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# dsptrd

```
NAME
DSPTRD - reduce a real symmetric matrix A stored in packed
form to symmetric tridiagonal form T by an orthogonal simi-
larity 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 simi-
larity 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.

(N*(N+1)/2)
AP      (input/output) DOUBLE PRECISION array, dimension
On entry, the upper or lower triangle of the sym-
metric 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 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.  D       (output) DOUBLE PRECISION array,
dimension (N) The diagonal elements of the tridiago-
nal matrix T: D(i) = A(i,i).

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

INFO    (output) 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'

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 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'

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).
```