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. 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.