Solve Linear Equations: Gauss-Seidel Method - Computational Physics Formulation & Simulation

Program 1

Gauss-Seidel Method

Python 3.11 Ready

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Mathematical Problem Formulation

Solve the following system of linear equations using the Gauss-Seidel method until convergence to a tolerance of 0.00001. Use initial guesses $ x = 0, y = 0, z = 0 $. Input the augmented matrix (coefficients and constants) in a single array.

Solve the following system of linear equations

$ 4x - y + z = 15 $

$ -x + 4y - z = 10 $

$ 2x - y + 3z = 10 $

Theoretical Background & Computational Mechanics

Algorithm:

Initial Guess:

Start with an initial guess for the unknowns (in this case, $ x = y = z = 0 $).

Iteratively Solve for Each Variable:

Use the current or updated values of the variables as soon as they are available.

From the system:

$ x^{(k+1)} = \frac{1}{4} (15 + y^{(k)} - z^{(k)}) $

$ y^{(k+1)} = \frac{1}{4} (10 + x^{(k+1)} + z^{(k)}) $

$ z^{(k+1)} = \frac{1}{3} (10 - 2x^{(k+1)} + y^{(k+1)}) $

Convergence Check:

Stop when the difference between consecutive values of $ x $, $ y $, and $ z $ is less than a predefined tolerance (e.g., $ 10^{-5} $).

Program 2

Gauss-Seidel Method

Python 3.11 Ready

import numpy as np
A = np.array([
    [10, -1, 2, 0, 0],
    [-1, 11, -1, 3, 0],
    [2, -1, 10, -1, 0],
    [0, 3, -1, 8, -2],
    [0, 0, 0, -2, 6]
])

B = np.array([6, 25, -11, 15, 10], dtype=float)
n = len(B)
x = np.zeros(n)   # Initial guesses (all zeros)
tolerance = 0.00001    # Tolerance for convergence
max_iterations = 1000

print("Gauss-Seidel Iterations:\n")

for iteration in range(max_iterations):
    x_old = x.copy()  # keep old values for comparison

# Loop through each equation
    for i in range(n):
        # Calculate the sum for known terms
        sum1 = 0
        for j in range(n):
            if j != i:
                sum1 += A[i][j] * x[j]
        # Update the current variable
        x[i] = (B[i] - sum1) / A[i][i]

# Print current iteration results
    print(f"Iteration {iteration+1}: {np.round(x, 5)}")

# Check if solution has converged
    if np.max(np.abs(x - x_old)) < tolerance:
        print(f"\nConverged in {iteration+1} iterations.")
        break

print("\nFinal Solution:")
for i in range(n):
    print(f"x{i+1} = {x[i]:.5f}")

# Verify using numpy solver
print("\nCheck with numpy.linalg.solve:")
sol = np.linalg.solve(A, B)
print(sol)

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Mathematical Problem Formulation

Gauss-Seidel Iterative Method Problem

Solve the following system of linear equations using the Gauss-Seidel method:

\[ \begin{cases} 10x_1 - x_2 + 2x_3 = 6 \\ -x_1 + 11x_2 - x_3 + 3x_4 = 25 \\ 2x_1 - x_2 + 10x_3 - x_4 = -11 \\ 3x_2 - x_3 + 8x_4 - 2x_5 = 15 \\ -2x_4 + 6x_5 = 10 \end{cases} \]

Use the following specifications for the iterative solution:

Initial guess: $x_1 = x_2 = x_3 = x_4 = x_5 = 0$
Convergence tolerance: $10^{-5}$
Maximum iterations: 1000

Compute the solution vector $\mathbf{x} = [x_1, x_2, x_3, x_4, x_5]^T$ using the Gauss-Seidel iteration and verify it using a direct solver.

Algorithm:

A = Coefficient matrix

B = Right-hand side constants

We start with all zeros as initial guesses.

For each iteration:

Each unknown is updated using the latest available values.
The formula used is:

$ x_i = \frac{B_i - \sum_{j \neq i} a_{ij} x_j}{a_{ii}} $

The process repeats until the change is smaller than the tolerance.

Program 3

Gauss-Seidel Method

Python 3.11 Ready

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

Gauss-Seidel Iterative Method Problem

Solve the following system of linear equations using the Gauss-Seidel method:

\[ \begin{cases} 10x_1 - x_2 + 2x_3 = 6 \\ -x_1 + 11x_2 - x_3 + 3x_4 = 25 \\ 2x_1 - x_2 + 10x_3 - x_4 = -11 \\ 3x_2 - x_3 + 8x_4 - 2x_5 = 15 \\ -2x_4 + 6x_5 = 10 \end{cases} \]

Use the following specifications for the iterative solution:

Initial guess: $x_1 = x_2 = x_3 = x_4 = x_5 = 0$
Convergence tolerance: $10^{-5}$
Maximum iterations: 1000

Compute the solution vector $\mathbf{x} = [x_1, x_2, x_3, x_4, x_5]^T$ using the Gauss-Seidel iteration and verify it using a direct solver.

Alternate Method using Numpy Array Slice.

Theoretical Principles & Computational Physics Guide

Gauss-Seidel Method: Matrix Formulation & Linear Algebraic Methods

Dr. Alorika Chatterjee & Dr. Anirban Shaw Science & Physics Learners Python 3 / NumPy / SciPy

1. Core Principles & Theoretical Framework

Linear Transformation: Matrices represent multi-variable coupled physical equations, rotation operators, and quantum observables.

Coordinate Transformations: Maps physical state vectors $\mathbf{x} \mapsto \mathbf{A}\mathbf{x}$ preserving or scaling metric spaces.

Numerical Stability: Employs pivoting and condition number monitoring to avoid numerical singularity during inversion.

2. Mathematical Formulation & Algorithmic Pipeline

Linear algebraic operator transformation:

$$ \mathbf{A}\mathbf{x} = \mathbf{b} \iff \sum_{j=1}^n A_{ij} x_j = b_i, \quad \det(\mathbf{A}) \neq 0 $$

1. Construct Matrix/Vector → 2. Linear Operator Execution → 3. Verify Norm / Residual $\|\mathbf{A}\mathbf{x} - \mathbf{b}\|$

3. Numerical Mechanics & Implementation Logic

Memory Layout: Structure row and column arrays ensuring dimensional conformability.
Algorithmic Execution: Apply row operations or inner product contractions iteratively.
Condition Checking: Verify determinant or diagonal dominance to guarantee reliable numerical convergence.

4. Relevant Computational Practice Exercises

Exercise 1

Implement Gauss-Seidel Method for a $3 \times 3$ system and verify the result against analytical solutions.

Exercise 2

Measure the computational scaling of the algorithm as matrix dimension $N$ increases from $10$ to $100$.

Exercise 3

Test edge cases involving singular or ill-conditioned matrices and handle potential division-by-zero errors.

Standard Literature & References:

Gilbert Strang, "Linear Algebra and Its Applications", Cengage Learning (2006).
William H. Press et al., "Numerical Recipes", Cambridge University Press (2007).

Ch. 9.3 Gauss-Seidel Method

Gauss-Seidel Method

Mathematical Problem Formulation

Theoretical Background & Computational Mechanics

Gauss-Seidel Method

Mathematical Problem Formulation

Gauss-Seidel Iterative Method Problem

Algorithm:

Gauss-Seidel Method

Mathematical Problem Formulation

Gauss-Seidel Iterative Method Problem

Alternate Method using Numpy Array Slice.

1. Core Principles & Theoretical Framework

2. Mathematical Formulation & Algorithmic Pipeline

3. Numerical Mechanics & Implementation Logic

4. Relevant Computational Practice Exercises