Numerical Methods

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Numerical Methods — Key Facts for GATE Engineering Mathematics

Core Topics:

Topic	Key Methods
Errors	Absolute, relative, round-off, truncation
Solution of Equations	Bisection, Newton-Raphson, Regula-Falsi
Interpolation	Newton’s Forward/Backward, Lagrange
Numerical Integration	Trapezoidal, Simpson’s 1/3 and 3/8
Solution of ODEs	Euler’s, RK4, Taylor’s
Numerical Differentiation	Forward, backward, central differences

Key Formulas:

• Newton-Raphson: x_{n+1} = x_n - f(x_n)/f’(x_n)
• Trapezoidal: ∫f(x)dx ≈ (h/2)[y₀ + 2y₁ + 2y₂ + … + 2y_{n-1} + y_n]
• Simpson’s 1/3: ∫f(x)dx ≈ (h/3)[y₀ + 4y₁ + 2y₂ + 4y₃ + … + y_n] (n must be even)
• RK4: k₁ = hf(xₙ, yₙ); k₂ = hf(xₙ+h/2, yₙ+k₁/2); k₃ = hf(xₙ+h/2, yₙ+k₂/2); k₄ = hf(xₙ+h, yₙ+k₃); y_{n+1} = yₙ + (k₁+2k₂+2k₃+k₄)/6

⚡ GATE Tip: In GATE, Simpson’s rule questions appear frequently — remember n must be even for Simpson’s 1/3 rule.

---

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Numerical Methods — Detailed Study Guide

Errors in Numerical Computation

Types of Errors

Error Type	Definition	Source
Absolute Error		E
Relative Error	RE =	E
Percentage Error	RE × 100	—
Round-off Error	Truncating digits	Limited precision
Truncation Error	Approximating infinite process	Taylor series truncation
Inherent Error	Input data already has error	Measurement

Significant Figures and Rounding

• Round half up (4.5 → 5)
• 3.14159265 to 4 sig figs → 3.142

Taylor Series and Truncation Error

Taylor Series: $$f(x) = f(a) + \frac{f’(a)}{1!}(x-a) + \frac{f”(a)}{2!}(x-a)^2 + … + \frac{f^{(n)}(a)}{n!}(x-a)^n + R_n$$

Remainder Term: $$R_n = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}, \quad \xi \in (a, x)$$

⚡ GATE Formula: If we use n+1 terms of Taylor series, truncation error ≈ (next term)

Solution of Algebraic and Transcendental Equations

Bisection Method (Bolzano Method)

Conditions:

• f(x) is continuous on [a, b]
• f(a) × f(b) < 0 (opposite signs)

Algorithm:

1. c = (a+b)/2
2. If f(a)×f(c) < 0: b = c; else a = c
3. Repeat until convergence

Error Estimate: |x - c| < (b-a)/2ⁿ after n iterations

⚡ GATE Properties: Always converges (slow), but guaranteed if sign changes.

Newton-Raphson Method

Formula: $$x_{n+1} = x_n - \frac{f(x_n)}{f’(x_n)}$$

Geometric Meaning: Tangent line intersects x-axis

Conditions:

• f(x) is continuous and differentiable
• f’(x) ≠ 0 near root

Convergence: Quadratic (error roughly squared each iteration)

⚡ GATE Example: Find √2 using Newton-Raphson. Solution: f(x) = x² - 2, f’(x) = 2x x_{n+1} = x_n - (x_n² - 2)/(2x_n) = (x_n + 2/x_n)/2 Starting with x₀ = 1: x₁ = 1.5, x₂ = 1.4167, x₃ = 1.4142…

Regula-Falsi Method (False Position)

Formula: $$x = \frac{af(b) - bf(a)}{f(b) - f(a)}$$

Comparison:

Method	Bisection	Regula-Falsi	Newton-Raphson
Convergence	Linear	Linear	Quadratic
Always converges	Yes	Yes	No (may diverge)
Speed	Slow	Slow	Fast
Uses derivative	No	No	Yes

---

Interpolation

Newton’s Forward Difference Formula

Forward Difference Table:

x    y    Δy    Δ²y   Δ³y
x₀   y₀
x₁   y₁   Δy₀
x₂   y₂   Δy₁   Δ²y₀
x₃   y₃   Δy₂   Δ²y₁  Δ³y₀

Newton’s Forward Formula: $$f(x) = f(x_0) + p\Delta f(x_0) + \frac{p(p-1)}{2!}\Delta^2 f(x_0) + …$$

where p = (x - x₀)/h, h = step size

⚡ GATE Point: Use forward differences when x is near the beginning of the data table.

Newton’s Backward Difference Formula

Uses backward differences (∇): $$\nabla y_i = y_i - y_{i-1}$$

Backward Formula: $$f(x) = f(x_n) + q\nabla f(x_n) + \frac{q(q+1)}{2!}\nabla^2 f(x_n) + …$$

where q = (x - xₙ)/h

⚡ GATE Point: Use backward differences when x is near the end of the data table.

Lagrange’s Interpolation

Formula: $$f(x) = \sum_{i=0}^{n} y_i L_i(x)$$

where: $$L_i(x) = \prod_{j=0, j \neq i}^{n} \frac{x - x_j}{x_i - x_j}$$

Advantage: No need for equally spaced points Use when: Data points are unevenly spaced

⚡ GATE Example: Interpolate at x=10 given (5,12), (8,15), (12,20). Solution using Lagrange formula.

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Numerical Methods — Complete Notes for GATE

Numerical Integration

Trapezoidal Rule

For n intervals (n+1 points), step h = (b-a)/n: $$\int_a^b f(x)dx \approx \frac{h}{2}\left[y_0 + 2(y_1 + y_2 + … + y_{n-1}) + y_n\right]$$

Error: E_T = -(b-a)h²/12 × f”(ξ)

Simpson’s 1/3 Rule

Condition: n must be even number of intervals!

$$\int_a^b f(x)dx \approx \frac{h}{3}\left[y_0 + 4y_1 + 2y_2 + 4y_3 + … + 4y_{n-1} + y_n\right]$$

Error: E_S = -(b-a)h⁴/180 × f⁴(ξ)

⚡ GATE Rule: If n is not even, use trapezoidal or split interval.

Simpson’s 3/8 Rule

For 3 intervals (4 points): $$\int_a^b f(x)dx \approx \frac{3h}{8}[y_0 + 3y_1 + 3y_2 + y_3]$$

Error: E = -(b-a)h⁴/80 × f⁴(ξ)

Comparison of Integration Methods

Method	Order of Error	Application
Trapezoidal	O(h²)	Simple, always applicable
Simpson’s 1/3	O(h⁴)	Fast convergence, n must be even
Simpson’s 3/8	O(h⁴)	When n = 3, 6, 9…
Boole’s Rule	O(h⁷)	Highest accuracy among these

Numerical Solution of ODEs

Euler’s Method

$$y_{n+1} = y_n + hf(x_n, y_n)$$

Error: O(h) — very inaccurate for practical use

Modified Euler’s Method

$$y_{n+1} = y_n + \frac{h}{2}[f(x_n, y_n) + f(x_{n+1}, y_{n+1})]$$

Also called Euler-Cauchy or Heun’s method.

Runge-Kutta 4th Order (RK4)

The Standard RK4 Method:

Given y’ = f(x, y), y(x₀) = y₀, step h:

k₁ = h·f(xₙ, yₙ)
k₂ = h·f(xₙ + h/2, yₙ + k₁/2)
k₃ = h·f(xₙ + h/2, yₙ + k₂/2)
k₄ = h·f(xₙ + h, yₙ + k₃)

y_{n+1} = yₙ + (k₁ + 2k₂ + 2k₃ + k₄)/6

⚡ GATE Properties:

• RK4 has O(h⁴) error per step (very accurate)
• Most commonly used method in engineering
• For most ODEs in GATE: Use RK4

Taylor’s Series Method

$$y_{n+1} = y_n + hy’ + \frac{h^2}{2!}y” + \frac{h^3}{3!}y''' + …$$

Where y’, y”, y”… are computed from the differential equation.

Solution of Linear Systems — Iterative Methods

Gauss-Seidel Method

For Ax = b, rewrite as: $$x_i^{(k+1)} = \frac{1}{a_{ii}}\left[b_i - \sum_{j < i} a_{ij}x_j^{(k+1)} - \sum_{j > i} a_{ij}x_j^{(k)}\right]$$

Convergence Condition: Matrix should be diagonally dominant

Advantages: Uses updated values immediately, faster convergence than Jacobi

Jacobi Method

Similar but doesn’t use updated values immediately (slower).

⚡ GATE Comparison:

Method	Jacobi	Gauss-Seidel
Speed	Slower	Faster
Updates	Simultaneous	Sequential
Convergence	May converge when G-S diverges	More commonly used

Numerical Differentiation

Finite Difference Formulas

First Derivative:

Type	Formula	Error
Forward	(y₁ - y₀)/h	O(h)
Backward	(y₀ - y₋₁)/h	O(h)
Central	(y₁ - y₋₁)/(2h)	O(h²)

Second Derivative (Central): $$\frac{d^2y}{dx^2} \approx \frac{y_1 - 2y_0 + y_{-1}}{h^2}, \quad \text{error } O(h^2)$$

GATE-Style Practice Questions

1. Using Newton-Raphson, find root of x³ - x - 1 = 0 starting from x₀ = 1.
   First iteration gives:
   (a) 1.5 (b) 1.33 (c) 1.25 (d) 1.75
   
   Answer: (b) 1.33
   Solution: f(1) = -1, f'(1) = 3
             x₁ = 1 - (-1)/3 = 1.33

2. The error in Simpson's rule is of order:
   (a) h (b) h² (c) h³ (d) h⁴
   
   Answer: (d) h⁴
   Solution: Simpson's 1/3 rule error is O(h⁴)

3. Interpolation is used when:
   (a) Data has errors (b) Value at intermediate point is needed
   (c) Data is exact (d) Integration is required
   
   Answer: (b) Value at intermediate point is needed

4. Gauss-Seidel method is applicable to:
   (a) All systems (b) Only diagonally dominant systems
   (c) Only tridiagonal systems (d) Only symmetric systems
   
   Answer: (b) Only diagonally dominant systems
   (More precisely: converges for diagonally dominant or symmetric positive definite)

5. For ∫₀¹ f(x)dx with h = 0.25, using Simpson's 1/3 rule, number of intervals is:
   (a) 2 (b) 4 (c) 8 (d) Any even number
   
   Answer: (b) 4
   Solution: Interval = (1-0)/0.25 = 4 intervals (n=4, even ✓)

⚡ GATE Strategy: For Numerical Methods in GATE, expect 2-4 questions. Focus on Newton-Raphson convergence conditions, Simpson’s rule requirements (n must be even), and RK4 method for ODEs.

---

Content adapted based on your selected roadmap duration. Switch tiers using the selector above.