E105: Function Interpolation

Author(s): F. James	Library: KERNLIB
Submitter: G.A. Erskine	Submitted: 19.07.1973
Language: Fortran	Revised: 27.11.1984

Function subprogram DIVDIF interpolates in a table of arguments and function values , using an interpolating polynomial of specified degree which passes through tabular points which are symmetrically-positioned around the interpolation argument. The table arguments need not be equidistant.

Structure:

FUNCTION subprogram
User Entry Names: DIVDIF
Files Referenced: Printer
External References: KERMTR, ABEND

Usage:

In any arithmetic expression,

DIVDIF(F,A,N,X,M)

has an approximate value of .

F

(REAL) One-dimensional array. F(j) must be equal to the value at A(j) of the function to be interpolated, .

A

(REAL) One-dimensional array. A(j) must be equal to the table argument .

N

(INTEGER) Number of values in arrays F and A.

X

(REAL) Argument at which the interpolating polynomial is to be evaluated.

M

(INTEGER) Requested degree of the interpolating polynomial. If M exceeds the interpolation is carried out using a polynomial of degree instead of M. The original value of M is unchanged on exit.

Method:

Newton's divided difference formula is used. Except when X lies near one end of the table (in which case unsymmetrically-situated interpolation points are used), the interpolation procedure is as follows:
M odd:
An interpolating polynomial passing through successive points symmetrically placed with respect to X is used.
M even:
The mean of two interpolating polynomials is used, each passing through successive points , one polynomial having an extra point to the left of X, the other having an extra point to the right of X.
If X lies too close to either end of the table for symmetrically-positioned tabular values to be used, the values at the end of the table are used. If X lies outside the range of the table, the interpolation becomes an extrapolation, with corresponding loss of accuracy.

Restrictions:

The argument values must be in either strictly increasing order or strictly decreasing order. No error message is printed if this is not true.

Error handling:

Error E105.1: or . DIVDIF is set equal to zero and a message is printed unless subroutine KERSET (N001) has been called.

Notes:

See also the write-up for POLINT (E100).