M371 - Alexiades Review2: for the Midterm
in addition to the material for Quiz 1 Material

floating point numbers and arithmetic

measures and types of error

Taylor expansion

finding roots

linear systems

nonlinear systems

finite-differences for derivatives
Concepts

finite precision and consequencies, machine epsilon, dangers in machine arithmetic

roundoff error: cause, consequencies, strategies for minimizing loss of significant digits

absolute / relative error of an approximation, big O notation

well / ill conditioned process (having small / large rel.error amplification factor)

norm axioms ; standard vector, matrix, function norms

well / ill conditioned matrix, condition number

absolute / relative error bounds in solving Ax=b

iterations: order of convergence, stopping criteria

Jacobian of a vector function F: ℝⁿ → ℝⁿ   ( the n×n matrix   J(x) = [ ∂F_i(x) / ∂x_j ] )
Methods

nested form of polynomial for efficient evaluation

find roots by bisection, Newton-Raphson, secant, fixed point

solve Ax=b:   reduction to REF + backsubstitution, LU factorization, pivoting, relevant Matlab functions

Newton method for a nonlinear system F(x) = 0 : solve the (linear) system  J(x_n) Δx = −F(x_n) for Δx , then x_n+1 = x_n + Δx

finite-difference approximation of derivatives: forward, backward, centered, their approximation order, roundoff error

For each Method should know: what it is for, idea, advantages/disadvantages, and:

Method	know	discretization error	"roundoff" error	order/rate
Taylor expansion	algo	en = f(n+1)(ξ) / (n+1)!  (x-xo)n+1
bisection  (for F(x)=0)	algo	en = (b-a) / 2n		linear
fixed point  (for x=g(x))	algo	|en+1| ≤ K |en| ,   K=Lipschitz constant	ε / (1−K)	linear
Newton-Raphson (for F(x)=0)	algo	|en+1| ≤ C |en|2		quadratic
secant  (for F(x)=0)	idea	|en+1| ≤ C |en|p ,  p ≈ 1.62		p ≈ 1.62
forward FD for f '(x)	algo	O(h)	ε / h	1st order
backward FD for f '(x)	algo	O(h)	ε / h	1st order
centered FD for f '(x)	algo	O(h2)	ε / h	2nd order