Floating Point Numbers: LU Decomposition and Errors

This lecture covers the representation of floating point numbers, machine epsilon, round-off errors, LU decomposition with and without pivoting, permutation matrices, Hilbert matrix example, matrix norm, condition number, and perturbation to the solution in the context of computational physics. It explains the instability of LU decomposition without pivoting, the importance of machine epsilon, and the consequences of round-off errors. The lecture also delves into the use of permutation matrices to improve LU decomposition, the behavior of Hilbert matrices, and the implications of the condition number on the accuracy of solutions.