!>
!>     Q = H(ilo) H(ilo+1) . . . H(ihi-1).
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - tau * v * v**H
!>
!>  where tau is a complex scalar, and v is a complex vector with
!>  v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
!>  exit in A(i+2:ihi,i), and tau in TAU(i).
!>
!>  The contents of A are illustrated by the following example, with
!>  n = 7, ilo = 2 and ihi = 6:
!>
!>  on entry,                        on exit,
!>
!>  ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
!>  (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
!>  (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
!>  (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
!>  (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
!>  (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
!>  (                         a )    (                          a )
!>
!>  where a denotes an element of the original matrix A, h denotes a
!>  modified element of the upper Hessenberg matrix H, and vi denotes an
!>  element of the vector defining H(i).
!>

Definition at line 148 of file zgehd2.f.

Author

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