CHPTRD(3)      LAPACK routine of NEC Numeric Library Collection      CHPTRD(3)



NAME
       CHPTRD

SYNOPSIS
       SUBROUTINE CHPTRD (UPLO, N, AP, D, E, TAU, INFO)



PURPOSE
            CHPTRD reduces a complex Hermitian matrix A stored in packed form to
            real symmetric tridiagonal form T by a unitary similarity
            transformation: Q**H * A * Q = T.




ARGUMENTS
           UPLO      (input)
                     UPLO is CHARACTER*1
                     = 'U':  Upper triangle of A is stored;
                     = 'L':  Lower triangle of A is stored.

           N         (input)
                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           AP        (input/output)
                     AP is COMPLEX array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the Hermitian matrix
                     A, packed columnwise in a linear array.  The j-th column of A
                     is stored in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
                     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.

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

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

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value






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 AP,
             overwriting A(1:i-1,i+1), and tau is stored 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 AP,
             overwriting A(i+2:n,i), and tau is stored in TAU(i).



LAPACK routine                  31 October 2017                      CHPTRD(3)