hetd2(3)

NAME
hetd2 - {he,sy}td2: reduction to tridiagonal, level 2

SYNOPSIS

  Functions
subroutine chetd2 (uplo, n, a, lda, d, e, tau, info)
CHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm).
subroutine dsytd2 (uplo, n, a, lda, d, e, tau, info)
DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).
subroutine ssytd2 (uplo, n, a, lda, d, e, tau, info)
SSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).
subroutine zhetd2 (uplo, n, a, lda, d, e, tau, info)
ZHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm).

Detailed Description

Function Documentation

  subroutine chetd2 (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( * ) tau, integer info)

CHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm).

Purpose:

     CHETD2 reduces a complex Hermitian matrix A to real symmetric
     tridiagonal form T by a unitary similarity transformation:
     Q**H * A * Q = T.

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.  N >= 0.

    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 diagonal and first superdiagonal
              of A are overwritten by the corresponding elements of the
              tridiagonal matrix T, and the elements above the first
              superdiagonal, with the array TAU, represent the unitary
              matrix Q as a product of elementary reflectors; if UPLO
              = 'L', the diagonal and first subdiagonal of A are over-
              written by the corresponding elements of the tridiagonal
              matrix T, 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).

    D

              D is REAL array, dimension (N)
              The diagonal elements of the tridiagonal matrix T:
              D(i) = A(i,i).

    E

              E is REAL array, dimension (N-1)
              The off-diagonal elements of the tridiagonal matrix T:
              E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

    TAU

              TAU is COMPLEX array, dimension (N-1)
              The scalar factors of the elementary reflectors (see Further
              Details).

    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:

      If UPLO = 'U', the matrix Q is represented as a product of elementary
      reflectors
         Q = H(n-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
      A(1:i-1,i+1), and tau in TAU(i).
      If UPLO = 'L', the matrix Q is represented as a product of elementary
      reflectors
         Q = H(1) H(2) . . . H(n-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 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
      and tau in TAU(i).
      The contents of A on exit are illustrated by the following examples
      with n = 5:
      if UPLO = 'U':                       if UPLO = 'L':
        (  d   e   v2  v3  v4 )              (  d                  )
        (      d   e   v3  v4 )              (  e   d              )
        (          d   e   v4 )              (  v1  e   d          )
        (              d   e  )              (  v1  v2  e   d      )
        (                  d  )              (  v1  v2  v3  e   d  )
      where d and e denote diagonal and off-diagonal elements of T, and vi
      denotes an element of the vector defining H(i).

Definition at line 174 of file chetd2.f.

  subroutine dsytd2 (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) tau, integer info)

DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).

Purpose:

     DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
     form T by an orthogonal similarity transformation: Q**T * A * Q = T.

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.  N >= 0.

    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 diagonal and first superdiagonal
              of A are overwritten by the corresponding elements of the
              tridiagonal matrix T, and the elements above the first
              superdiagonal, with the array TAU, represent the orthogonal
              matrix Q as a product of elementary reflectors; if UPLO
              = 'L', the diagonal and first subdiagonal of A are over-
              written by the corresponding elements of the tridiagonal
              matrix T, 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).

    D

              D is DOUBLE PRECISION array, dimension (N)
              The diagonal elements of the tridiagonal matrix T:
              D(i) = A(i,i).

    E

              E is DOUBLE PRECISION array, dimension (N-1)
              The off-diagonal elements of the tridiagonal matrix T:
              E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

    TAU

              TAU is DOUBLE PRECISION array, dimension (N-1)
              The scalar factors of the elementary reflectors (see Further
              Details).

    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:

      If UPLO = 'U', the matrix Q is represented as a product of elementary
      reflectors
         Q = H(n-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
      A(1:i-1,i+1), and tau in TAU(i).
      If UPLO = 'L', the matrix Q is represented as a product of elementary
      reflectors
         Q = H(1) H(2) . . . H(n-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 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
      and tau in TAU(i).
      The contents of A on exit are illustrated by the following examples
      with n = 5:
      if UPLO = 'U':                       if UPLO = 'L':
        (  d   e   v2  v3  v4 )              (  d                  )
        (      d   e   v3  v4 )              (  e   d              )
        (          d   e   v4 )              (  v1  e   d          )
        (              d   e  )              (  v1  v2  e   d      )
        (                  d  )              (  v1  v2  v3  e   d  )
      where d and e denote diagonal and off-diagonal elements of T, and vi
      denotes an element of the vector defining H(i).

Definition at line 172 of file dsytd2.f.

  subroutine ssytd2 (character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) tau, integer info)

SSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).

Purpose:

     SSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
     form T by an orthogonal similarity transformation: Q**T * A * Q = T.

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.  N >= 0.

    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 diagonal and first superdiagonal
              of A are overwritten by the corresponding elements of the
              tridiagonal matrix T, and the elements above the first
              superdiagonal, with the array TAU, represent the orthogonal
              matrix Q as a product of elementary reflectors; if UPLO
              = 'L', the diagonal and first subdiagonal of A are over-
              written by the corresponding elements of the tridiagonal
              matrix T, 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).

    D

              D is REAL array, dimension (N)
              The diagonal elements of the tridiagonal matrix T:
              D(i) = A(i,i).

    E

              E is REAL array, dimension (N-1)
              The off-diagonal elements of the tridiagonal matrix T:
              E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

    TAU

              TAU is REAL array, dimension (N-1)
              The scalar factors of the elementary reflectors (see Further
              Details).

    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:

      If UPLO = 'U', the matrix Q is represented as a product of elementary
      reflectors
         Q = H(n-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
      A(1:i-1,i+1), and tau in TAU(i).
      If UPLO = 'L', the matrix Q is represented as a product of elementary
      reflectors
         Q = H(1) H(2) . . . H(n-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 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
      and tau in TAU(i).
      The contents of A on exit are illustrated by the following examples
      with n = 5:
      if UPLO = 'U':                       if UPLO = 'L':
        (  d   e   v2  v3  v4 )              (  d                  )
        (      d   e   v3  v4 )              (  e   d              )
        (          d   e   v4 )              (  v1  e   d          )
        (              d   e  )              (  v1  v2  e   d      )
        (                  d  )              (  v1  v2  v3  e   d  )
      where d and e denote diagonal and off-diagonal elements of T, and vi
      denotes an element of the vector defining H(i).

Definition at line 172 of file ssytd2.f.

  subroutine zhetd2 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( * ) tau, integer info)

ZHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm).

Purpose:

     ZHETD2 reduces a complex Hermitian matrix A to real symmetric
     tridiagonal form T by a unitary similarity transformation:
     Q**H * A * Q = T.

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.  N >= 0.

    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 diagonal and first superdiagonal
              of A are overwritten by the corresponding elements of the
              tridiagonal matrix T, and the elements above the first
              superdiagonal, with the array TAU, represent the unitary
              matrix Q as a product of elementary reflectors; if UPLO
              = 'L', the diagonal and first subdiagonal of A are over-
              written by the corresponding elements of the tridiagonal
              matrix T, 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).

    D

              D is DOUBLE PRECISION array, dimension (N)
              The diagonal elements of the tridiagonal matrix T:
              D(i) = A(i,i).

    E

              E is DOUBLE PRECISION array, dimension (N-1)
              The off-diagonal elements of the tridiagonal matrix T:
              E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

    TAU

              TAU is COMPLEX*16 array, dimension (N-1)
              The scalar factors of the elementary reflectors (see Further
              Details).

    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:

      If UPLO = 'U', the matrix Q is represented as a product of elementary
      reflectors
         Q = H(n-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
      A(1:i-1,i+1), and tau in TAU(i).
      If UPLO = 'L', the matrix Q is represented as a product of elementary
      reflectors
         Q = H(1) H(2) . . . H(n-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 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
      and tau in TAU(i).
      The contents of A on exit are illustrated by the following examples
      with n = 5:
      if UPLO = 'U':                       if UPLO = 'L':
        (  d   e   v2  v3  v4 )              (  d                  )
        (      d   e   v3  v4 )              (  e   d              )
        (          d   e   v4 )              (  v1  e   d          )
        (              d   e  )              (  v1  v2  e   d      )
        (                  d  )              (  v1  v2  v3  e   d  )
      where d and e denote diagonal and off-diagonal elements of T, and vi
      denotes an element of the vector defining H(i).

Definition at line 174 of file zhetd2.f.

Author

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