The Conjugate Gradient Method for Solving Linear Systems
Linear Systems & Numerical Methods DS practice problem on Onlearn.
Difficulty: medium.
Topics: The Conjugate Gradient Method for Solving Linear Systems, Residual Vector, A-conjugacy, Symmetric Positive Definite (SPD) Matrices, Krylov Subspaces, Condition Number, Numerical Analysis, Linear Algebra, Optimization, Scientific Computing, Applied Mathematics, Iterative Solvers, Preconditioning Techniques, Sparse Matrix Computations, Unconstrained Optimization, Convergence Theory.
Task: Implement the Conjugate Gradient Method for Solving Linear Systems Your task is to implement the Conjugate Gradient (CG) method, an efficient iterative algorithm for solving large, sparse, symmetric, positive definite linear systems. Given a matrix A and a vector b, the algorithm will solve for x in the system \( Ax = b \). Write a function conjugate gradient(A, b, n, x0=None, tol=1e 8) that performs the Conjugate Gradient method as follows: A: A symmetric, positive definite matrix representing the linear system. b: The vector on the right side of the equation. n: Maximum number of iterations. x0: Initial guess for the solution vector. tol: Tolerance for stopping criteria. The function should return the solution vector x.