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).
    !>

  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).
    !>

  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).
    !>

  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).
    !>

Author

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