Lagrange interpolation in Python

Question

I want to interpolate a polynomial with the Lagrange method, but this code doesn't work:

def interpolate(x_values, y_values):
    def _basis(j):
        p = [(x - x_values[m])/(x_values[j] - x_values[m]) for m in xrange(k + 1) if m != j]
        return reduce(operator.mul, p)

    assert len(x_values) != 0 and (len(x_values) == len(y_values)), 'x and y cannot be empty and must have the same length'

    k = len(x_values)
    return sum(_basis(j) for j in xrange(k))

I followed Wikipedia, but when I run it I receive an IndexError at line 3!

Thanks

Answer 1

Check the indices, Wikipedia says "k+1 data points", but you're setting k = len(x_values) where it should be k = len(x_values) - 1 if you followed the formula exactly.

Answer 2

Try

def interpolate(x, x_values, y_values):
    def _basis(j):
        p = [(x - x_values[m])/(x_values[j] - x_values[m]) for m in xrange(k) if m != j]
        return reduce(operator.mul, p)
    assert len(x_values) != 0 and (len(x_values) == len(y_values)), 'x and y cannot be empty and must have the same length'
    k = len(x_values)
    return sum(_basis(j)*y_values[j] for j in xrange(k))

You can confirm it as follows:

>>> interpolate(1,[1,2,4],[1,0,2])
1.0
>>> interpolate(2,[1,2,4],[1,0,2])
0.0
>>> interpolate(4,[1,2,4],[1,0,2])
2.0
>>> interpolate(3,[1,2,4],[1,0,2])
0.33333333333333331

So the result is the interpolated value based on the polynomial that goes through the points given. In this case, the 3 points define a parabola and the first 3 tests show that the stated y_value is returned for the given x_value.