Solving Linear Equations with MATLAB Matrices: 3 Methods, Exploring Different Solution Paths

# Solving Linear Equations with MATLAB: 3 Methods, Exploring Diverse Solution Paths

## 1. Overview of Linear Equation Systems

Linear equation systems are common mathematical problems that have broad applications in science, engineering, and finance. There are various methods to solve linear equation systems, each with its own advantages and disadvantages.

In MATLAB, the main approaches to solving linear equation systems are direct methods, iterative methods, and Singular Value Decomposition (SVD). Direct methods simplify the coefficient matrix into an upper triangular or diagonal matrix through a series of elementary transformations, and then solve the equation system by back substitution. Iterative methods repeatedly update the approximate values of the unknowns until certain convergence criteria are met. SVD decomposes the coefficient matrix into the product of three matrices, and then uses the decomposed matrices to solve the equation system.

## 2. Methods for Solving Linear Equations in MATLAB

MATLAB provides various methods for solving linear equation systems, including direct methods, iterative methods, and Singular Value Decomposition (SVD).

### 2.1 Direct Methods

Direct methods simplify the coefficient matrix into an upper triangular or diagonal matrix through a series of elementary row operations, and then solve the equation system by back substitution from top to bottom.

#### 2.1.1 Gauss Elimination Method

Gauss Elimination is a classical direct method. Its basic idea is to transform the coefficient matrix into an upper triangular matrix through row operations, and then solve it by back substitution from top to bottom.

**Code Example:**

% Coefficient matrix
A = [2, 1, 1; 4, 3, 2; 8, 7, 4];

% Right-hand side constant vector
b = [4; 10; 22];

% Solving using Gauss Elimination
x = gauss(A, b);

% Displaying the solution vector
disp(x);

**Logical Analysis:**

* The `gauss()` function implements Gauss Elimination to solve linear equations.
* The function first performs elementary row operations on the coefficient matrix to simplify it into an upper triangular matrix.
* Then, it back substitutes to solve for the unknowns from top to bottom.

#### 2.1.2 LU Decomposition Method

LU Decomposition is a direct method that decomposes the coefficient matrix into the product of a lower triangular matrix and an upper triangular matrix, then solves each of these triangular matrices' systems.

**Code Example:**

% Coefficient matrix
A = [2, 1, 1; 4, 3, 2; 8, 7, 4];

% Right-hand side constant vector
b = [4; 10; 22];

% LU Decomposition
[L, U] = lu(A);

% Solving Ly = b
y = L \ b;

% Solving Ux = y
x = U \ y;

% Displaying the solution vector
disp(x);

**Logical Analysis:**

* The `lu()` function decomposes the coefficient matrix into lower triangular matrix `L` and upper triangular matrix `U`.
* It then solves `Ly = b` and `Ux = y` separately to obtain the unknowns' solution vector `x`.

### 2.2 Iterative Methods

Iterative methods update the approximate solutions of unknowns through continuous iterations until certain convergence criteria are met.

#### 2.2.1 Jacobi Iterative Method

Jacobi Iterative Method is an iterative method that breaks the linear equation system into multiple subsystems and then solves each subsystem one by one.

**Code Example:**

% Coefficient matrix
A = [2, 1, 1; 4, 3, 2; 8, 7, 4];

% Right-hand side constant vector
b = [4; 10; 22];

% Solving using Jacobi Iterative Method
x = jacobi(A, b, 100, 1e-6);

% Displaying the solution vector
disp(x);

**Logical Analysis:**

* The `jacobi()` function implements the Jacobi Iterative Method to solve linear equations.
* The function sets the maximum number of iterations and convergence precision.
* During each iteration, the function updates the approximate solutions of each unknown until the convergence conditions are met.

#### 2.2.2 Gauss-Seidel Iterative Method

Gauss-Seidel Iterative Method is an iterative method that, unlike the Jacobi method, uses the latest approximate solution to update each unknown, rather than the approximation from the previous iteration.

**Code Example:**

% Coefficient matrix
A = [2, 1, 1; 4, 3, 2; 8, 7, 4];

% Right-hand side constant vector
b = [4; 10; 22];

% Solving using Gauss-Seidel Iterative Method
x = gaussSeidel(A, b, 100, 1e-6);

% Displaying the solution vector
disp(x);

**Logical Analysis:**

* The `gaussSeidel()` function implements the Gauss-Seidel Iterative Method to solve linear equations.
* The function sets the maximum number of iterations and convergence precision.
* During each iteration, the function updates the approximate solutions