Question

Design and implement a MATLAB program that receives a coefficient matrix [A]n*n and a vector of constants {b}n*1 as inputs, and generates a vector {x}n*1, as a solution to a system of n linear algebraic equations represented in matrix form as [A]{x}={b}. Your program should handle division by zero (pivoting). Compare your results with the results obtained from MATLAB built-in left-division x = A \ b and/or matrix inversion x = inv(A) * b functions.

Answer 1

The student is tasked with creating a MATLAB program to solve a system of linear algebraic equations, implementing Gaussian elimination with partial pivoting and comparing the solution to MATLAB's built-in methods.

Step-by-step explanation:

The task is to design and implement a MATLAB program to solve a system of linear algebraic equations [A]{x}={b}, where [A] is a coefficient matrix and {b} is a constants vector. This can be done by using various methods such as Gaussian elimination with partial pivoting to avoid division by zero errors. Once the program is implemented, the solution {x} should be compared with MATLAB's built-in solutions using left-division or matrix inversion functions.

The program implementation should start by receiving the matrix [A] and vector {b} as inputs. Then it should perform row operations to form an upper triangular matrix, and apply back substitution to find the solution vector {x}. Throughout the process, pivoting is necessary to prevent division by zero, ensuring numerical stability.

Here is a simple scaffold for the MATLAB implementation:

function x = solveLinearSystem(A, b)
[m,n] = size(A);
if m ~= n
error('Matrix A must be square.');
end
% Implementation of Gaussian elimination with pivoting
% ...
% After obtaining an upper triangular matrix and solving for x
x = backSubstitution(U, b_);
end

It is important to note that the built-in MATLAB functions for solving linear systems, A \ b and inv(A) * b, are efficient and robust, utilizing advanced algorithms that handle pivoting and other numerical issues.