Question

Determine the smallest positive root of the following equation: 
f(x) ≡ x3 – 9x2
- x + 9 = 0
to three significant digits using
(a) Regula-falsi method (b) Newton-Raphson method
(c) Bisectionmethod (d) Secant method

02 Aug 2023
Answer :
Word Count : 856

To find the smallest positive root of the equation f(x) = x^3 - 9x^2 - x + 9 = 0, we can use the Regula-Falsi method, Newton-Raphson method, Bisection method, and Secant method. Let's apply each method and find the root to three significant digits:

(a) Regula-Falsi Method:
In the Regula-Falsi method, we start with two initial points, a and b, such that f(a) and f(b) have opposite signs. Then, we calculate the point of intersection of the x-axis using the formula:

c = (af(b) - bf(a)) / (f(b) - f(a))

We update the values of a and b based on the sign of f(c) to narrow down the interval containing the root. We iterate this process until we reach the desired accuracy.

Let's assume the initial interval [a, b] as [1, 2].

Iteration 1:
f(a) = f(1) ≈ -8
f(b) = f(2) ≈ 7
c = (1 * 7 - 2 * -8) / (7 - (-8)) ≈ 0.7778

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