lagrange_value Subroutine

Given DATA_NUM distinct abscissas, T_DATA(1:DATA_NUM), 
the I-th Lagrange polynomial L(I)(T) is defined as the polynomial of 
degree DATA_NUM - 1 which is 1 at T_DATA(I) and 0 at the DATA_NUM - 1 
other abscissas.

A formal representation is:

  L(I)(T) = Product ( 1 <= J <= DATA_NUM, I /= J )
   ( T - T(J) ) / ( T(I) - T(J) )

This routine accepts a set of INTERP_NUM values at which the Lagrange
polynomials should be evaluated.

Given data values P_DATA at each of the abscissas, the value of the
Lagrange interpolating polynomial at each of the interpolation points
is then simple to compute by matrix multiplication:

  P_INTERP(1:INTERP_NUM) = 
    P_DATA(1:DATA_NUM) * L_INTERP(1:DATA_NUM,1:INTERP_NUM)

or, in the case where P is multidimensional:

  P_INTERP(1:DIM_NUM,1:INTERP_NUM) =
    P_DATA(1:DIM_NUM,1:DATA_NUM) * L_INTERP(1:DATA_NUM,1:INTERP_NUM)

This code is distributed under the GNU LGPL license.

03 December 2007

John Burkardt

Input, integer ( kind = 4 ) DATA_NUM, the number of data points.
DATA_NUM must be at least 1.

Input, real ( kind = 8 ) T_DATA(DATA_NUM), the data points.

Input, integer ( kind = 4 ) INTERP_NUM, the number of 
interpolation points.

Input, real ( kind = 8 ) T_INTERP(INTERP_NUM), the 
interpolation points.

Output, real ( kind = 8 ) L_INTERP(DATA_NUM,INTERP_NUM), the values 
of the Lagrange polynomials at the interpolation points.