gehrd(3)

NAME
gehrd - gehrd: reduction to Hessenberg

SYNOPSIS

  Functions

subroutine cgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info)
CGEHRD
subroutine dgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info)
DGEHRD
subroutine sgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info)
SGEHRD
subroutine zgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info)
ZGEHRD

Detailed Description

Function Documentation

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

CGEHRD

Purpose:

    !>
    !> CGEHRD reduces a complex general matrix A to upper Hessenberg form H by
    !> an 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 <= 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
    !>          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). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
    !>          zero.
    !>

    WORK

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

    LWORK

    !>          LWORK is INTEGER
    !>          The length of the array WORK.  LWORK >= max(1,N).
    !>          For good performance, LWORK should generally be larger.
    !>
    !>          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:

    !>
    !>  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).
    !>
    !>  This file is a slight modification of LAPACK-3.0's CGEHRD
    !>  subroutine incorporating improvements proposed by Quintana-Orti and
    !>  Van de Geijn (2006). (See CLAHR2.)
    !>

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

DGEHRD

Purpose:

    !>
    !> DGEHRD 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 <= 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
    !>          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). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
    !>          zero.
    !>

    WORK

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

    LWORK

    !>          LWORK is INTEGER
    !>          The length of the array WORK.  LWORK >= max(1,N).
    !>          For good performance, LWORK should generally be larger.
    !>
    !>          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:

    !>
    !>  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).
    !>
    !>  This file is a slight modification of LAPACK-3.0's DGEHRD
    !>  subroutine incorporating improvements proposed by Quintana-Orti and
    !>  Van de Geijn (2006). (See DLAHR2.)
    !>

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

SGEHRD

Purpose:

    !>
    !> SGEHRD 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 <= 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
    !>          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). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
    !>          zero.
    !>

    WORK

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

    LWORK

    !>          LWORK is INTEGER
    !>          The length of the array WORK.  LWORK >= max(1,N).
    !>          For good performance, LWORK should generally be larger.
    !>
    !>          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:

    !>
    !>  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).
    !>
    !>  This file is a slight modification of LAPACK-3.0's SGEHRD
    !>  subroutine incorporating improvements proposed by Quintana-Orti and
    !>  Van de Geijn (2006). (See SLAHR2.)
    !>

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

ZGEHRD

Purpose:

    !>
    !> ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by
    !> an 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 <= 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
    !>          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). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
    !>          zero.
    !>

    WORK

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

    LWORK

    !>          LWORK is INTEGER
    !>          The length of the array WORK.  LWORK >= max(1,N).
    !>          For good performance, LWORK should generally be larger.
    !>
    !>          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:

    !>
    !>  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).
    !>
    !>  This file is a slight modification of LAPACK-3.0's ZGEHRD
    !>  subroutine incorporating improvements proposed by Quintana-Orti and
    !>  Van de Geijn (2006). (See ZLAHR2.)
    !>

Author

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