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).

Definition at line 198 of file clatrd.f.

  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).

Definition at line 197 of file dlatrd.f.

  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).

Definition at line 197 of file slatrd.f.

  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).

Definition at line 198 of file zlatrd.f.

Author

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