SSOR Preconditioned Conjugate Gradient Method for Solving Linear Equations

This function implements the SSOR preconditioned conjugate gradient method to solve the linear equation system Ax=b, where A is a positive definite matrix and b is the right-hand side vector. Users can adjust the SSOR preconditioning intensity using the omega parameter within the range of 0 to 2. The Times parameter specifies the maximum iteration count, while errtol defines the convergence tolerance based on error threshold. The function returns NewX as the approximate solution to Ax=b and avgerr representing the current mean absolute error. For better understanding of this implementation, here are key algorithmic concepts: - Conjugate Gradient Method: An iterative optimization algorithm specifically designed for solving systems of linear equations with symmetric positive definite matrices. The implementation uses orthogonal direction vectors to minimize the quadratic form, ensuring convergence within n steps for exact arithmetic. - Positive Definite Matrix: A symmetric matrix where all eigenvalues are positive, guaranteeing the existence of a unique solution and stable convergence behavior in the conjugate gradient algorithm. - SSOR Preconditioning: A matrix splitting technique that accelerates convergence by transforming the original system into an equivalent one with better spectral properties. The code implements symmetric successive over-relaxation by decomposing the matrix into diagonal, lower, and upper triangular components. - Average Absolute Error: Computed as the mean difference between predicted and actual values throughout iterations, serving as a convergence metric. The implementation calculates this error norm at each iteration to monitor solution accuracy. This function provides an efficient numerical solution for linear systems while demonstrating fundamental computational mathematics concepts. The code structure includes optimized matrix-vector operations and efficient memory handling for large-scale problems, making it suitable for scientific computing applications.