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PubNumAnalysisIpynb/InterpolationLib.py
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InterpolationLib.py
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#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Thu Jan 30
@author: Simone Deparis
"""
import numpy as np
# Defining the mxn Vandermonde matrix
def VandermondeMatrix(x, m=0):
# Input
# x : +1 array with interpolation nodes
# m : degree of the polynomial. If empty, chooses m=size(x)-1
# Output
# Matrix of Vandermonde of size m x n
n = x.size - 1
if (m == 0):
m = n
# The following is a trick to put toghether the matrix A. Don't need to learn by heart ...
# np.tile(x, (n+1, 1)) : repete n+1 times of the array x
# .T : transpose
# np.linspace(0,n,n+1) : [0,...,n+1]
# np.tile : again: repete n+1 times the array [0,...,n+1]
# np.power : element by element power funcion
A = np.power( np.tile(x, (m+1, 1)).T , np.tile(np.linspace(0,m,m+1), (n+1, 1)))
return A
# Defining the Lagrange basis functions
def LagrangeBasis(t,k,z):
# the input variables are:
# t : the interpolatory points
# k : which basis function
# z : where to evaluate the function
# careful, there are n+1 interpolation points!
n = t.size - 1
# init result to one, of same type and size as z
result = np.zeros_like(z) + 1
# first few checks on k:
if (type(k) != int) or (t.size < 1) or (k > n) or (k < 0):
raise ValueError('Lagrange basis needs a positive integer k, smaller than the size of t')
# loop on n to compute the product
for j in range(0,n+1) :
if (j == k) :
continue
if (t[k] == t[j]) :
raise ValueError('Lagrange basis: all the interpolation points need to be distinct')
result = result * (z - t[j]) / (t[k] - t[j])
return result
# Piece-wise linear interpolation, equidistributed points
def PiecewiseLinearInterpolation(a,b,N,f,z):
# the input variables are:
# a,b : x[0] = a, x[n] = b
# f : the corresponding data at the points x
# z : where to evaluate the function
def localLinearInterpolation (a, H, x, y, zk):
# first find out in which interval we are. Easy here since equidistributed point
i = int( (zk-a)//H )
# if we are not in the interval, return constant function
if x[i] == zk :
return y[i]
# use formula for local linear interpolation
return y[i] + (y[i+1] - y[i])/(x[i+1] - x[i])* (zk - x[i])
# careful, there are N+1 points
H = (b-a)/N
if np.isclose(H,0) :
raise ValueError('PiecewiseLinearInterpolation : a and b are too close')
x = np.linspace(a,b,N+1)
# if f is a function, we need to evaluate it at alle the locations
from types import FunctionType
if isinstance(f, FunctionType) :
y = f(x)
else :
y = f
# if we want to evaluate the interpolating function in a single point
if np.ndim(z) == 0 :
return localLinearInterpolation (b, H, x, y, z)
# if we are here, it means that we need to evaluate the interpolation on many points.
# init result to zero, of same type and size as z
result = np.zeros_like(z)
# The following part is not optimized
for k in range(z.size) :
result[k] = localLinearInterpolation (a, H, x, y, z[k])
return result
# Piece-wise quadratic interpolation, equidistributed points
def PiecewiseQuadraticInterpolation(a,b,N,f,z):
# the input variables are:
# a,b : x[0] = a, x[n] = b
# f : the corresponding data at the points x
# only works if fis a fonction
# z : where to evaluate the function
def localQuadraticInterpolation (a, H, x, f, zk):
# first find out in which interval we are. Easy here since equidistributed point
i = int( (zk-a)//H )
# if we are not in the interval, return constant function
if x[i] == zk :
return f(x[i])
t = np.array([ x[i], (x[i]+x[i+1])/2, x[i+1]] )
# use formula for local quadratic interpolation
return LagrangeInterpolation(t,f(t),zk)
# careful, there are N+1 points
H = (b-a)/N
if np.isclose(H,0) :
raise ValueError('PiecewiseQuadraticInterpolation : a and b are too close')
x = np.linspace(a,b,N+1)
# if f is a function, we need to evaluate it at alle the locations
# from types import FunctionType
# if ~isinstance(f, FunctionType) :
# raise ValueError('PiecewiseQuadraticInterpolation : works only with a given function')
# if we want to evaluate the interpolating function in a single point
if np.ndim(z) == 0 :
return localQuadraticInterpolation (b, H, x, f, z)
# if we are here, it means that we need to evaluate the interpolation on many points.
# init result to zero, of same type and size as z
result = np.zeros_like(z)
# The following part is not optimized
for k in range(z.size) :
result[k] = localQuadraticInterpolation (a, H, x, f, z[k])
return result
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