Lesson 1.6: Numerical Methods (Errors, Iterative Methods, Interpolation, Numerical Integration, ODE Solutions)

Numerical Methods are widely used in engineering problem-solving where analytical solutions are difficult. GATE and PSU exams test topics like error analysis, iterative solutions, interpolation, numerical integration, and solving ODEs numerically.

🔹 1. Errors in Numerical Methods

Types of Errors:

• Round-off Error: Due to finite precision in computers

• Truncation Error: Approximating an infinite process with a finite one

• Absolute Error: ∣xtrue−xapprox∣|x_{\text{true}} – x_{\text{approx}}|∣xtrue​−xapprox​∣

• Relative Error: ∣xtrue−xapprox∣∣xtrue∣\frac{|x_{\text{true}} – x_{\text{approx}}|}{|x_{\text{true}}|}∣xtrue​∣∣xtrue​−xapprox​∣​

Applications:

• Ensuring accuracy in engineering calculations

• Error bounds in iterative methods

🔹 2. Iterative Methods for Equations

Goal: Solve $f(x)=0$ approximately.

• Bisection Method:
  Repeatedly halve interval [a, b] where $f(a)f(b)<0$

• Newton-Raphson Method:

xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n – \frac{f(x_n)}{f'(x_n)}xn+1​=xn​−f′(xn​)f(xn​)​

• Secant Method:

xn+1=xn−f(xn)xn−xn−1f(xn)−f(xn−1)x_{n+1} = x_n – f(x_n) \frac{x_n – x_{n-1}}{f(x_n) – f(x_{n-1})}xn+1​=xn​−f(xn​)f(xn​)−f(xn−1​)xn​−xn−1​​

Applications: Root-finding in structural, thermal, and fluid problems

🔹 3. Interpolation

Purpose: Estimate unknown value within a known dataset.

• Linear Interpolation: Between two points

y=y0+(x−x0)(y1−y0)x1−x0y = y_0 + \frac{(x-x_0)(y_1 – y_0)}{x_1 – x_0}y=y0​+x1​−x0​(x−x0​)(y1​−y0​)​

• Polynomial Interpolation:

  • Newton’s forward/backward difference

  • Lagrange polynomial

Applications:

• Stress-strain tables

• Thermodynamic property tables

• Material property estimation

🔹 4. Numerical Differentiation & Integration

• Differentiation: Approximates derivative using discrete points

  • Forward, Backward, Central difference formulas

• Integration (Quadrature): Approximate definite integrals

  • Trapezoidal Rule:

∫abf(x)dx≈h2[f(a)+f(b)]\int_a^b f(x) dx \approx \frac{h}{2}[f(a) + f(b)]∫ab​f(x)dx≈2h​[f(a)+f(b)]

• Simpson’s 1/3 Rule:

∫abf(x)dx≈h3[f(x0)+4f(x1)+f(x2)]\int_a^b f(x) dx \approx \frac{h}{3}[f(x_0) + 4f(x_1) + f(x_2)]∫ab​f(x)dx≈3h​[f(x0​)+4f(x1​)+f(x2​)]

Applications: Area under curve, work, energy, heat transfer calculations

🔹 5. Numerical Solution of ODEs

• Euler’s Method:

yn+1=yn+hf(xn,yn)y_{n+1} = y_n + h f(x_n, y_n)yn+1​=yn​+hf(xn​,yn​)

• Runge-Kutta Methods: Higher accuracy, commonly 4th order:

yn+1=yn+16(k1+2k2+2k3+k4)y_{n+1} = y_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)yn+1​=yn​+61​(k1​+2k2​+2k3​+k4​)

• Applications:

  • Vibration analysis

  • Thermal conduction modeling

  • Fluid dynamics simulations

🔹 6. Solved Examples (PYQ Style)

Example 1 (GATE ME 2019):
Find root of x2−4=0x^2 – 4 = 0x2−4=0 using Newton-Raphson, initial guess x0=3x_0 = 3x0​=3.
👉 Solution: x1=3−56=2.167x_1 = 3 – \frac{5}{6} = 2.167x1​=3−65​=2.167, next iteration → x ≈ 2

Example 2 (PSU Exam):
Use Trapezoidal rule to evaluate ∫01(1+x2)dx\int_0^1 (1+x^2) dx∫01​(1+x2)dx with h=0.5.
👉 Solution:
$I\approx 0.5/2[1+2.25+2\ast 2]=1.333$

🔹 7. Practice Exercises

1. Compute relative error if xtrue=2x_{\text{true}}=2xtrue​=2 and xapprox=1.95x_{\text{approx}}=1.95xapprox​=1.95.

2. Solve x3−x−2=0x^3 – x – 2=0x3−x−2=0 using Bisection method (2 iterations).

3. Interpolate y at x=2.5 given points (2,4) and (3,9).

4. Numerically integrate f(x)=x2f(x) = x^2f(x)=x2 from 0 to 1 using Simpson’s 1/3 rule.

5. Solve dydx=x+y\frac{dy}{dx} = x+ydxdy​=x+y, y(0)=1 using Euler method with h=0.1.

🔹 8. Summary

• Errors: Round-off, truncation, absolute, relative

• Iterative Methods: Bisection, Newton-Raphson, Secant

• Interpolation: Linear, polynomial (Lagrange/Newton)

• Numerical Differentiation & Integration: Approximates derivatives/integrals

• ODE Solutions: Euler & Runge-Kutta methods, used in engineering simulations