
DSPTRD(l) ) DSPTRD(l)
NAME
DSPTRD  reduce a real symmetric matrix A stored in packed form to symmetric tridiagonal
form T by an orthogonal similarity transformation
SYNOPSIS
SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO )
CHARACTER UPLO
INTEGER INFO, N
DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * )
PURPOSE
DSPTRD reduces a real symmetric matrix A stored in packed form to symmetric tridiagonal
form T by an orthogonal similarity transformation: Q**T * A * Q = T.
ARGUMENTS
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise
in a linear array. The jth column of A is stored in the array AP as follows: if
UPLO = 'U', AP(i + (j1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i +
(j1)*(2*nj)/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 orthogonal matrix Q as a product of elementary reflec
tors; 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. D (output) DOUBLE
PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i).
E (output) DOUBLE PRECISION array, dimension (N1)
The offdiagonal 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) DOUBLE PRECISION array, dimension (N1)
The scalar factors of the elementary reflectors (see Further Details).
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
FURTHER DETAILS
If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors
Q = H(n1) . . . H(2) H(1).
Each H(i) has the form
H(i) = I  tau * v * v'
where tau is a real scalar, and v is a real vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i1) is stored on exit in AP, overwriting A(1:i1,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(n1).
Each H(i) has the form
H(i) = I  tau * v * v'
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 AP, overwriting A(i+2:n,i), and
tau is stored in TAU(i).
LAPACK version 3.0 15 June 2000 DSPTRD(l) 
