How to Calculate System of Linear Equations Python Without Numpy

Introduction

A system of linear equations consists of two or more linear equations made up of two or more variables. Solving such a system means finding the values of the variables that satisfy all equations simultaneously.

In Python, there are several methods to solve systems of linear equations without using NumPy. These methods include:

• Substitution method
• Elimination method
• Matrix method (using lists)

This guide will focus on the matrix method implemented using Python's built-in capabilities.

Methods for Solving Linear Equations

Substitution Method

The substitution method involves solving one equation for one variable and substituting that expression into the other equation(s).

Elimination Method

The elimination method involves adding or subtracting equations to eliminate one variable, solving for another, and then substituting back to find the remaining variables.

Matrix Method

The matrix method represents the system of equations in matrix form and uses matrix operations to find the solution. This method is particularly useful when dealing with large systems of equations.

Python Implementation Without NumPy

Python provides several ways to solve systems of linear equations without using NumPy. One common approach is to use the matrix method with Python's built-in lists and loops.

Example Implementation

def solve_linear_system(A, B):
    n = len(B)

    # Forward elimination
    for i in range(n):
        # Partial pivoting
        max_row = i
        for k in range(i+1, n):
            if abs(A[k][i]) > abs(A[max_row][i]):
                max_row = k
        A[i], A[max_row] = A[max_row], A[i]
        B[i], B[max_row] = B[max_row], B[i]

        # Make the diagonal element 1
        pivot = A[i][i]
        for j in range(i, n):
            A[i][j] /= pivot
        B[i] /= pivot

        # Eliminate other rows
        for k in range(n):
            if k != i and A[k][i] != 0:
                factor = A[k][i]
                for j in range(i, n):
                    A[k][j] -= factor * A[i][j]
                B[k] -= factor * B[i]

    return B

# Example usage
A = [[2, 1, -1], [-3, -1, 2], [-2, 1, 2]]
B = [8, -11, -3]
solution = solve_linear_system(A, B)
print("Solution:", solution)

This code implements Gaussian elimination with partial pivoting to solve a system of linear equations. The function takes the coefficient matrix A and the constant vector B as inputs and returns the solution vector.

Worked Example

Let's solve the following system of linear equations using the Python implementation above:

2x + y - z = 8
-3x - y + 2z = -11
-2x + y + 2z = -3

The coefficient matrix A and constant vector B are:

A = [[2, 1, -1], [-3, -1, 2], [-2, 1, 2]]
B = [8, -11, -3]

Running the solve_linear_system function with these inputs gives the solution:

Solution: [2.0, 3.0, -1.0]

This means x = 2, y = 3, and z = -1 is the solution to the system.