Exercise 5 - Newton's method

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Exercise 5 - Newton's method

Let a and b two real numbers such that a < b. Let I = [a, b]. Let f : R-> R a differentiable function defined and strictly monotonic on the interval I, and such that f(a) f(b) < 0 .

It can easily be shown that f has a unique zero in I (let it be denoted by z).

Let h : R-> R defined by h(x) = x - f(x) / f'(x).

Let (u_n)_{n in N} the sequence defined by u₀ = b if both f and f' are increasing, or both decreasing, u₀ = a else, and by u_n+1 = h(u_n) for any natural number n.

It can be shown that if f' is differentiable on I, f" continuous on I and f" with no zero in I, then the sequence (u_n)_{n in N} converges to z. Then an approximate value for z can be computed by the Newton's method, which consists of computing the successive elements in the sequence (u_n)_{n in N} until a value considered closed enough to z is reached.

1

Using the higher-order function  deriv , write a higher-order function (named  newton ) which, given a function f from real numbers to real numbers and two real numbers a and b, computes by the Newton's method an approximation of the unique zero f has on the closed interval defined by a and b (f will be assumed to satisfy the conditions stated above).

2

Compute an approximation of the square root of 2 by the Newton's method.

Exercise 5 - Newton's method

Documentation and user's manual

Table of contents

OCaml programs