latrz(3)

NAME
latrz - latrz: RZ factor step

SYNOPSIS

  Functions

subroutine clatrz (m, n, l, a, lda, tau, work)
CLATRZ factors an upper trapezoidal matrix by means of unitary transformations.
subroutine dlatrz (m, n, l, a, lda, tau, work)
DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
subroutine slatrz (m, n, l, a, lda, tau, work)
SLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
subroutine zlatrz (m, n, l, a, lda, tau, work)
ZLATRZ factors an upper trapezoidal matrix by means of unitary transformations.

Detailed Description

Function Documentation

  subroutine clatrz (integer m, integer n, integer l, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) tau, complex, dimension( * ) work)

CLATRZ factors an upper trapezoidal matrix by means of unitary transformations.

Purpose:

    !>
    !> CLATRZ factors the M-by-(M+L) complex upper trapezoidal matrix
    !> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z by means
    !> of unitary transformations, where  Z is an (M+L)-by-(M+L) unitary
    !> matrix and, R and A1 are M-by-M upper triangular matrices.
    !>

Parameters
    M

    !>          M is INTEGER
    !>          The number of rows of the matrix A.  M >= 0.
    !>

    N

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

    L

    !>          L is INTEGER
    !>          The number of columns of the matrix A containing the
    !>          meaningful part of the Householder vectors. N-M >= L >= 0.
    !>

    A

    !>          A is COMPLEX array, dimension (LDA,N)
    !>          On entry, the leading M-by-N upper trapezoidal part of the
    !>          array A must contain the matrix to be factorized.
    !>          On exit, the leading M-by-M upper triangular part of A
    !>          contains the upper triangular matrix R, and elements N-L+1 to
    !>          N of the first M rows of A, with the array TAU, represent the
    !>          unitary matrix Z as a product of M elementary reflectors.
    !>

    LDA

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

    TAU

    !>          TAU is COMPLEX array, dimension (M)
    !>          The scalar factors of the elementary reflectors.
    !>

    WORK

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

    !>
    !>  The factorization is obtained by Householder's method.  The kth
    !>  transformation matrix, Z( k ), which is used to introduce zeros into
    !>  the ( m - k + 1 )th row of A, is given in the form
    !>
    !>     Z( k ) = ( I     0   ),
    !>              ( 0  T( k ) )
    !>
    !>  where
    !>
    !>     T( k ) = I - tau*u( k )*u( k )**H,   u( k ) = (   1    ),
    !>                                                 (   0    )
    !>                                                 ( z( k ) )
    !>
    !>  tau is a scalar and z( k ) is an l element vector. tau and z( k )
    !>  are chosen to annihilate the elements of the kth row of A2.
    !>
    !>  The scalar tau is returned in the kth element of TAU and the vector
    !>  u( k ) in the kth row of A2, such that the elements of z( k ) are
    !>  in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
    !>  the upper triangular part of A1.
    !>
    !>  Z is given by
    !>
    !>     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
    !>

  subroutine dlatrz (integer m, integer n, integer l, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work)

DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.

Purpose:

    !>
    !> DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
    !> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z, by means
    !> of orthogonal transformations.  Z is an (M+L)-by-(M+L) orthogonal
    !> matrix and, R and A1 are M-by-M upper triangular matrices.
    !>

Parameters
    M

    !>          M is INTEGER
    !>          The number of rows of the matrix A.  M >= 0.
    !>

    N

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

    L

    !>          L is INTEGER
    !>          The number of columns of the matrix A containing the
    !>          meaningful part of the Householder vectors. N-M >= L >= 0.
    !>

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA,N)
    !>          On entry, the leading M-by-N upper trapezoidal part of the
    !>          array A must contain the matrix to be factorized.
    !>          On exit, the leading M-by-M upper triangular part of A
    !>          contains the upper triangular matrix R, and elements N-L+1 to
    !>          N of the first M rows of A, with the array TAU, represent the
    !>          orthogonal matrix Z as a product of M elementary reflectors.
    !>

    LDA

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

    TAU

    !>          TAU is DOUBLE PRECISION array, dimension (M)
    !>          The scalar factors of the elementary reflectors.
    !>

    WORK

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

    !>
    !>  The factorization is obtained by Householder's method.  The kth
    !>  transformation matrix, Z( k ), which is used to introduce zeros into
    !>  the ( m - k + 1 )th row of A, is given in the form
    !>
    !>     Z( k ) = ( I     0   ),
    !>              ( 0  T( k ) )
    !>
    !>  where
    !>
    !>     T( k ) = I - tau*u( k )*u( k )**T,   u( k ) = (   1    ),
    !>                                                 (   0    )
    !>                                                 ( z( k ) )
    !>
    !>  tau is a scalar and z( k ) is an l element vector. tau and z( k )
    !>  are chosen to annihilate the elements of the kth row of A2.
    !>
    !>  The scalar tau is returned in the kth element of TAU and the vector
    !>  u( k ) in the kth row of A2, such that the elements of z( k ) are
    !>  in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
    !>  the upper triangular part of A1.
    !>
    !>  Z is given by
    !>
    !>     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
    !>

  subroutine slatrz (integer m, integer n, integer l, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work)

SLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.

Purpose:

    !>
    !> SLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
    !> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z, by means
    !> of orthogonal transformations.  Z is an (M+L)-by-(M+L) orthogonal
    !> matrix and, R and A1 are M-by-M upper triangular matrices.
    !>

Parameters
    M

    !>          M is INTEGER
    !>          The number of rows of the matrix A.  M >= 0.
    !>

    N

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

    L

    !>          L is INTEGER
    !>          The number of columns of the matrix A containing the
    !>          meaningful part of the Householder vectors. N-M >= L >= 0.
    !>

    A

    !>          A is REAL array, dimension (LDA,N)
    !>          On entry, the leading M-by-N upper trapezoidal part of the
    !>          array A must contain the matrix to be factorized.
    !>          On exit, the leading M-by-M upper triangular part of A
    !>          contains the upper triangular matrix R, and elements N-L+1 to
    !>          N of the first M rows of A, with the array TAU, represent the
    !>          orthogonal matrix Z as a product of M elementary reflectors.
    !>

    LDA

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

    TAU

    !>          TAU is REAL array, dimension (M)
    !>          The scalar factors of the elementary reflectors.
    !>

    WORK

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

    !>
    !>  The factorization is obtained by Householder's method.  The kth
    !>  transformation matrix, Z( k ), which is used to introduce zeros into
    !>  the ( m - k + 1 )th row of A, is given in the form
    !>
    !>     Z( k ) = ( I     0   ),
    !>              ( 0  T( k ) )
    !>
    !>  where
    !>
    !>     T( k ) = I - tau*u( k )*u( k )**T,   u( k ) = (   1    ),
    !>                                                 (   0    )
    !>                                                 ( z( k ) )
    !>
    !>  tau is a scalar and z( k ) is an l element vector. tau and z( k )
    !>  are chosen to annihilate the elements of the kth row of A2.
    !>
    !>  The scalar tau is returned in the kth element of TAU and the vector
    !>  u( k ) in the kth row of A2, such that the elements of z( k ) are
    !>  in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
    !>  the upper triangular part of A1.
    !>
    !>  Z is given by
    !>
    !>     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
    !>

  subroutine zlatrz (integer m, integer n, integer l, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16, dimension( * ) work)

ZLATRZ factors an upper trapezoidal matrix by means of unitary transformations.

Purpose:

    !>
    !> ZLATRZ factors the M-by-(M+L) complex upper trapezoidal matrix
    !> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z by means
    !> of unitary transformations, where  Z is an (M+L)-by-(M+L) unitary
    !> matrix and, R and A1 are M-by-M upper triangular matrices.
    !>

Parameters
    M

    !>          M is INTEGER
    !>          The number of rows of the matrix A.  M >= 0.
    !>

    N

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

    L

    !>          L is INTEGER
    !>          The number of columns of the matrix A containing the
    !>          meaningful part of the Householder vectors. N-M >= L >= 0.
    !>

    A

    !>          A is COMPLEX*16 array, dimension (LDA,N)
    !>          On entry, the leading M-by-N upper trapezoidal part of the
    !>          array A must contain the matrix to be factorized.
    !>          On exit, the leading M-by-M upper triangular part of A
    !>          contains the upper triangular matrix R, and elements N-L+1 to
    !>          N of the first M rows of A, with the array TAU, represent the
    !>          unitary matrix Z as a product of M elementary reflectors.
    !>

    LDA

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

    TAU

    !>          TAU is COMPLEX*16 array, dimension (M)
    !>          The scalar factors of the elementary reflectors.
    !>

    WORK

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details:

    !>
    !>  The factorization is obtained by Householder's method.  The kth
    !>  transformation matrix, Z( k ), which is used to introduce zeros into
    !>  the ( m - k + 1 )th row of A, is given in the form
    !>
    !>     Z( k ) = ( I     0   ),
    !>              ( 0  T( k ) )
    !>
    !>  where
    !>
    !>     T( k ) = I - tau*u( k )*u( k )**H,   u( k ) = (   1    ),
    !>                                                 (   0    )
    !>                                                 ( z( k ) )
    !>
    !>  tau is a scalar and z( k ) is an l element vector. tau and z( k )
    !>  are chosen to annihilate the elements of the kth row of A2.
    !>
    !>  The scalar tau is returned in the kth element of TAU and the vector
    !>  u( k ) in the kth row of A2, such that the elements of z( k ) are
    !>  in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
    !>  the upper triangular part of A1.
    !>
    !>  Z is given by
    !>
    !>     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
    !>

Author

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