Numerical Analysis

Taylor's Theorem with Remainder - Fixed Point Convergence - If g is continuously differentiable, g(r)=r, and ‖g'(r)‖<1, then the fixed point iteration converges linearly with rate S to the fixed point r. S=‖g'(r)‖ Newton's Method - Approximates the root of non-linear functions by finding the root of the tangent line. In the second iteration, plug in the approximation from the first iteration as "x_n". Repeat recursively. Newton's Method Convergence - Newton's Method converges quadratically to a root x if f'(r)≠0 at rate M. M=lim(e_(i+1))/e_i² Derivation of Newton's Method - f'(x_0)(x-x_0)=0-f(x_0) x-x_0=-f(x_0)/f'(x_0) x=x_0-f(x_0)/f'(x_0) Secant Method - Approximates the root of the function by taking the secant line through the last two approximation. The new guess is the point in which the secant line intersects the x-axis. Start with two initial approximations p₀ and p₁. The next guess, p₂, is the x-intercept of the line joining (p₀,f(p₀) and (p₁,f(p₁). The approximation p₃ is the x-intercept of the line joining (p₁,f(p₁)) and (p₂,f(p₂)), and so on. Method of False Position - Combination of the Secant Method and the Bisection Method. It uses the Bisection Method to ensure that the two previous approximations used for the secant line have opposite signs.