Download
1 / 21
210 likes | 330 Vues
This content focuses on optimization and learning algorithms related to root finding for the function g(x). Key algorithms such as Newton's method and the Secant method are explored, highlighting their superlinear convergence rates. The importance of initialization, termination, approximate differentiation, and numerical considerations are covered. Various figures illustrate the iterative processes of both methods, their divergence, and oscillation behavior. Additionally, the convergence rates of different methods, including Bisection, Newton's, and Secant methods, are discussed, providing a comprehensive understanding of effective root-finding strategies.
E N D
CS B553: Algorithms for Optimization and Learning Root finding
g(x) x Roots of g
Key Ideas • Newton’s method • Secant method • Superlinear convergence rates • Initialization and termination • Approximate differentiation • Numerical considerations
Figure 10 Newton’s method g(x) x0 x In a neighborhood of a root, the line tangent to thegraph crosses the x axis near the root
Figure 10 Newton’s method g(x) x1 x In a neighborhood of a root, the line tangent to thegraph crosses the x axis near the root… iterate!
Figure 10 Newton’s method g(x) x2 x In a neighborhood of a root, the line tangent to thegraph crosses the x axis near the root… iterate!
Figure 11 Divergence x1 x g(x)
Figure 11 Divergence x1 x2 x g(x)
Figure 11 Divergence x3 x1 x2 x g(x)
Figure 11 Divergence x3 x1 x2 x4 x g(x)
Figure 11 Divergence x5 x3 x1 x2 x4 x g(x)
Figure 12 Oscillation x
Figure 12 Oscillation x
Figure 12 Oscillation x
Figure 12 Oscillation x
Figure 13 Secant method g(x) x0 x1 x Idea: Use line through two points on graph as approximation ofthe derivative
Figure 13 Secant method g(x) x0 x1 x2 x Idea: Use line through two points on graph as approximation ofthe derivative
Figure 13 Secant method g(x) x3 x0 x1 x2 x Idea: Use line through two points on graph as approximation ofthe derivative
Figure 13 Secant method g(x) x3 x0 x1 x2 x Idea: Use line through two points on graph as approximation ofthe derivative
Orders of convergence • Bisection: linear • Newton’s method: quadratic • Secant method: order 1.6 • Only bisection has guaranteed convergence (given appropriate initial interval) • Newton’s method needs derivatives • Most “out of the box” subroutines take a hybrid approach
Figure 14 Basins of attraction in complex plane: x5-1=0
