Numerical Methods Simulations

A collection of interactive 3D visualizations and simulations to help you master concepts in numerical methods.

Roots of Equations - Theory & Concepts - Newton Raphson

Methods for finding roots of algebraic and transcendental equations.

Newton-Raphson Method Visualization

Finding the root of $f(x) = x^2 - 4$ using the Newton-Raphson method with initial guess $x_0 = 4$ .

Iteration Step: 0

Step 0 (Initial)Step 5

Iteration 0 Calculations

Current Guess (x_i)

4.0000

f(x_i) = 12.0000

Derivative (Slope)

8.0000

f'(x_i) = 8.0000

Formula

x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i)}

x_{i+1} = 4.0000 - \frac{12.0000}{8.0000}

Next Estimate (x_i+1)

2.500000

Roots of Equations - Theory & Concepts - Secant Method

Methods for finding roots of algebraic and transcendental equations.

Secant Method Visualization

Finding the root of $f(x) = e^{-x} - x$ using the Secant method with initial guesses $x_{-1} = 0$ and $x_0 = 1$ .

Iteration Step: 0

Step 0 (Initial)Step 4

Iteration 0 Calculations

Previous Guess (x_i-1)

0.00000

f(x_i-1) = 1.00000

Current Guess (x_i)

1.00000

f(x_i) = -0.63212

Formula

x_{i+1} = x_i - \frac{f(x_i)(x_{i-1} - x_i)}{f(x_{i-1}) - f(x_i)}

Next Estimate (x_i+1)

0.612700

Systems of Linear Equations - Theory & Concepts - Gauss Elimination

Numerical techniques to solve simultaneous linear algebraic equations.

Gauss Elimination Process

Step-by-step visualization of solving a 3x3 system of linear equations using naive Gauss elimination.

Initial Augmented Matrix [A|B]

3.0000	-0.1000	-0.2000	7.8500
0.1000	7.0000	-0.3000	-19.3000
0.3000	-0.2000	10.0000	71.4000

Step 1 of 7

iCurrent Action

Identify pivot element a_11 (3).

Solution Vector x

x_1

x_2

x_3

Curve Fitting and Interpolation - Theory & Concepts - Linear Regression

Techniques for approximating discrete data points with continuous functions.

Least-Squares Linear Regression

Drag the points or click on the graph to add new points. Observe how the best-fit line $y = a_0 + a_1x$ minimizes the sum of squared residuals.

Click to add points (max 15). Drag to move.

Regression Statistics

Equation of the Line

y = 0.329 + 1.032x

Intercept (

a_0

)

0.3286

Slope (

a_1

)

1.0321

R-squared (

r^2

)

0.9820

Sum of Squares (

S_r

)

0.5482

Point	x	y
1	1.00	1.50
2	2.00	2.20
3	3.00	3.10
4	4.00	4.80
5	5.00	5.50
6	6.00	6.90
7	7.00	7.20

Numerical Integration - Theory & Concepts - Trapezoidal Rule

Numerical methods for evaluating definite integrals.

Composite Trapezoidal Rule

Approximating the integral of $f(x)$ from $a=0$ to $b=0.8$ using multiple segments.

Number of Segments (n): 2

1 Segment20 Segments

Integration Results

Step Size (h)

0.4000

Exact Value

1.640533

Approximate Integral (Area)

1.068800

Formula

I \approx \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]

True Percent Relative Error:

34.8504%

As n increases, the error approaches zero.

Ordinary Differential Equations - Theory & Concepts - Euler Method

Numerical solutions for initial-value and boundary-value problems of ordinary differential equations.

Euler's Method Visualization

Solving the initial value problem $\frac{dy}{dx} = -2x^3 + 12x^2 - 20x + 8.5$ , with $y(0) = 1$ .

Step Size (h): 0.50

0.1 (More Accurate)2.0 (Less Accurate)

Results at x = 4

True y(4)

3.0000

Euler Approx y(4)

7.0000

True Percent Relative Error:

133.33%

Euler's Formula

y_{i+1} = y_i + f(x_i, y_i)h

Euler's method truncates the Taylor series after the first derivative. Decreasing the step size

h

reduces this truncation error and improves accuracy.