SYTD2 - 5.2 English - 68552

Reduction of a real symmetric matrix a to real symmetric tridiagonal form (unblocked algorithm)

Purpose:

    Reduction of a real symmetric matrix a to real symmetric tridiagonal form T by an
orthogonal similarity transformation(unblocked algorithm): Q**T * A * Q = T.

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).

Parameters:

• uplo – [in]

  uplo is char*

  = ‘U’: Upper triangle of a is stored;

  = ‘L’: Lower triangle of a is stored.

• n – [in]

  n is integer*

  The order of the matrix a. n >= 0.

• a – [inout]

  a is float/double 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 – [in]

  lda is integer*

  The leading dimension of the array a. lda >= fla_max(1,n).

• d – [out]

  d is float/double array, dimension (n)

  The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).

• e – [out]

  e is float/double 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 – [out]

  tau is COMPLEX/COMPLEX*16 array, dimension (n-1)

  The scalar factors of the elementary reflectors (see Further Details).

• INFO – [out]

  INFO is INTEGER

  = 0: successful exit

  < 0: if INFO = -i, the i-th argument had an illegal value.