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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( * )



     #include <sunperf.h>

     void dsptrd(char uplo, int n, double *dap, double *d, double
               *e, double *tau, int *info) ;

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.

     AP        (input/output) DOUBLE PRECISION  array,  dimension
               (N*(N+1)/2)
               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  superdiago-
               nal of A are overwritten by the corresponding ele-
               ments of the tridiagonal matrix T,  and  the  ele-
               ments  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 tri-
               diagonal 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 (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 ille-
               gal 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).

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