Engineering Mathematics II
Numerical Analysis
Module II: Numerical Analysis. This module introduces numerical techniques for solving equations, interpolation, integration and differential equations. These methods approximate exact mathematical results when analytical solutions are difficult or impossible to obtain. Numerical methods form the core of scientific computing and simulation.
1. Solution of Equations (Root Finding)
Many engineering problems require solving nonlinear equations of the formf(x) = 0. When an exact solution is not available, iterative numerical methods provide approximations.
1.1 Bisection Method
- Based on the Intermediate Value Theorem.
- If f(a) and f(b) have opposite signs, a root lies in the interval.
- The interval is repeatedly halved to narrow down the root.
Find the root of f(x) = x³ - x - 2 in [1, 2] using one iteration.
Midpoint m = (1 + 2) / 2 = 1.5.
Since f(1) < 0 and f(1.5) > 0, new interval is [1, 1.5].
Diagram: bisection-method.png
1.2 Regula Falsi (False Position)
An improved version of bisection that uses a secant line to estimate the root.
- Iteration formula:
x_new = (a*f(b) - b*f(a)) / (f(b) - f(a)) - Converges faster than bisection for many problems.
Diagram: regula-falsi.png
1.3 Newton Raphson Method
Uses tangent lines to converge very quickly to a root when the initial guess is close.
- Formula:
x_(n+1) = x_n - f(x_n)/f'(x_n) - Quadratic convergence (very fast).
- Requires derivative computation.
Find root of f(x) = x² − 2 with x₀ = 1.5.
f(x) = x² - 2,f'(x) = 2x.
x₁ = 1.5 - (1.5² - 2)/(3) = 1.4167.
Diagram: newton-raphson-geometry.png
2. Finite Differences
Finite differences approximate derivatives and form the foundation for interpolation formulas.
- Forward difference:
Δf(x) = f(x+h) - f(x) - Backward difference:
∇f(x) = f(x) - f(x-h) - Central difference:
δf(x) = f(x+h/2) - f(x-h/2) - Difference tables help visualize polynomial data patterns.
Diagram: difference-table.png
3. Interpolation
Interpolation constructs a polynomial that passes through given data points, allowing estimation at intermediate values.
3.1 Lagrange Interpolation
- Polynomial of degree (n - 1) for n data points.
- Formula:
P(x) = Σ y_i * Π (x - x_j)/(x_i - x_j)
3.2 Newton Forward Interpolation
- Used when x values are equally spaced.
- Forward difference table required.
3.3 Newton Backward Interpolation
- Used for estimating values near the end of the table.
3.4 Gauss Central Interpolation
- Useful when x = 0 lies near the middle of the data set.
Diagram: interpolation-cartoon.png
4. Numerical Integration
When exact integrals cannot be computed, numerical rules approximate the area under a curve.
4.1 Trapezoidal Rule
- Approximates the region under a curve using trapeziums.
- Formula:
I ≈ h/2 * [f(x0) + 2(f(x1)+...+f(x(n-1))) + f(xn)]
4.2 Simpson's One Third Rule
- Uses parabolic arcs to approximate the curve.
- Number of intervals must be even.
- Formula:
I ≈ h/3 [f(x0) + 4Σf(odd) + 2Σf(even) + f(xn)]
Diagram: simpson-rule-illustration.png
5. Numerical Solution of Differential Equations
Initial value problems (IVP) of the formdy/dx = f(x, y)often require numerical solutions. Two commonly used methods are Euler and Runge Kutta.
5.1 Euler's Method
- Simplest numerical method.
- Formula:
y_(n+1) = y_n + h * f(x_n, y_n) - Accuracy is low but easy to compute.
5.2 Runge Kutta Methods
RK methods provide higher accuracy by evaluating slopes multiple times within each interval.
- RK2: midpoint method.
- RK4: most widely used, very accurate for small h.
Diagram: rk4-flowchart.png