Gaussian Elimination with Partial Pivoting

Gaussian elimination is a classical method for solving linear systems of equations, while Gaussian elimination with partial pivoting serves as an improved version that enhances numerical stability. This method selects the element with the largest absolute value in the current column as the pivot during each elimination step, thereby avoiding rounding errors caused by division with extremely small values. The implementation approach primarily consists of three key steps: First, traverse each column to locate the pivot row by comparing the absolute values of elements from the diagonal downward in the current column; then swap the current row with the pivot row to position the pivot element on the diagonal; finally perform standard Gaussian elimination operations, using the pivot row to eliminate corresponding column elements in the rows below. When implementing in MATLAB, developers can leverage matrix operation features to simplify the code. Special attention should be paid to handling both the coefficient matrix and the right-hand side vector simultaneously during row exchanges. This method's advantage lies in effectively controlling error propagation, making it particularly suitable for solving ill-conditioned systems or cases where diagonal elements approach zero. Compared with standard Gaussian elimination, it demonstrates significantly better numerical stability. In practical applications, the computational complexity of partial pivoting elimination remains the same as the conventional method at O(n³), but with substantially improved reliability. This technique finds widespread application in engineering computations, scientific simulations, and other fields requiring solutions to large-scale equation systems. The MATLAB implementation typically involves using built-in functions for element comparison and row operations, while maintaining proper indexing for matrix manipulations.