I am studying function approximation and while trying to understand/implement polynomial interpolation I've found an example here. I find the code below a good example to understand what is actually going on instead of using ready functions, however it doesn't run:
Defining the interpolation algorithm. Essentially, we are trying to come up with a representation of true f as a linear combination of basis functions (psi-s).
import sympy as sym
def interpolation(f, psi, points):
N = len(psi) - 1 #order of the approximant polynomial
A = sym.zeros((N+1, N+1)) # initiating the square matrix, whose regular element is psi evaluated at each nodes
b = sym.zeros((N+1, 1)) # original function f evaluated at the selected nodes
psi_sym = psi # save symbolic expression
# Turn psi and f into Python functions
x = sym.Symbol('x')
psi = [sym.lambdify([x], psi[i]) for i in range(N+1)]
f = sym.lambdify([x], f)
for i in range(N+1):
for j in range(N+1):
A[i,j] = psi[j](points[i])
b[i,0] = f(points[i])
c = A.LUsolve(b) #finding the accurate weights for each psi
# c is a sympy Matrix object, turn to list
c = [sym.simplify(c[i,0]) for i in range(c.shape[0])]
u = sym.simplify(sum(c[i,0]*psi_sym[i] for i in range(N+1)))
return u, c
True function f:= 10(x-1)^2 -1
, nodes: x0:= 1 + 1/3
and x1:= 1 + 2/3
. Interval: [1,2]
.
x = sym.Symbol('x')
f = 10*(x-1)**2 - 1
psi = [1, x] # approximant polynomial of order 1 (linear approximation)
Omega = [1, 2] #interval
points = [1 + sym.Rational(1,3), 1 + sym.Rational(2,3)]
u, c = interpolation(f, psi, points)
comparison_plot(f, u, Omega)
The code doesn't run. The error occurs in line
A = sym.zeros((N+1, N+1))
Error message: ValueError: (2, 2) is not an integer
But A isn't supposed to be an integer, it is a square matrix whose each element is psi evaluated at each node. f = A*c.