Question

edit: I’m not looking for you to debug this code. If you are familiar with this well-known algorithm, then you may be able to help. Please note that the algorithm produces the coefficients correctly.

This code for cubic spline interpolation is producing linear splines and I can’t seem to figure out why (yet). The algorithm comes from Burden’s Numerical Analysis, which is just about identical to the pseudo code here, or you can find that book from a link in the comments (see chapter 3, it’s worth having anyway). The code is producing the correct coefficients; I believe that I am misunderstanding the implementation. Any feedback is greatly appreciated. Also, i’m new to programming, so any feedback on how bad my coding is also welcome. I tried uploading pics of the linear system in terms of h, a, and c, but as a new user i can not. If you want a visual of the tridiagonal linear system that the algorithm solves, and which is set up by the var alpha, see the link in the comments for the book, see chap 3. The system is strictly diagonally dominant, so we know there exists a unique solution c0,…,cn. Once we know the ci values, the other coefficients follow.

import matplotlib.pyplot as plt

# need some zero vectors...
def zeroV(m):
    z = [0]*m
    return(z)

 #INPUT: n; x0, x1, ... ,xn; a0 = f(x0), a1 =f(x1), ... , an = f(xn).
 def cubic_spline(n, xn, a, xd):
    """function cubic_spline(n,xn, a, xd) interpolates between the knots
       specified by lists xn and a. The function computes the coefficients
       and outputs the ranges of the piecewise cubic splines."""        

    h = zeroV(n-1)

    # alpha will be values in a system of eq's that will allow us to solve for c
    # and then from there we can find b, d through substitution.
    alpha = zeroV(n-1)

    # l, u, z are used in the method for solving the linear system
    l = zeroV(n+1)
    u = zeroV(n)
    z = zeroV(n+1)

    # b, c, d will be the coefficients along with a.
    b = zeroV(n)     
    c = zeroV(n+1)
    d = zeroV(n)    

for i in range(n-1):
    # h[i] is used to satisfy the condition that 
    # Si+1(xi+l) = Si(xi+l) for each i = 0,..,n-1
    # i.e., the values at the knots are "doubled up"
    h[i] = xn[i+1]-xn[i]  

for i in range(1, n-1):
    # Sets up the linear system and allows us to find c.  Once we have 
    # c then b and d follow in terms of it.
    alpha[i] = (3./h[i])*(a[i+1]-a[i])-(3./h[i-1])*(a[i] - a[i-1])

# I, II, (part of) III Sets up and solves tridiagonal linear system...
# I   
l[0] = 1      
u[0] = 0      
z[0] = 0

# II
for i in range(1, n-1):
    l[i] = 2*(xn[i+1] - xn[i-1]) - h[i-1]*u[i-1]
    u[i] = h[i]/l[i]
    z[i] = (alpha[i] - h[i-1]*z[i-1])/l[i]

l[n] = 1
z[n] = 0
c[n] = 0

# III... also find b, d in terms of c.
for j in range(n-2, -1, -1):      
    c[j] = z[j] - u[j]*c[j+1]
    b[j] = (a[j+1] - a[j])/h[j] - h[j]*(c[j+1] + 2*c[j])/3.
    d[j] = (c[j+1] - c[j])/(3*h[j])   

# This is my only addition, which is returning values for Sj(x). The issue I'm having
# is related to this implemention, i suspect.
for j in range(n-1): 
    #OUTPUT:S(x)=Sj(x)= aj + bj(x - xj) + cj(x - xj)^2 + dj(x - xj)^3; xj <= x <= xj+1)
    return(a[j] + b[j]*(xd - xn[j]) + c[j]*((xd - xn[j])**2) + d[j]*((xd - xn[j])**3))

For the bored, or overachieving…

Here is code for testing, the interval is x: [1, 9], y:[0, 19.7750212]. The test function is xln(x), so we start 1 and increase by .1 up to 9.

ln = [] 
ln_dom = [] 
cub = [] 
step = 1. 
X=[1., 9.] 
FX=[0, 19.7750212]
while step <= 9.:
    ln.append(step*log(step))
    ln_dom.append(step)
    cub.append(cubic_spline(2, x, fx, step))
    step += 0.1

…and for plotting:

plt.plot(ln_dom, cub, color='blue')
plt.plot(ln_dom, ln, color='red')
plt.axis([1., 9., 0, 20], 'equal')
plt.axhline(y=0, color='black')
plt.axvline(x=0, color='black')
plt.show()

Answer 1

Ok, got this working. The problem was in my implementation. I got it working with a different approach, where the splines are constructed individually instead of continuously. This is fully functioning cubic spline interpolation by method of first constructing the coefficients of the spline polynomials (which is 99% of the work), then implementing them. Obviously this is not the only way to do it. I may work on a different approach and post that if there is interest. One thing that would clarify the code would be an image of the linear system that is solved, but i can’t post pics until my rep gets up to 10. If you want to go deeper into the algorithm, see the text book link in the comments above.

import matplotlib.pyplot as plt
from pylab import arange
from math import e
from math import pi
from math import sin
from math import cos
from numpy import poly1d

# need some zero vectors...
def zeroV(m):
    z = [0]*m
    return(z)

#INPUT: n; x0, x1, ... ,xn; a0 = f(x0), a1 =f(x1), ... , an = f(xn).
def cubic_spline(n, xn, a):
"""function cubic_spline(n,xn, a, xd) interpolates between the knots
   specified by lists xn and a. The function computes the coefficients
   and outputs the ranges of the piecewise cubic splines."""        

    h = zeroV(n-1)

    # alpha will be values in a system of eq's that will allow us to solve for c
    # and then from there we can find b, d through substitution.
    alpha = zeroV(n-1)

    # l, u, z are used in the method for solving the linear system
    l = zeroV(n+1)
    u = zeroV(n)
    z = zeroV(n+1)

    # b, c, d will be the coefficients along with a.
    b = zeroV(n)     
    c = zeroV(n+1)
    d = zeroV(n)    

    for i in range(n-1):
        # h[i] is used to satisfy the condition that 
        # Si+1(xi+l) = Si(xi+l) for each i = 0,..,n-1
        # i.e., the values at the knots are "doubled up"
        h[i] = xn[i+1]-xn[i]  

    for i in range(1, n-1):
        # Sets up the linear system and allows us to find c.  Once we have 
        # c then b and d follow in terms of it.
        alpha[i] = (3./h[i])*(a[i+1]-a[i])-(3./h[i-1])*(a[i] - a[i-1])

    # I, II, (part of) III Sets up and solves tridiagonal linear system...
    # I   
    l[0] = 1      
    u[0] = 0      
    z[0] = 0

    # II
    for i in range(1, n-1):
        l[i] = 2*(xn[i+1] - xn[i-1]) - h[i-1]*u[i-1]
        u[i] = h[i]/l[i]
        z[i] = (alpha[i] - h[i-1]*z[i-1])/l[i]

    l[n] = 1
    z[n] = 0
    c[n] = 0

    # III... also find b, d in terms of c.
    for j in range(n-2, -1, -1):      
        c[j] = z[j] - u[j]*c[j+1]
        b[j] = (a[j+1] - a[j])/h[j] - h[j]*(c[j+1] + 2*c[j])/3.
        d[j] = (c[j+1] - c[j])/(3*h[j]) 

    # Now that we have the coefficients it's just a matter of constructing
    # the appropriate polynomials and graphing.
    for j in range(n-1):
        cub_graph(a[j],b[j],c[j],d[j],xn[j],xn[j+1])

    plt.show()

def cub_graph(a,b,c,d, x_i, x_i_1):
    """cub_graph takes the i'th coefficient set along with the x[i] and x[i+1]'th
       data pts, and constructs the polynomial spline between the two data pts using
       the poly1d python object (which simply returns a polynomial with a given root."""

    # notice here that we are just building the cubic polynomial piece by piece
    root = poly1d(x_i,True)
    poly = 0
    poly = d*(root)**3
    poly = poly + c*(root)**2
    poly = poly + b*root
    poly = poly + a

    # Set up our domain between data points, and plot the function
    pts = arange(x_i,x_i_1, 0.001)
    plt.plot(pts, poly(pts), '-')
    return

If you want to test it, here’s some data you can use to get started, which comes from the
function 1.6e^(-2x)sin(3*pi*x) between 0 and 1:

# These are our data points
x_vals = [0, 1./6, 1./3, 1./2, 7./12, 2./3, 3./4, 5./6, 11./12, 1]

# Set up the domain
x_domain = arange(0,2, 1e-2)

fx = zeroV(10)

# Defines the function so we can get our fx values
def sine_func(x):
    return(1.6*e**(-2*x)*sin(3*pi*x))

for i in range(len(x_vals)):
    fx[i] = sine_func(x_vals[i])

# Run cubic_spline interpolant.
cubic_spline(10,x_vals,fx)