🚀 Numerical Analysis Suite (محاسبات عددی)

A professional, high-performance Python framework for solving classic Numerical Analysis problems — engineered for academic rigor and built to stand out on a GitHub portfolio.

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✨ Why This Project

This isn't just a homework dump — it's a modular, object-oriented solver engine where every assignment is implemented as an independent, testable, reusable component. Each problem ships with its own convergence analysis, error tracking, and (where applicable) visualization, wrapped in a clean CLI driven by a single student ID.

Key Features:

• 🔁 Analytical + numerical derivative support for root-finding (no finite-difference guesswork)
• 📊 Convergence tables and iteration logs (tabulate-powered) for every method
• 🌡️ Full 2D PDE solver with Gauss-Seidel relaxation and heatmap visualization
• 🎛️ Personalized parameters per student — no two runs are identical
• 📈 Matplotlib visualizations for root convergence, integration regions, and temperature fields
• 🧩 Zero external solver dependencies for core logic — the math is implemented from scratch, not just called from a library

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📌 Table of Contents

• Why This Project
• Project Architecture
• Student ID Parameterization System
• Assignments & Solutions Breakdown
  • Problem 1: Newton-Raphson Root Finding (Hyperbolic)
  • Problem 2: Newton-Raphson Polynomial Root Finding
  • Problem 3: Non-Linear Spring Energy Equilibrium
  • Problem 4: Logarithmic Root Determination
  • Problem 5: Numerical Quadrature (Definite Integration)
  • Problem 6: 2D Heat Transfer PDE (Gauss-Seidel Method)
  • Problem 7: LC Ladder Network Dynamics & Eigenvalues
• Sample Output
• Installation & Execution
• License

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🏗 Project Architecture

numerical-analysis-suite/
│
├── numerical_suite.py       # Main object-oriented solver engine
├── main.py                  # CLI runner with custom student ID input
├── requirements.txt         # Dependencies (numpy, scipy, matplotlib, tabulate)
└── README.md                # Project documentation & mathematical formulations

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🔢 Student ID Parameterization System

To ensure personalized parameters for each student, all assignment variables are derived from the 4 rightmost digits of the Student ID: $$\mathbf{abcd} = [a, b, c, d]$$

For instance, given ID suffix 1234:

• $a = 1, b = 2, c = 3, d = 4$
• $9.\mathbf{bcda} = 9.2341$
• $k_1 = 4\mathbf{c}00\mathbf{b} = 43002$

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🧮 Assignments & Solutions Breakdown

Problem 1: Newton-Raphson Root Finding (Hyperbolic)

📐 Math Formulation

Find the root of the hyperbolic equation: $$f(x) = \tanh(x^2 - 9.\mathbf{bcda}) = 0$$

Analytical Derivative: $$f'(x) = 2x \cdot \text{sech}^2(x^2 - 9.\mathbf{bcda}) = 2x \left(1 - \tanh^2(x^2 - 9.\mathbf{bcda})\right)$$

Newton-Raphson Iteration Formula: $$x_{k+1} = x_k - \frac{\tanh(x_k^2 - 9.\mathbf{bcda})}{2x_k \left(1 - \tanh^2(x_k^2 - 9.\mathbf{bcda})\right)}$$

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Problem 2: Newton-Raphson Polynomial Root Finding

📐 Math Formulation

Solve for $x$ in the $4^{\text{th}}$-degree polynomial: $$f(x) = 0.007\mathbf{d} \cdot x^4 - 0.284\mathbf{a} \cdot x^3 + 3.355\mathbf{c} \cdot x^2 - 12.183\mathbf{b} \cdot x + 5 = 0$$

First Derivative: $$f'(x) = 4(0.007\mathbf{d})x^3 - 3(0.284\mathbf{a})x^2 + 2(3.355\mathbf{c})x - 12.183\mathbf{b}$$

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Problem 3: Non-Linear Spring Energy Equilibrium

📐 Math Formulation

A dropped mass $m$ compresses a non-linear spring with force law $F = -(k_1 d' + k_2 d'^{3/2})$. Energy conservation yields: $$g(d') = \frac{2}{5} k_2 d'^{5/2} + \frac{1}{2} k_1 d'^2 - m g d' - m g h = 0$$

Derivative with respect to $d'$: $$g'(d') = k_2 d'^{3/2} + k_1 d' - m g$$

Parameters:

• $k_1 = 4\mathbf{c}00\mathbf{b}$ N/m
• $k_2 = 4\mathbf{d}$ N/m$^{3/2}$
• $h = 0.4\mathbf{c}$ m
• $m = 98$ kg, $g = 9.81$ m/s$^2$

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Problem 4: Logarithmic Root Determination

📐 Math Formulation

Solve: $$\ln(x^2) = 0.7\mathbf{acdb} \implies \ln(x^2) - 0.7\mathbf{acdb} = 0$$

Exact Analytical Solution: $$x = \pm \exp\left(\frac{0.7\mathbf{acdb}}{2}\right)$$

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Problem 5: Numerical Quadrature (Definite Integration)

📐 Math Formulation

Evaluate the definite integral: $$I = \int_{0}^{1} \frac{\mathbf{a.bc} \cdot x}{2 e^x - e^{-x}} , dx$$

Methods Implemented:

1. Composite Trapezoidal Rule
2. Composite Simpson's $1/3$ Rule
3. Adaptive Gaussian Quadrature (scipy.integrate.quad)

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Problem 6: 2D Heat Transfer PDE (Gauss-Seidel Method)

📐 Math Formulation

Steady-state 2D heat conduction governed by Laplace's Equation: $$\nabla^2 T = \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$$

Discretized finite-difference grid equation for node $(i,j)$: $$T_{i,j}^{(k+1)} = \frac{1}{4} \left( T_{i-1,j}^{(k+1)} + T_{i+1,j}^{(k)} + T_{i,j-1}^{(k+1)} + T_{i,j+1}^{(k)} \right)$$

Boundary Conditions:

• Top: $25^\circ\text{C}$
• Bottom: $75^\circ\text{C}$
• Left: $100^\circ\text{C}$
• Right: $0^\circ\text{C}$

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Problem 7: LC Ladder Network Dynamics & Eigenvalues

📐 Math Formulation

Mesh current differential equations for 3-stage LC filter: $$\mathbf{L} \mathbf{\ddot{q}} + \mathbf{S} \mathbf{q} = \mathbf{v}(t)$$

System Matrix $\mathbf{A}$: $$\mathbf{A} = \mathbf{L}^{-1} \mathbf{S}$$

Eigenvalue Problem: $$\mathbf{A} \mathbf{v} = \omega^2 \mathbf{v}$$

Parameters:

• $I(t) = 2.\mathbf{bcd'a} \sin(\mathbf{bc} \cdot t)$
• $\mathbf{L} = [0.\mathbf{bd'ca}, 0.\mathbf{cd'ab}, 0.\mathbf{ad'bc}]$
• $\mathbf{C} = [0.\mathbf{d'cba}, 0.\mathbf{d'abc}, 0.\mathbf{bcd'a}]$

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📊 Sample Output

$ python main.py --id 1234

╔══════════════════════════════════════════════════╗
║   NUMERICAL ANALYSIS SUITE — Student ID: 1234     ║
╚══════════════════════════════════════════════════╝

[Problem 1] Newton-Raphson (Hyperbolic Root)
┌───────┬────────────┬────────────┬──────────────┐
│ Iter  │   x_k      │   f(x_k)   │   Error      │
├───────┼────────────┼────────────┼──────────────┤
│   0   │  3.000000  │  0.999999  │      —       │
│   1   │  2.849213  │  0.998842  │  1.5079e-01  │
│   2   │  2.812004  │  0.001203  │  3.7209e-02  │
│  ...  │    ...     │    ...     │     ...      │
└───────┴────────────┴────────────┴──────────────┘
✔ Converged in 6 iterations | Root: x* = 2.808921

[Problem 6] 2D Heat Transfer PDE (Gauss-Seidel)
✔ Converged after 143 sweeps | Tolerance: 1e-6
→ Temperature field saved to heatmap_output.png

[Problem 7] LC Ladder Network Eigenvalue Analysis
Eigenfrequencies (rad/s): [1204.5, 3872.1, 6510.8]
→ Mode shapes saved to eigenmodes_output.png

Actual figures (convergence plots, heatmaps, mode shapes) are generated in the output/ directory on each run.

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💻 Installation & Execution

1. Clone Repository:

   git clone https://github.com/your-username/numerical-analysis-suite.git
   cd numerical-analysis-suite

2. Install Dependencies:

   pip install -r requirements.txt

3. Run Solver Suite:

   python main.py --id <your_student_id>

👨‍💻 Author

Armin Ilat
Electrical Engineering Student

Interests:
Electrical Engineering · Scientific Computing · Programming · Digital Communications · Control Systems · Engineering Simulation

🔗 Links

• LinkedIn: linkedin.com/in/armin-ilat
• GitHub: github.com/Armin-Il
• YouTube: @VoltVerse-Electrical

📄 License

This project is licensed under the MIT License.

Developed for Shahid Beheshti University (SBU) Numerical Analysis Course.