Contents


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 * 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 symmetric tridiagonal form T by an orthogonal simi-
     larity 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)
               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)
               = 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).