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SUBROUTINE <a name="DSPTRD.1"></a><a href="dsptrd.f.html#DSPTRD.1">DSPTRD</a>( UPLO, N, AP, D, E, TAU, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> CHARACTER UPLO
INTEGER INFO, N
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="DSPTRD.18"></a><a href="dsptrd.f.html#DSPTRD.1">DSPTRD</a> reduces a real symmetric matrix A stored in packed form to
</span><span class="comment">*</span><span class="comment"> symmetric tridiagonal form T by an orthogonal similarity
</span><span class="comment">*</span><span class="comment"> transformation: Q**T * A * Q = T.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> UPLO (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> = 'U': Upper triangle of A is stored;
</span><span class="comment">*</span><span class="comment"> = 'L': Lower triangle of A is stored.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The order of the matrix A. N >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
</span><span class="comment">*</span><span class="comment"> On entry, the upper or lower triangle of the symmetric matrix
</span><span class="comment">*</span><span class="comment"> A, packed columnwise in a linear array. The j-th column of A
</span><span class="comment">*</span><span class="comment"> is stored in the array AP as follows:
</span><span class="comment">*</span><span class="comment"> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
</span><span class="comment">*</span><span class="comment"> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
</span><span class="comment">*</span><span class="comment"> On exit, if UPLO = 'U', the diagonal and first superdiagonal
</span><span class="comment">*</span><span class="comment"> of A are overwritten by the corresponding elements of the
</span><span class="comment">*</span><span class="comment"> tridiagonal matrix T, and the elements above the first
</span><span class="comment">*</span><span class="comment"> superdiagonal, with the array TAU, represent the orthogonal
</span><span class="comment">*</span><span class="comment"> matrix Q as a product of elementary reflectors; if UPLO
</span><span class="comment">*</span><span class="comment"> = 'L', the diagonal and first subdiagonal of A are over-
</span><span class="comment">*</span><span class="comment"> written by the corresponding elements of the tridiagonal
</span><span class="comment">*</span><span class="comment"> matrix T, and the elements below the first subdiagonal, with
</span><span class="comment">*</span><span class="comment"> the array TAU, represent the orthogonal matrix Q as a product
</span><span class="comment">*</span><span class="comment"> of elementary reflectors. See Further Details.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> D (output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment"> The diagonal elements of the tridiagonal matrix T:
</span><span class="comment">*</span><span class="comment"> D(i) = A(i,i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> E (output) DOUBLE PRECISION array, dimension (N-1)
</span><span class="comment">*</span><span class="comment"> The off-diagonal elements of the tridiagonal matrix T:
</span><span class="comment">*</span><span class="comment"> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> TAU (output) DOUBLE PRECISION array, dimension (N-1)
</span><span class="comment">*</span><span class="comment"> The scalar factors of the elementary reflectors (see Further
</span><span class="comment">*</span><span class="comment"> Details).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> INFO (output) INTEGER
</span><span class="comment">*</span><span class="comment"> = 0: successful exit
</span><span class="comment">*</span><span class="comment"> < 0: if INFO = -i, the i-th argument had an illegal value
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Further Details
</span><span class="comment">*</span><span class="comment"> ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If UPLO = 'U', the matrix Q is represented as a product of elementary
</span><span class="comment">*</span><span class="comment"> reflectors
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Q = H(n-1) . . . H(2) H(1).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Each H(i) has the form
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> H(i) = I - tau * v * v'
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where tau is a real scalar, and v is a real vector with
</span><span class="comment">*</span><span class="comment"> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
</span><span class="comment">*</span><span class="comment"> overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If UPLO = 'L', the matrix Q is represented as a product of elementary
</span><span class="comment">*</span><span class="comment"> reflectors
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Q = H(1) H(2) . . . H(n-1).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Each H(i) has the form
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> H(i) = I - tau * v * v'
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where tau is a real scalar, and v is a real vector with
</span><span class="comment">*</span><span class="comment"> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
</span><span class="comment">*</span><span class="comment"> overwriting A(i+2:n,i), and tau is stored in TAU(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Parameters ..
</span> DOUBLE PRECISION ONE, ZERO, HALF
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0,
$ HALF = 1.0D0 / 2.0D0 )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Local Scalars ..
</span> LOGICAL UPPER
INTEGER I, I1, I1I1, II
DOUBLE PRECISION ALPHA, TAUI
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Subroutines ..
</span> EXTERNAL DAXPY, <a name="DLARFG.107"></a><a href="dlarfg.f.html#DLARFG.1">DLARFG</a>, DSPMV, DSPR2, <a name="XERBLA.107"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>
<span class="comment">*</span><span class="comment"> ..
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