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.
Información del documento
- Subido en
- 26 de diciembre de 2023
- Número de páginas
- 12
- Escrito en
- 2023/2024
- Tipo
- Examen
- Contiene
- Preguntas y respuestas
