03 December 2008

Interpolation in OCaml

I just wrote some code to do polynomial interpolation in O'Caml. The full darcs repository is at http://web.mit.edu/farr/www/onumerics/; here's the guts of the code. Eventually (depending on my time and level of continued interest), this may turn into a more general O'Caml numerical library. As usual, it's released under the GPL.

(*  interp.ml: Interpolation.   
    Copyright (C) 2008 Will M. Farr 

    This program is free software: you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
    the Free Software Foundation, either version 3 of the License, or
    (at your option) any later version.

    This program is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    GNU General Public License for more details.

    You should have received a copy of the GNU General Public License
    along with this program.  If not, see .

let interp xs ys x = 
  let n = Array.length xs in 
  assert(Array.length ys = n);
  let ytab = Array.make n ys in 
  for i = 1 to n - 1 do 
    let m = n - i in 
    let last_col = ytab.(i-1) and 
        this_col = Array.make m 0.0 in 
    for j = 0 to m - 1 do 
      let xlow = xs.(j) and 
          xhigh = xs.(j+i) and 
          ylow = last_col.(j) and 
          yhigh = last_col.(j+1) in 
      this_col.(j) <- (x-.xlow)*.yhigh/.(xhigh-.xlow) +. (x-.xhigh)*.ylow/.(xlow-.xhigh)
    ytab.(i) <- this_col

Here's how it works. Initially, we have two arrays of points, xs and ys, and a point at which we want to evaluate the interpolated polynomial, x. This means we have an array of 0-order interpolating polynomials (the constants ys). We can use a linear interpolation between neighboring points to construct the values of first-order interpolating polynomials at x. Then we can linearly interpolate between the neighboring first-order polynomials to construct second-order interpolating polynomials. We iterate this process until we have a single value, which is the value of the n-point interpolating polynomial at x. In the case of four points, here is a graphical representation of the procedure:

x0: y0
x1: y1     y012
       y12      y0123
x2: y2     y123
x3: y3 
The y's subscripts indicate the points involved in the interpolation. Each y is a linear interpolation of the y values above and below it to the left at the corresponding x points.

11 June 2008

Very Basic Matrix Library for PLT Scheme

I just uploaded to PLaneT a very simple matrix library. The library provides a matrix datastructure, iterators over matrices and vectors, algebraic operations on matrices, and matrix-vector and matrix-matrix products. There is also an initial module import language which provides all of scheme except the +, -, * and / operations, which instead are generalized for all reasonable combinations of scalar, vector and matrix operands.

I feel like there is a need for this sort of library in PLT Scheme, even though its set of operations is pretty impoverished. Maybe people can build on it to do actual linear algebra in Scheme. I always find myself re-implementing these sorts of trivial matrix operations (since they're simple, it doesn't take very long); maybe now I'll just grab the PLaneT library instead.

By the way, I'm particularly proud of the sequence nature of the matrix struct, and the for/vector, for/matrix, and in-matrix forms. These take advantage of the new machinery for user-defined iterations and comprehensions in PLT Scheme version 4 (have a look at the source code if you're interested in how it's done). (Version 4, by the way, represents a tremendous improvement from version 3, in every part of the system, but most particularly in the documentation tools---have a look at the new documentation for my library here to see what I mean.)