This segment introduces incomplete Cholesky factorization as a powerful preconditioning technique. The speaker explains the method's core idea: simulating a Cholesky factorization but ignoring fill-in (new non-zero entries) during the process. This results in a sparse approximate factorization that can be efficiently used as a preconditioner. The speaker highlights the method's surprising effectiveness and its reliance on the relationship between the sparsity patterns of the original matrix and its Cholesky factor. This lecture covers iterative solvers for linear systems (Ax=b). Preconditioners improve convergence by simplifying A (e.g., sparsity constraints, incomplete Cholesky factorization, domain decomposition). Alternative iterative methods exist (e.g., splitting A=M-N), particularly for non-symmetric positive definite matrices (normal equations, MinRes, etc.). Finally, conjugate gradient methods extend to nonlinear function minimization.