latrd(3)

NAME
latrd - latrd: step in hetrd

SYNOPSIS

  Functions

subroutine clatrd (uplo, n, nb, a, lda, e, tau, w, ldw)
CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.
subroutine dlatrd (uplo, n, nb, a, lda, e, tau, w, ldw)
DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.
subroutine slatrd (uplo, n, nb, a, lda, e, tau, w, ldw)
SLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.
subroutine zlatrd (uplo, n, nb, a, lda, e, tau, w, ldw)
ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

Detailed Description

Function Documentation

  subroutine clatrd (character uplo, integer n, integer nb, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) e, complex, dimension( * ) tau, complex, dimension( ldw, * ) w, integer ldw)

CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

Purpose:

    !>
    !> CLATRD reduces NB rows and columns of a complex Hermitian matrix A to
    !> Hermitian tridiagonal form by a unitary similarity
    !> transformation Q**H * A * Q, and returns the matrices V and W which are
    !> needed to apply the transformation to the unreduced part of A.
    !>
    !> If UPLO = 'U', CLATRD reduces the last NB rows and columns of a
    !> matrix, of which the upper triangle is supplied;
    !> if UPLO = 'L', CLATRD reduces the first NB rows and columns of a
    !> matrix, of which the lower triangle is supplied.
    !>
    !> This is an auxiliary routine called by CHETRD.
    !>

Parameters
    UPLO

    !>          UPLO is CHARACTER*1
    !>          Specifies whether the upper or lower triangular part of the
    !>          Hermitian matrix A is stored:
    !>          = 'U': Upper triangular
    !>          = 'L': Lower triangular
    !>

    N

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

    NB

    !>          NB is INTEGER
    !>          The number of rows and columns to be reduced.
    !>

    A

    !>          A is COMPLEX array, dimension (LDA,N)
    !>          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
    !>          n-by-n upper triangular part of A contains the upper
    !>          triangular part of the matrix A, and the strictly lower
    !>          triangular part of A is not referenced.  If UPLO = 'L', the
    !>          leading n-by-n lower triangular part of A contains the lower
    !>          triangular part of the matrix A, and the strictly upper
    !>          triangular part of A is not referenced.
    !>          On exit:
    !>          if UPLO = 'U', the last NB columns have been reduced to
    !>            tridiagonal form, with the diagonal elements overwriting
    !>            the diagonal elements of A; the elements above the diagonal
    !>            with the array TAU, represent the unitary matrix Q as a
    !>            product of elementary reflectors;
    !>          if UPLO = 'L', the first NB columns have been reduced to
    !>            tridiagonal form, with the diagonal elements overwriting
    !>            the diagonal elements of A; the elements below the diagonal
    !>            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).
    !>

    E

    !>          E is REAL array, dimension (N-1)
    !>          If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
    !>          elements of the last NB columns of the reduced matrix;
    !>          if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
    !>          the first NB columns of the reduced matrix.
    !>

    TAU

    !>          TAU is COMPLEX array, dimension (N-1)
    !>          The scalar factors of the elementary reflectors, stored in
    !>          TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
    !>          See Further Details.
    !>

    W

    !>          W is COMPLEX array, dimension (LDW,NB)
    !>          The n-by-nb matrix W required to update the unreduced part
    !>          of A.
    !>

    LDW

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  If UPLO = 'U', the matrix Q is represented as a product of elementary
    !>  reflectors
    !>
    !>     Q = H(n) H(n-1) . . . H(n-nb+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(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
    !>  and tau in TAU(i-1).
    !>
    !>  If UPLO = 'L', the matrix Q is represented as a product of elementary
    !>  reflectors
    !>
    !>     Q = H(1) H(2) . . . H(nb).
    !>
    !>  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 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
    !>  and tau in TAU(i).
    !>
    !>  The elements of the vectors v together form the n-by-nb matrix V
    !>  which is needed, with W, to apply the transformation to the unreduced
    !>  part of the matrix, using a Hermitian rank-2k update of the form:
    !>  A := A - V*W**H - W*V**H.
    !>
    !>  The contents of A on exit are illustrated by the following examples
    !>  with n = 5 and nb = 2:
    !>
    !>  if UPLO = 'U':                       if UPLO = 'L':
    !>
    !>    (  a   a   a   v4  v5 )              (  d                  )
    !>    (      a   a   v4  v5 )              (  1   d              )
    !>    (          a   1   v5 )              (  v1  1   a          )
    !>    (              d   1  )              (  v1  v2  a   a      )
    !>    (                  d  )              (  v1  v2  a   a   a  )
    !>
    !>  where d denotes a diagonal element of the reduced matrix, a denotes
    !>  an element of the original matrix that is unchanged, and vi denotes
    !>  an element of the vector defining H(i).
    !>

  subroutine dlatrd (character uplo, integer n, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) e, double precision, dimension( * ) tau, double precision, dimension( ldw, * ) w, integer ldw)

DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.

Purpose:

    !>
    !> DLATRD reduces NB rows and columns of a real symmetric matrix A to
    !> symmetric tridiagonal form by an orthogonal similarity
    !> transformation Q**T * A * Q, and returns the matrices V and W which are
    !> needed to apply the transformation to the unreduced part of A.
    !>
    !> If UPLO = 'U', DLATRD reduces the last NB rows and columns of a
    !> matrix, of which the upper triangle is supplied;
    !> if UPLO = 'L', DLATRD reduces the first NB rows and columns of a
    !> matrix, of which the lower triangle is supplied.
    !>
    !> This is an auxiliary routine called by DSYTRD.
    !>

Parameters
    UPLO

    !>          UPLO is CHARACTER*1
    !>          Specifies whether the upper or lower triangular part of the
    !>          symmetric matrix A is stored:
    !>          = 'U': Upper triangular
    !>          = 'L': Lower triangular
    !>

    N

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

    NB

    !>          NB is INTEGER
    !>          The number of rows and columns to be reduced.
    !>

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA,N)
    !>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
    !>          n-by-n upper triangular part of A contains the upper
    !>          triangular part of the matrix A, and the strictly lower
    !>          triangular part of A is not referenced.  If UPLO = 'L', the
    !>          leading n-by-n lower triangular part of A contains the lower
    !>          triangular part of the matrix A, and the strictly upper
    !>          triangular part of A is not referenced.
    !>          On exit:
    !>          if UPLO = 'U', the last NB columns have been reduced to
    !>            tridiagonal form, with the diagonal elements overwriting
    !>            the diagonal elements of A; the elements above the diagonal
    !>            with the array TAU, represent the orthogonal matrix Q as a
    !>            product of elementary reflectors;
    !>          if UPLO = 'L', the first NB columns have been reduced to
    !>            tridiagonal form, with the diagonal elements overwriting
    !>            the diagonal elements of A; the elements below the diagonal
    !>            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 >= (1,N).
    !>

    E

    !>          E is DOUBLE PRECISION array, dimension (N-1)
    !>          If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
    !>          elements of the last NB columns of the reduced matrix;
    !>          if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
    !>          the first NB columns of the reduced matrix.
    !>

    TAU

    !>          TAU is DOUBLE PRECISION array, dimension (N-1)
    !>          The scalar factors of the elementary reflectors, stored in
    !>          TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
    !>          See Further Details.
    !>

    W

    !>          W is DOUBLE PRECISION array, dimension (LDW,NB)
    !>          The n-by-nb matrix W required to update the unreduced part
    !>          of A.
    !>

    LDW

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  If UPLO = 'U', the matrix Q is represented as a product of elementary
    !>  reflectors
    !>
    !>     Q = H(n) H(n-1) . . . H(n-nb+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(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
    !>  and tau in TAU(i-1).
    !>
    !>  If UPLO = 'L', the matrix Q is represented as a product of elementary
    !>  reflectors
    !>
    !>     Q = H(1) H(2) . . . H(nb).
    !>
    !>  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 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
    !>  and tau in TAU(i).
    !>
    !>  The elements of the vectors v together form the n-by-nb matrix V
    !>  which is needed, with W, to apply the transformation to the unreduced
    !>  part of the matrix, using a symmetric rank-2k update of the form:
    !>  A := A - V*W**T - W*V**T.
    !>
    !>  The contents of A on exit are illustrated by the following examples
    !>  with n = 5 and nb = 2:
    !>
    !>  if UPLO = 'U':                       if UPLO = 'L':
    !>
    !>    (  a   a   a   v4  v5 )              (  d                  )
    !>    (      a   a   v4  v5 )              (  1   d              )
    !>    (          a   1   v5 )              (  v1  1   a          )
    !>    (              d   1  )              (  v1  v2  a   a      )
    !>    (                  d  )              (  v1  v2  a   a   a  )
    !>
    !>  where d denotes a diagonal element of the reduced matrix, a denotes
    !>  an element of the original matrix that is unchanged, and vi denotes
    !>  an element of the vector defining H(i).
    !>

  subroutine slatrd (character uplo, integer n, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( * ) e, real, dimension( * ) tau, real, dimension( ldw, * ) w, integer ldw)

SLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.

Purpose:

    !>
    !> SLATRD reduces NB rows and columns of a real symmetric matrix A to
    !> symmetric tridiagonal form by an orthogonal similarity
    !> transformation Q**T * A * Q, and returns the matrices V and W which are
    !> needed to apply the transformation to the unreduced part of A.
    !>
    !> If UPLO = 'U', SLATRD reduces the last NB rows and columns of a
    !> matrix, of which the upper triangle is supplied;
    !> if UPLO = 'L', SLATRD reduces the first NB rows and columns of a
    !> matrix, of which the lower triangle is supplied.
    !>
    !> This is an auxiliary routine called by SSYTRD.
    !>

Parameters
    UPLO

    !>          UPLO is CHARACTER*1
    !>          Specifies whether the upper or lower triangular part of the
    !>          symmetric matrix A is stored:
    !>          = 'U': Upper triangular
    !>          = 'L': Lower triangular
    !>

    N

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

    NB

    !>          NB is INTEGER
    !>          The number of rows and columns to be reduced.
    !>

    A

    !>          A is REAL array, dimension (LDA,N)
    !>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
    !>          n-by-n upper triangular part of A contains the upper
    !>          triangular part of the matrix A, and the strictly lower
    !>          triangular part of A is not referenced.  If UPLO = 'L', the
    !>          leading n-by-n lower triangular part of A contains the lower
    !>          triangular part of the matrix A, and the strictly upper
    !>          triangular part of A is not referenced.
    !>          On exit:
    !>          if UPLO = 'U', the last NB columns have been reduced to
    !>            tridiagonal form, with the diagonal elements overwriting
    !>            the diagonal elements of A; the elements above the diagonal
    !>            with the array TAU, represent the orthogonal matrix Q as a
    !>            product of elementary reflectors;
    !>          if UPLO = 'L', the first NB columns have been reduced to
    !>            tridiagonal form, with the diagonal elements overwriting
    !>            the diagonal elements of A; the elements below the diagonal
    !>            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 >= (1,N).
    !>

    E

    !>          E is REAL array, dimension (N-1)
    !>          If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
    !>          elements of the last NB columns of the reduced matrix;
    !>          if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
    !>          the first NB columns of the reduced matrix.
    !>

    TAU

    !>          TAU is REAL array, dimension (N-1)
    !>          The scalar factors of the elementary reflectors, stored in
    !>          TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
    !>          See Further Details.
    !>

    W

    !>          W is REAL array, dimension (LDW,NB)
    !>          The n-by-nb matrix W required to update the unreduced part
    !>          of A.
    !>

    LDW

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  If UPLO = 'U', the matrix Q is represented as a product of elementary
    !>  reflectors
    !>
    !>     Q = H(n) H(n-1) . . . H(n-nb+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(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
    !>  and tau in TAU(i-1).
    !>
    !>  If UPLO = 'L', the matrix Q is represented as a product of elementary
    !>  reflectors
    !>
    !>     Q = H(1) H(2) . . . H(nb).
    !>
    !>  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 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
    !>  and tau in TAU(i).
    !>
    !>  The elements of the vectors v together form the n-by-nb matrix V
    !>  which is needed, with W, to apply the transformation to the unreduced
    !>  part of the matrix, using a symmetric rank-2k update of the form:
    !>  A := A - V*W**T - W*V**T.
    !>
    !>  The contents of A on exit are illustrated by the following examples
    !>  with n = 5 and nb = 2:
    !>
    !>  if UPLO = 'U':                       if UPLO = 'L':
    !>
    !>    (  a   a   a   v4  v5 )              (  d                  )
    !>    (      a   a   v4  v5 )              (  1   d              )
    !>    (          a   1   v5 )              (  v1  1   a          )
    !>    (              d   1  )              (  v1  v2  a   a      )
    !>    (                  d  )              (  v1  v2  a   a   a  )
    !>
    !>  where d denotes a diagonal element of the reduced matrix, a denotes
    !>  an element of the original matrix that is unchanged, and vi denotes
    !>  an element of the vector defining H(i).
    !>

  subroutine zlatrd (character uplo, integer n, integer nb, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) e, complex*16, dimension( * ) tau, complex*16, dimension( ldw, * ) w, integer ldw)

ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

Purpose:

    !>
    !> ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to
    !> Hermitian tridiagonal form by a unitary similarity
    !> transformation Q**H * A * Q, and returns the matrices V and W which are
    !> needed to apply the transformation to the unreduced part of A.
    !>
    !> If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a
    !> matrix, of which the upper triangle is supplied;
    !> if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a
    !> matrix, of which the lower triangle is supplied.
    !>
    !> This is an auxiliary routine called by ZHETRD.
    !>

Parameters
    UPLO

    !>          UPLO is CHARACTER*1
    !>          Specifies whether the upper or lower triangular part of the
    !>          Hermitian matrix A is stored:
    !>          = 'U': Upper triangular
    !>          = 'L': Lower triangular
    !>

    N

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

    NB

    !>          NB is INTEGER
    !>          The number of rows and columns to be reduced.
    !>

    A

    !>          A is COMPLEX*16 array, dimension (LDA,N)
    !>          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
    !>          n-by-n upper triangular part of A contains the upper
    !>          triangular part of the matrix A, and the strictly lower
    !>          triangular part of A is not referenced.  If UPLO = 'L', the
    !>          leading n-by-n lower triangular part of A contains the lower
    !>          triangular part of the matrix A, and the strictly upper
    !>          triangular part of A is not referenced.
    !>          On exit:
    !>          if UPLO = 'U', the last NB columns have been reduced to
    !>            tridiagonal form, with the diagonal elements overwriting
    !>            the diagonal elements of A; the elements above the diagonal
    !>            with the array TAU, represent the unitary matrix Q as a
    !>            product of elementary reflectors;
    !>          if UPLO = 'L', the first NB columns have been reduced to
    !>            tridiagonal form, with the diagonal elements overwriting
    !>            the diagonal elements of A; the elements below the diagonal
    !>            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).
    !>

    E

    !>          E is DOUBLE PRECISION array, dimension (N-1)
    !>          If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
    !>          elements of the last NB columns of the reduced matrix;
    !>          if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
    !>          the first NB columns of the reduced matrix.
    !>

    TAU

    !>          TAU is COMPLEX*16 array, dimension (N-1)
    !>          The scalar factors of the elementary reflectors, stored in
    !>          TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
    !>          See Further Details.
    !>

    W

    !>          W is COMPLEX*16 array, dimension (LDW,NB)
    !>          The n-by-nb matrix W required to update the unreduced part
    !>          of A.
    !>

    LDW

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  If UPLO = 'U', the matrix Q is represented as a product of elementary
    !>  reflectors
    !>
    !>     Q = H(n) H(n-1) . . . H(n-nb+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(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
    !>  and tau in TAU(i-1).
    !>
    !>  If UPLO = 'L', the matrix Q is represented as a product of elementary
    !>  reflectors
    !>
    !>     Q = H(1) H(2) . . . H(nb).
    !>
    !>  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 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
    !>  and tau in TAU(i).
    !>
    !>  The elements of the vectors v together form the n-by-nb matrix V
    !>  which is needed, with W, to apply the transformation to the unreduced
    !>  part of the matrix, using a Hermitian rank-2k update of the form:
    !>  A := A - V*W**H - W*V**H.
    !>
    !>  The contents of A on exit are illustrated by the following examples
    !>  with n = 5 and nb = 2:
    !>
    !>  if UPLO = 'U':                       if UPLO = 'L':
    !>
    !>    (  a   a   a   v4  v5 )              (  d                  )
    !>    (      a   a   v4  v5 )              (  1   d              )
    !>    (          a   1   v5 )              (  v1  1   a          )
    !>    (              d   1  )              (  v1  v2  a   a      )
    !>    (                  d  )              (  v1  v2  a   a   a  )
    !>
    !>  where d denotes a diagonal element of the reduced matrix, a denotes
    !>  an element of the original matrix that is unchanged, and vi denotes
    !>  an element of the vector defining H(i).
    !>

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