BCA Semester 4 – Numerical Methods – Unit 1: Solution of Nonlinear Equations (10Hrs.)

Newton–Raphson Method Given: The equation is \[ f(x) = x^2 - 5x + 6 \] We are asked to compute the root in the vicinity of \(x = 5\) using the Newton–Raphson method. --- Step 1: Formula of Newton–

(A) Algorithm: Bisection Method Step 1: Define the nonlinear function \(f(x)\). Step 2: Choose initial guesses \(a\) and \(b\) such that \[ f(a)\cdot f(b) < 0 \] (This ensures a root lies between \(

Given: - True value of \( \pi \): \[ \pi{\text{true}} = 3.1415926 \] - Approximate value of \( \pi \): \[ \pi{\text{approx}} = 3.1428571 \] --- (a) Absolute Error \[ \text{Absolute Error} = \left| \p

Part 1: When We Do Not Use Newton-Raphson Method Newton–Raphson (N-R) method may fail or is not recommended when: 1. Derivative is Zero or Near Zero - If \(f'(x) = 0\) at or near the root, iterat

1. Definition of Error In numerical computation, error is the difference between the true value and the approximate value of a quantity: \[ \text{Error} = \text{True Value} - \text{Approximate Value}

1. Definitions (a) Absolute Error (AE): The difference between the true value \(x{\text{true}}\) and the approximate value \(x{\text{approx}}\) of a quantity: \[ \text{Absolute Error} = |x{\text{tru

1. Type of Equations Applicable Newton's method (or Newton-Raphson method) is applicable to nonlinear equations of the form: \[ f(x) = 0 \] where: 1. \(f(x)\) is differentiable in the neighborhood of

1. Interpolation Techniques for Root Finding When the derivative \(f'(x)\) cannot be found, derivative-based methods like Newton-Raphson are not applicable. In such cases, interpolation techniques c