Given a function f(x) on floating number x and two numbers ‘a’ and ‘b’ such that f(a)*f(b) < 0 and f(x) is continuous in [a, b]. Here f(x) represents algebraic or transcendental equation. Find root of function in interval [a, b] (Or find a value of x such that f(x) is 0).
Input: A function of x, for example x3 – x2 + 2. And two values: a = -200 and b = 300 such that f(a)*f(b) < 0, i.e., f(a) and f(b) have opposite signs. Output: The value of root is : -1.00 OR any other value close to root.
We strongly recommend to refer below post as a prerequisite of this post.
Solution of Algebraic and Transcendental Equations | Set 1 (The Bisection Method)
In this post The Method Of False Position is discussed. This method is also known as Regula Falsi or The Method of Chords.
Similarities with Bisection Method:
- Same Assumptions: This method also assumes that function is continuous in [a, b] and given two numbers 'a' and 'b' are such that f(a) * f(b) < 0.
- Always Converges: like Bisection, it always converges, usually considerably faster than Bisection--but sometimes very much more slowly than Bisection.
Differences with Bisection Method:
It differs in the fact that we make a chord joining the two points [a, f(a)] and [b, f(b)]. We consider the point at which the chord touches the x axis and named it as c.
- Write equation of the line connecting the two points.
y – f(a) = ( (f(b)-f(a))/(b-a) )*(x-a) Now we have to find the point which touches x axis. For that we put y = 0. so x = a - (f(a)/(f(b)-f(a))) * (b-a) x = (a*f(b) - b*f(a)) / (f(b)-f(a)) This will be our c that is c = x.
- If f(c) == 0, then c is the root of the solution.
- Else f(c) != 0
- If value f(a)*f(c) < 0 then root lies between a and c. So we recur for a and c
- Else If f(b)*f(c) < 0 then root lies between b and c. So we recur b and c.
- Else given function doesn't follow one of assumptions.
Since root may be a floating point number and may converge very slow in worst case, we iterate for a very large number of times such that the answer becomes closer to the root.
Following is the implementation.
The value of root is : -1
This method always converges, usually considerably faster than Bisection. But worst case can be very slow.
We will soon be discussing other methods to solve algebraic and transcendental equations.