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).
    !>

  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).
    !>

  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).
    !>

  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).
    !>

Author

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