E104: Multidimensional Linear Interpolation

Author(s): C. Letertre	Library: KERNLIB
Submitter: B. Schorr	Submitted: 17.05.1971
Language: Fortran	Revised: 27.11.1984

Function subprogram FINT uses repeated linear interpolation to evaluate a function ) 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 be equally spaced.

Structure:

FUNCTION subprogram
User Entry Names: FINT
Files Refernced: Printer
External References: KERMTR, ABEND

Usage:

In any arithmetic expression,

FINT(N,X,NA,A,F)

has an approximate value of .

N

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

X

(REAL) One-dimensional array. , must contain the coordinates of the point at which the interpolation is to be performed.

NA

(INTEGER) One-dimensional array. For must be equal to the number of numerical values of variable which are stored in array A.

A

(REAL) One-dimensional array of length not less than the sum of . The first NA(1) elements of A must contain numerical values of the first variable in strictly increasing order, the next NA(2) elements of A must contain numerical values of the second variable 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:
.

If any coordinate of the given point X lies outside the range of the corresponding table arguments, the interpolation for this coordinate is replaced by an extrapolation based on the two nearest table arguments, with consequent loss of accuracy.

Method:

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

Let be the tabulated values of x. Let be the tabulated values of y.
Let i and j be the subscripts for which .
Then compute:

Restrictions:

1. . FINT is set equal to zero if N is not in this range.
2. 
3. The table arguments for each variable must be in strictly increasing order.

There is no test for conditions 2 or 3.

Error handling:

E104.1: or . 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 for , 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)
      ...