E104 Multidimensional Linear Interpolation

Function subprogram FINT uses repeated linear interpolation to evaluate a function $f(x$₁,x₂,...,x_n ) of n variables which has been tabulated at the nodes of an n-dimensional rectangular grid. It is not necessary that the table arguments corresponding to any coordinate $x$_i be equally spaced.

Structure:

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

Usage:

In any arithmetic expression, FINT(N,X,NA,A,F)

has an approximate value of $f(X$₁,X₂,...,X_n) .

N

( INTEGER) Number of coordinates n required to define the function f.

X

( REAL) One-dimensional array. $X(j),\hspace{0.17em}(j=1,2,...,N)$ , must contain the coordinates of the point at which the interpolation is to be performed.

NA

( INTEGER) One-dimensional array. For $j=1,2,...,N,\hspace{0.17em}NA(j)$ must be equal to the number of numerical values of variable $x$_j which are stored in array A.

A

( REAL) One-dimensional array of length not less than the sum of $NA(1),...,NA(N)$ . The first NA(1) elements of A must contain numerical values $a$₁₁, a₁₂,... of the first variable $x$₁ in strictly increasing order, the next NA(2) elements of A must contain numerical values $a$₂₁, a₂₂,... of the second variable $x$₂ in strictly increasing order, and so on.

F

( REAL) Multidimensional array with dimensions NA(1), NA(2), ..., NA(N), containing values of the function f at the nodes of the rectangular grid defined by A:
$F(i,j,...,m)=\; f(a$_1i,a_2j,...,a_nm),(i=1,2,...,NA(1),...;m=1,2,...,NA(N)) .

Method:

Repeated linear interpolation with respect to variables $x$₁,x₂,... within the grid cell which contains the given point X. For n=2, with $(x$₁,x₂) replaced by $(x,y)$ for clarity, the procedure is equivalent to the following:

Let $a$₁,a₂,... be the tabulated values of x. Let $b$₁,b₂,... be the tabulated values of y.
Let i and j be the subscripts for which $a$_i≤x<a_i+1, b_j≤y <b_j+1 .
Then compute:

$t$	$=$	$(x-a)/(a$i+1-a),
$g$j	$=$	$(1-t)f(a$i,bj)+t f(ai+1,bj),
$g$j+1	$=$	$(1-t)f(a$i,bj+1+t f(ai+1,bj+1),
$u$	$=$	$(y-b)/(b$j+1-b),
$f$appr	$=$	$(1-u)\; g$j+ u gj+1

Restrictions:

1. $1\le N\; \le 5$ . FINT is set equal to zero if N is not in this range.
2. $NA(j)\; \ge 1,\hspace{0.17em}(j=1,2,...,N).$

3. The table arguments for each variable must be in strictly increasing order.

Error handling:

E104.1: or $N>5$ . FINT is set equal to zero, and a message is printed unless subroutine KERSET (N001) has been called.

Examples:

Given a function of two variables $g(x,y)$ defined by a FUNCTION subprogram G, to construct a table of values of $f$_km= g(k,logm) for $k=1,2,...,10,\; m=1,2,...,15$ , and to interpolate in this table to set GINT equal to an approximate value of $g(1.7,2.9)$ . The program is written in a form which allows generalization to functions of more than two variables.

      PARAMETER (NA1=10,NA2=15)
      DIMENSION X(2),NA(2),A(NA1+NA2),F(NA1,NA2)
      DATA NA/NA1,NA2/
C     STORE ARGUMENT ARRAY
      K1=0
      K2=K1+NA1
      DO 1 J = 1,MAX(NA1,NA2)
        IF (J .LE. NA1) A(J+K1)=SQRT(FLOAT(J))
        IF (J .LE. NA2) A(J+K2)=LOG(FLOAT(J))
    1 CONTINUE
C     STORE FUNCTION ARRAY
      DO 3 J1 = 1,NA1
        DO 2 J2 = 1,NA2
          F(J1,J2)=G(A(J1+K1),A(J2+K2))
    2   CONTINUE
    3 CONTINUE
C     INTERPOLATE IN TABLE
      X(1)=1.7
      X(2)=2.9
      GINT=FINT(2,X,NA,A,F)
      ...

E105