False Position Method

This is the oldest method for finding the real root of a nonlinear equation f(x) = 0 and closely resembles the bisection method. In this method, also known as regula falsi or the method of chords, we choose two points a and b such that f(a) and f(b) are of opposite signs. Hence, a root must lie in between these points. Now, the equation of the chord joining the two points [a, f(a)] and [b, f(b)] is given by

{y – f(a)}/{x – a} = {f(b) – f(a)}/{b – a}

The method consists in replacing the part of the curve between the points [a, f(a)] and [b, f(b)] by means of the chord joining these points, and taking the point of intersection of the chord with the x-axis as an approximation to the root. The point of intersection in the present case is obtained by putting y = 0 in first equation. Thus, we obtain

x₁ = a – {(b – a)f(a)}/{f(b) – f(a)} = {af(b) – bf(a)}/{f(b) – f(a)}

Which is the first approximation to the root of f(x) = 0. If now f(x₁) and f(a) are of opposite signs, then the root lies between a and x₁, and we replace b by x₁ in second equation, and obtain the next approximation. Otherwise, we replace a by x₁ and generate the next approximation. The procedure is repeated till the root is obtained to the desired accuracy. Figure gives a graphical representation of the method.

Example: - Find a real root of the equation f(x) = x³ - 2x - 5 = 0.

Solⁿ: -

We find f(2) = -1 and f(3) = 16. Hence a = 2 , b = 3, and a root lies between 2 and 3. Using False Position Method Formula:

x₁ = {2(16) - 3(-1)}/{16 - (-1)} = 2.058823529

Now, f(x₁) = -0.390799917 and hence the root lies between 2.058823529 and 3.0. Using False Position Method Formula:

x₂ = {2.058823529(16) - 3(-0.390799917)}/{16 - (-0.390799917)} = 2.08126366

Since f(x₂) = -0.147204057, it follows that the root lies between 2.08126366 and 3.0. Hence, we have

x₃ = {2.08126366(16) - 3(-0.147204057)}/{16 - (-0.147204057)} = 2.089639211

Proceeding in this way, we obtain successively:

x₄ = 2.092739575, x₅ = 2.09388371,

x₆ = 2.094305452, x₇ = 2.094460846, ………..

The correct value is 2.0945…., so that x₇ is correct to five significant figures.