Solving linear systems (A\b) in MATLAB - Time & Space Complexity

Time Complexity: Solving linear systems (A\b)

O(n^3)

Understanding Time Complexity

When solving linear systems using A\b in MATLAB, it's important to know how the time needed grows as the system size increases.

We want to understand how the work changes when the matrix gets bigger.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

    n = 1000;
    A = rand(n);
    b = rand(n,1);
    x = A \ b;
    

This code creates a random square matrix A and vector b, then solves for x in the equation A*x = b.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Matrix factorization and back substitution steps inside A\b.
• How many times: These steps involve nested loops over the matrix size, roughly repeating n times inside n times.

How Execution Grows With Input

As the matrix size n grows, the number of operations grows much faster because each element interacts with many others.

Input Size (n)	Approx. Operations
10	About 1,000
100	About 1,000,000
1000	About 1,000,000,000

Pattern observation: When n increases ten times, operations increase about a thousand times, showing a cubic growth.

Final Time Complexity

Time Complexity: O(n^3)

This means the time needed grows roughly with the cube of the matrix size, so doubling n makes the work about eight times bigger.

Common Mistake

[X] Wrong: "Solving A\b takes time proportional to n, like a simple loop."

[OK] Correct: The solution involves nested operations on the matrix, not just one pass, so it grows much faster than n.

Interview Connect

Understanding how solving linear systems scales helps you explain performance in real tasks like simulations or data analysis, showing you know how algorithms behave with bigger data.

Self-Check

"What if the matrix A is sparse (mostly zeros)? How would the time complexity change?"