Nonlinearity Root-finding Bisection Fixed Point Iteration Newton s Method Secant Method Conclusion. Nonlinear Systems
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1 Nonlinear Systems CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Doug James (and Justin Solomon) CS 205A: Mathematical Methods Nonlinear Systems 1 / 27
2 Part III: Nonlinear Problems Not all numerical problems can be solved with \ in Matlab. CS 205A: Mathematical Methods Nonlinear Systems 2 / 27
3 Question Have we already seen a nonlinear problem? CS 205A: Mathematical Methods Nonlinear Systems 3 / 27
4 Question Have we already seen a nonlinear problem? minimize A x 2 such that x 2 = 1 nonlinear! CS 205A: Mathematical Methods Nonlinear Systems 3 / 27
5 Root-Finding Problem Given: f : R n R m Find: x with f( x ) = 0 CS 205A: Mathematical Methods Nonlinear Systems 4 / 27
6 Root-Finding Applications Collision detection (graphics, astronomy) Graphics rendering (ray intersection) Robotics (kinematics) Optimization (line search) CS 205A: Mathematical Methods Nonlinear Systems 5 / 27
7 Issue: Regularizing Assumptions f(x) = { 1 when x 0 1 when x > 0 CS 205A: Mathematical Methods Nonlinear Systems 6 / 27
8 Issue: Regularizing Assumptions f(x) = g(x) = { 1 when x 0 1 when x > 0 { 1 when x Q 1 when x Q CS 205A: Mathematical Methods Nonlinear Systems 6 / 27
9 Typical Regularizing Assumptions Continuous f( x) f( y) as x y CS 205A: Mathematical Methods Nonlinear Systems 7 / 27
10 Typical Regularizing Assumptions Continuous f( x) f( y) as x y Lipschitz f( x) f( y) 2 c x y 2 for all x, y (same c) CS 205A: Mathematical Methods Nonlinear Systems 7 / 27
11 Typical Regularizing Assumptions Continuous f( x) f( y) as x y Lipschitz f( x) f( y) 2 c x y 2 for all x, y (same c) Differentiable Df( x) exists for all x CS 205A: Mathematical Methods Nonlinear Systems 7 / 27
12 Typical Regularizing Assumptions Continuous f( x) f( y) as x y Lipschitz f( x) f( y) 2 c x y 2 for all x, y (same c) Differentiable Df( x) exists for all x C k k derivatives exist and are continuous CS 205A: Mathematical Methods Nonlinear Systems 7 / 27
13 Today f : R R CS 205A: Mathematical Methods Nonlinear Systems 8 / 27
14 Property of Continuous Functions Intermediate Value Theorem Suppose that f : [a, b] R is continuous and that f(a) < u < f(b) or f(b) < u < f(a). Then, there exists z (a, b) such that f(z) = u CS 205A: Mathematical Methods Nonlinear Systems 9 / 27
15 Reasonable Input Continuous function f(x) l, r R with f(l) f(r) < 0 (why?) CS 205A: Mathematical Methods Nonlinear Systems 10 / 27
16 Bisection Algorithm 1. Compute c = l+r /2. 2. If f(c) = 0, return x = c. 3. If f(l) f(c) < 0, take r c. Otherwise take l c. 4. Return to step 1 until r l < ε; then return c. CS 205A: Mathematical Methods Nonlinear Systems 11 / 27
17 Bisection: Illustration f(x) f(x) > 0 l c x r x f(x) < 0 CS 205A: Mathematical Methods Nonlinear Systems 12 / 27
18 Two Important Questions 1. Does it converge? CS 205A: Mathematical Methods Nonlinear Systems 13 / 27
19 Two Important Questions 1. Does it converge? Yes! Unconditionally. CS 205A: Mathematical Methods Nonlinear Systems 13 / 27
20 Two Important Questions 1. Does it converge? Yes! Unconditionally. 2. How quickly? CS 205A: Mathematical Methods Nonlinear Systems 13 / 27
21 Convergence Analysis Examine E k with x k x < E k. CS 205A: Mathematical Methods Nonlinear Systems 14 / 27
22 Bisection: Linear Convergence E k E k for E k r k l k CS 205A: Mathematical Methods Nonlinear Systems 15 / 27
23 Fixed Points g(x ) = x CS 205A: Mathematical Methods Nonlinear Systems 16 / 27
24 Fixed Points g(x ) = x Question: Same as root-finding? CS 205A: Mathematical Methods Nonlinear Systems 16 / 27
25 Simple Strategy x k+1 = g(x k ) CS 205A: Mathematical Methods Nonlinear Systems 17 / 27
26 Convergence Criterion E k x k x = g(x k 1 ) g(x ) CS 205A: Mathematical Methods Nonlinear Systems 18 / 27
27 Convergence Criterion E k x k x = g(x k 1 ) g(x ) c x k 1 x if g is Lipschitz = ce k 1 CS 205A: Mathematical Methods Nonlinear Systems 18 / 27
28 Convergence Criterion E k x k x = g(x k 1 ) g(x ) c x k 1 x = ce k 1 if g is Lipschitz = E k c k E 0 0 as k (c < 1) CS 205A: Mathematical Methods Nonlinear Systems 18 / 27
29 Alternative Criterion Lipschitz near x with good starting point. CS 205A: Mathematical Methods Nonlinear Systems 19 / 27
30 Alternative Criterion Lipschitz near x with good starting point. e.g. C 1 with g (x ) < 1 CS 205A: Mathematical Methods Nonlinear Systems 19 / 27
31 Convergence Rate of Fixed Point When it converges... Always linear (why?) CS 205A: Mathematical Methods Nonlinear Systems 20 / 27
32 Convergence Rate of Fixed Point When it converges... Always linear (why?) Often quadratic! ( board) CS 205A: Mathematical Methods Nonlinear Systems 20 / 27
33 Approach for Differentiable f(x) f(x) x 0 x 1 x 2 x CS 205A: Mathematical Methods Nonlinear Systems 21 / 27
34 Newton s Method x k+1 = x k f(x k) f (x k ) CS 205A: Mathematical Methods Nonlinear Systems 22 / 27
35 Newton s Method x k+1 = x k f(x k) f (x k ) Fixed point iteration on g(x) x f(x) f (x) CS 205A: Mathematical Methods Nonlinear Systems 22 / 27
36 Convergence of Newton Simple Root A root x with f (x ) 0. CS 205A: Mathematical Methods Nonlinear Systems 23 / 27
37 Convergence of Newton Simple Root A root x with f (x ) 0. Quadratic convergence in this case! ( board) CS 205A: Mathematical Methods Nonlinear Systems 23 / 27
38 Issue Differentiation is hard! CS 205A: Mathematical Methods Nonlinear Systems 24 / 27
39 Secant Method x k+1 = x k f(x k)(x k x k 1 ) f(x k ) f(x k 1 ) CS 205A: Mathematical Methods Nonlinear Systems 25 / 27
40 Secant Method x k+1 = x k f(x k)(x k x k 1 ) f(x k ) f(x k 1 ) Trivia: Converges at rate CS 205A: Mathematical Methods Nonlinear Systems 25 / 27
41 Secant Method x k+1 = x k f(x k)(x k x k 1 ) f(x k ) f(x k 1 ) Trivia: Converges at rate ( Golden Ratio ) CS 205A: Mathematical Methods Nonlinear Systems 25 / 27
42 Hybrid Methods Want: Convergence rate of secant/newton with convergence guarantees of bisection CS 205A: Mathematical Methods Nonlinear Systems 26 / 27
43 Hybrid Methods Want: Convergence rate of secant/newton with convergence guarantees of bisection e.g. Dekker s Method: Take secant step if it is in the bracket, bisection step otherwise CS 205A: Mathematical Methods Nonlinear Systems 26 / 27
44 Single-Variable Conclusion Unlikely to solve exactly, so we settle for iterative methods Must check that method converges at all Convergence rates: Linear: Ek+1 CE k for some 0 C < 1 Superlinear: E k+1 CEk r for some r > 1 Quadratic: r = 2 Cubic: r = 3 Time per iteration also important Next CS 205A: Mathematical Methods Nonlinear Systems 27 / 27
Nonlinearity Root-finding Bisection Fixed Point Iteration Newton s Method Secant Method Conclusion. Nonlinear Systems
Nonlinear Systems CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Nonlinear Systems 1 / 24 Part III: Nonlinear Problems Not all numerical problems
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