E105 Function Interpolation

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

Structure:

FUNCTION subprogram
User Entry Names: DIVDIF
Files Referenced: Printer
External References: KERMTR (N001), ABEND (Z035)

Usage:

In any arithmetic expression, DIVDIF(F,A,N,X,M)

has an approximate value of $f(X)$ .

F

( REAL) One-dimensional array. F(j) must be equal to the value at A(j) of the function to be interpolated, $(j=1,2,...,N)$ .

A

( REAL) One-dimensional array. A(j) must be equal to the table argument $a$_j,(j=1,2,...,N) .

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 $M$_max=min(10,N-1) the interpolation is carried out using a polynomial of degree $M$_max

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 $M+1$ successive points $(a$_j,f_j) symmetrically placed with respect to X is used.
M even:
The mean of two interpolating polynomials is used, each passing through $M+1$ successive points $(a$_j,f_j) , 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 $M+1$

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 $A(1),A(2),...$ 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).
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E106