# How to present a polynomial interpolation if the first order information of function f(x) is given?

Suppose $f'(x_1),\ f'(x_2),\ f'(x_3)$ are given, how to give a polynomial interpolation $p(x)$ such that $p'(x_1)= f'(x_1),\ p'(x_2)=f'(x_2),\ p'(x_3)=f'(x_3)$? And how to give an error analysis?

• This is called Hermite interpolation, and should be discussed in most textbooks on numerical analysis (e.g., Gautschi's). Note that the polynomial is not uniquely specified unless you also have some function values $f(x_i)$. – Christian Clason Jan 15 '15 at 14:43
• Yes it will be uniquely determined if we set $p(x_1)=f(x_1)$. But it is not straightfoward to apply the Hermite interpolation since we do not know the informatio about $f(x_2)$ and $f(x_3)$? – mathstudy Jan 18 '15 at 3:59
• You didn't say in your questions that you set $p(x_1)=f(x_1)$. – nicoguaro Jan 20 '15 at 19:05

## 1 Answer

The problem boils down to the solution of a linear system a equations. You get one equation for the function value at $f(x_1)$, and three more equations from your derivative equations. Then, a polynomial of degree three is suitable.

A more brute-force, not so clever way is to use computer algebra such as SymPy for a symbolical solution:

from sympy import *

N = 3 # degree of polynomial - for N + 1 'pieces of information', use a polynomial of degree N

known_x = symbols(["x_%i"%i for i in range(N)])  # create symbols for supporting points
known_derivs = symbols(["d_%i"%i for i in range(N)])  # create symbols for supporting points
y1 = Symbol("y_1")  # the single zeroth derivative we know

# create symbols for the independent variable and the coefficients:
x = Symbol("x")
a = [ Symbol("a_%i"%i) for i in range(N+1) ]

# generate polynomial and the derivative
p = sum([ coeff*x**i for i, coeff in enumerate(a) ])
p_prime = diff(p, x, 1)

# define the equations to solve:
eqns = [ p_prime.subs(x, xi) - d for xi, d in zip(known_x, known_derivs) ]  # equation for the first derivative
eqns.append(p.subs(x, known_x[0]) - y1)

# solve for the coefficients and print solution:
solution = solve(eqns, a)
pretty_sol = [ str(coeff) + " = " + str(value) + "\n"
for coeff, value in zip(solution.keys(), solution.values()) ]
print("The coefficients are:\n" + "".join(pretty_sol))


This not very elegant, but seems to work even you added another equation for the derivative (so that could use a polynomial of degree 'N = 4' for the five equations).

• I think you mean "A polynomial of degree 3 is suitable"? – Rahul Jan 21 '15 at 5:05
• Yes, of course. Thank you. Edited in my reply. – AlexE Jan 21 '15 at 6:00
• Thanks, can you be more clear about your comment "The problem boils down to the solution of a linear system a equations"? It seems interesting, can it be possible to give a error analysis? – mathstudy Jan 22 '15 at 3:02