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dsptrd(3P)		    Sun Performance Library		    dsptrd(3P)

NAME
       dsptrd - reduce a real symmetric matrix A stored in packed form to sym‐
       metric tridiagonal form T by an orthogonal similarity transformation

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

       CHARACTER * 1 UPLO
       INTEGER N, INFO
       DOUBLE PRECISION AP(*), D(*), E(*), TAU(*)

       SUBROUTINE DSPTRD_64(UPLO, N, AP, D, E, TAU, INFO)

       CHARACTER * 1 UPLO
       INTEGER*8 N, INFO
       DOUBLE PRECISION AP(*), D(*), E(*), TAU(*)

   F95 INTERFACE
       SUBROUTINE SPTRD(UPLO, [N], AP, D, E, TAU, [INFO])

       CHARACTER(LEN=1) :: UPLO
       INTEGER :: N, INFO
       REAL(8), DIMENSION(:) :: AP, D, E, TAU

       SUBROUTINE SPTRD_64(UPLO, [N], AP, D, E, TAU, [INFO])

       CHARACTER(LEN=1) :: UPLO
       INTEGER(8) :: N, INFO
       REAL(8), DIMENSION(:) :: AP, D, E, TAU

   C INTERFACE
       #include <sunperf.h>

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

       void  dsptrd_64(char  uplo,  long  n, double *ap, double *d, double *e,
		 double *tau, long *info);

PURPOSE
       dsptrd reduces a real symmetric matrix A stored in packed form to  sym‐
       metric  tridiagonal  form T by an orthogonal similarity transformation:
       Q**T * A * Q = T.

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

       N (input) 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 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	corre‐
		 sponding  elements  of the tridiagonal matrix T, and the ele‐
		 ments above the first superdiagonal, with the array TAU, rep‐
		 resent	 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 subdi‐
		 agonal, 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 ele‐
		 ments 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)
		 = 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, overwrit‐
       ing 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, overwrit‐
       ing A(i+2:n,i), and tau is stored in TAU(i).

				  6 Mar 2009			    dsptrd(3P)
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