Introduction to Numerical Methods – definitions, errors, convergence, classifications

1. What are Numerical Methods?

Numerical methods are systematic, algorithmic procedures that produce approximate numerical solutions to mathematical problems that lack closed‑form answers or are too costly to solve analytically. They are indispensable in engineering, physics, finance, and computer science.

Key Characteristics

2. Types of Errors

Understanding error terminology is the foundation of any numerical analysis.

2.1 True (Exact) Error

$$E_{\text{true}}=x_{\text{exact}}-x_{\text{approx}}$$

2.2 Absolute Error

$$E_{\text{abs}}=|x_{\text{exact}}-x_{\text{approx}}|$$

2.3 Relative Error

$$E_{\text{rel}}=\frac{|x_{\text{exact}}-x_{\text{approx}}|}{|x_{\text{exact}}|}$$

2.4 Visualising Errors

The bar chart shows how the same approximation yields different numerical error measures.

3. Convergence, Stability, and Rate of Convergence

3.1 Convergence

A method converges if the sequence of approximations $\{x_k\}$ approaches the exact solution $x^{\text{*}}$ as $k\to \infty$.

$$\underset{k\to \infty }{\lim }{x}_k=x^{\text{*}}$$

3.2 Stability

A stable algorithm produces bounded output for bounded input, even in the presence of round‑off errors.

3.3 Rate of Convergence

If the error satisfies

$$|e_{k+1}|\approx C|e_k{|}^p,$$

the method is said to have order $p$.

• Linear ($p=1$) – e.g., simple iteration.
• Quadratic ($p=2$) – e.g., Newton‑Raphson.

3.4 Convergence Plot Example

The steep drop for Newton‑Raphson illustrates quadratic convergence.

4. Classification of Numerical Methods

Numerical techniques are grouped based on the type of problem they solve and the algorithmic approach.

flowchart TD
    A["Numerical Methods"] --> B["Root‑Finding"]
    A --> C["Linear Systems"]
    A --> D["Interpolation & Curve Fitting"]
    A --> E["Differentiation & Integration"]
    A --> F["Ordinary Differential Equations"]
    A --> G["Partial Differential Equations"]
    B --> B1["Bisection"]
    B --> B2["Newton‑Raphson"]
    B --> B3["Secant"]
    C --> C1["Direct (Gaussian Elimination)"]
    C --> C2["Iterative (Jacobi, Gauss‑Seidel)"]
    D --> D1["Lagrange Polynomial"]
    D --> D2["Least‑Squares"]
    E --> E1["Finite Difference"]
    E --> E2["Simpson, Trapezoidal"]
    F --> F1["Euler"]
    F --> F2["Runge‑Kutta"]
    G --> G1["Finite Difference"]
    G --> G2["Finite Element"]

5. Worked Example – Bisection Method

Problem: Find the root of $f(x)=2x^3-2x-5$ in the interval $[1,2]$ using the bisection method up to the 5th iteration.

5.1 Algorithm Steps

1. Verify $f(a)\cdot f(b)<0$.
2. Compute midpoint $c=(a+b)/2$.
3. Evaluate $f(c)$.
4. Replace the endpoint that has the same sign as $f(c)$ with $c$.
5. Repeat until desired tolerance or iteration count is reached.

5.2 Iteration Table

5.3 Approximate Root

After 5 iterations, the midpoint $c=1.7969$ approximates the root with absolute error $\approx 1.6\times 10^{-2}$.

5.4 Graphical Illustration

The plot shows the function curve and the successive midpoints converging to the root.

6. Advantages & Disadvantages of Common Introductory Methods

7. Real‑World Applications

7.1 eSewa Transaction Rounding

eSewa processes millions of micro‑transactions daily. When converting Nepali Rupee amounts to the nearest paisa for digital wallets, the system uses absolute error to ensure the rounded amount never exceeds the true value by more than 0.5 paisa, preserving financial integrity.

7.2 Daraz Order Queue Prioritization

Daraz’s order‑fulfilment engine employs a fixed‑point iteration to balance inventory levels and delivery slots. The iteration solves the equation

$$x=g(x)=\frac{D}{S+x},$$

where $D$ is demand and $S$ is current stock, converging to the optimal number of orders to dispatch per hour.

7.3 Ncell Signal Strength Mapping

Ncell uses interpolation (linear & spline) to estimate signal strength at locations where no direct measurement exists. The underlying error analysis (relative error) guarantees that the interpolated value stays within 5 % of the true field strength, guiding tower placement.

7.4 Worked Real Situation – Bank Loan Interest

A commercial bank calculates the effective annual rate (EAR) for a loan with monthly compounding using the root‑finding Newton‑Raphson method on

$$f(r)=(1+r/12{)}^{12}-(1+\text{EAR})=0.$$

Starting with $r_0=0.08$ (8 % nominal), the method converges in three iterations to $r\approx 0.0833$, giving an EAR of 8.70 %. This illustrates how numerical root‑finding directly impacts everyday financial products.

8. Summary of Core Concepts

• Numerical methods bridge the gap between theory and computation.
• Error analysis (true, absolute, relative) quantifies approximation quality.
• Convergence guarantees that repeated iterations approach the exact solution; the order tells how fast.
• Classification helps select the right tool: root‑finding, linear systems, interpolation, etc.
• Real‑world systems—from digital payments to telecom network planning—rely on these ideas every day.

Exam tip

• Remember the definitions: true vs absolute vs relative error; convergence vs stability; order of convergence.
• Derivation matters: Be able to derive the bisection formula and explain why the sign change condition guarantees a root.
• Algorithm trace: Practice writing iteration tables (like the bisection example) without a calculator; examiners love a clean, step‑by‑step trace.
• Comparison tables: Memorize at least three pros/cons for Bisection, Newton‑Raphson, and Secant; they often appear in short‑answer questions.
• Real‑world link: Cite one concrete application (e.g., eSewa rounding or Daraz order queue) to earn extra marks in essay‑type questions.