Convergence Rates on Root Finding

Transcription

1 Convergence Rates on Root Finding Com S 477/577 Oct 5, 004 A sequence x i R converges to ξ if for each ǫ > 0, there exists an integer Nǫ) such that x l ξ > ǫ for all l Nǫ). The Cauchy convergence criterion states that a sequence x i R is convergent if and only if for each ǫ > 0 there exists an Nǫ) such that x l x m < ǫ for all l,m Nǫ). Let a sequence x i R be generated by an iteration function Φ, that is, x i+1 = Φx i ), i = 0,1,,... Let ξ be a fixed point of Φ, that is, ξ = Φξ). Suppose that the sequence {x i } is generated in the neighborhood of ξ. The corresponding iteration method is said to be of at least pth order if there exists a neighborhood Nξ) of ξ such that for all x 0 Nξ) the generated sequence x i+1 = Φx i ), i = 0,1,..., satisfies x i+1 ξ C x i ξ p, where C < 1 if p = 1. In the case of first order convergence, for instance, we have Since C < 1, it follows that x i ξ C x i 1 ξ C x i ξ C i x 0 ξ. lim x i ξ = lim C i x 0 ξ = 0, i i namely, the sequence {x i } will converge to ξ. Now suppose Φ is sufficiently often differentiable in Nξ). If x i Nξ) and if Φ k) ξ) = 0 for k = 1,,...,p 1 but Φ p)ξ) 0, that is, ξ is a zero of order p, then it follows from the Taylor expansion that Because Φξ) = ξ, we obtain x i+1 = Φx i ) = Φξ) + Φp) ξ) x i ξ) p + O x i ξ) p+1). p! x i+1 ξ lim i x i ξ) p = Φp) ξ). p! For p =,3,..., the method is of precisely) pth order. The method is of first order if p = 1 and Φ ξ) < 1. When 0 < Φ ξ) < 1, the sequence {x i } will converge monotonically to ξ as shown in the left figure below. When 1 < Φ ξ) < 0, the sequence will alternate about ξ during convergence as shown in the right figure. 1

2 x φ ) x x φ ) x x i 0 x i+1 x i + ξ 0 x i x i + ξ x i+1 In the below we study the convergence rates of several root finding methods introduced before. 1 Quadratic Convergence of Newton s Method Newton s method has the iteration function Φx) = x fx) f x) with fξ) = 0. Suppose f is sufficiently continuously differentiable in some neighborhood Nξ). In the nondegenerate case, f ξ) 0. So we have Φξ) = ξ, Φ ξ) = 1 f x)) fx)f x) f x)) x=ξ = 0, since fξ) = 0, ) ) f x)f x) f x)f x) fx)f x) f x)) f x)f x) f x)) fx)f x) Φ ξ) = 4 f x)) x=ξ 3f f ) f ξ)) ξ) fξ) f ξ)) ξ) + fξ)f ξ) f ξ) = 3f f ξ)) ξ) = f ξ) = f ξ) f ξ) 0. f ξ) ) 4 ) 4, since fξ) = 0 So Newton s method is quadratically convergent. In the degenerate case, ξ is an m-fold zero of f, for some m > 1, that is, f i) ξ) = 0, for i = 0,1,...,m 1. We will leave to the students to determine the order of convergence in this case.

3 Linear Convergence of Regula Falsi For clarity of analysis we let x i = b i for all i. We make some simplification assumptions for the discussion of the convergence behavior: f exists and for some k the following conditions hold: a) x k < a k ; b) fx k ) < 0 and fa k ) > 0; c) f x) 0 for all x [x k,a k ]. x k x k +1 a k a) b) Under these assumptions, either fx k+1 ) = 0 or fx k+1 )fx k ) > 0 and consequently x i < x i+1 < a i+1 = a i. To see this, use the remainder formula for polynomial interpolation at x k and a k : fx) px) = x x k )x a k ) f η) for x [x k,a k ] and a suitable η [x k,a k ]. Under condition c), the above equation implies that fx) px) 0. In particular, fx k+1 ) px k+1 ) 0, which in turn implies that fx k+1 ) 0 since px k+1 ) = 0. Unless fx k+1 ) = 0, in which case the iteration stops at x k+1, we can see that conditions a), b), and c) hold for all i k. Therefore a i = a k = a and x i+1 = afx i) x i fa) fx i ) fa) for all i k. Furthermore, {x i } for i k form a monotone increasing sequence bounded by a. So lim i x i = ξ exists. Consequently, fξ) 0, fa) > 0, and ξ = afξ) ξfa) fξ) fa), which gives ξ a)fξ) = 0. But ξ < a since fξ) 0 < fa). Hence fξ) = 0 and {x i } converges to a zero of f. The above discussion enables us to look at the order of convergence through the iteration function afx) xfa) x i+1 = Φx i ), where Φx) = fx) fa). 3

4 Since fξ) = 0, we obtain that ) af ξ) fa) fa) + ξfa)f ξ) Φ ξ) = fa) = 1 f ξ a ξ) fξ) fa). By the mean value theorem, there exist η 1,η such that fξ) fa) ξ a fx i ) fξ) x i ξ = fa) ξ a = fx i) x i ξ = f η 1 ), ξ < η 1 < a; 1) = f η ), x i < η < ξ. ) Since f x) 0, f x) increases monotonically in [x i,a], So f η ) f ξ) f η 1 ). Meanwhile, condition ), x i < ξ, and fx i ) < 0 together imply that 0 < f η ). Therefore 0 < f ξ) f η 1 ). We have thus shown that 0 Φ ξ) = 1 f ξ) f η 1 ) < 1. So the regula falsi method converges linearly. From the previous discussion we see that the method of regula falsi will almost always end up with the one-sided convergence demonstrated before. 3 Superlinear Convergence of Secant Method In secant method, the iteration is in the form x i+1 = x i fx i), i = 0,1,... 3) To determine the local convergence rate, without loss of generality we assume that the sequence {x i } is in a small enough neighborhood of the zero ξ and that f is twice differentiable. Subtract ξ from both sides of 3): x i+1 ξ = x i ξ) fx i) = x i ξ) 1 f[x ) i,ξ], since f[x i,ξ] = fx i) fξ) = fx i) x i ξ x i ξ = x i ξ)x i 1 ξ) f[x i 1,x i ] f[x i,ξ] x i 1 ξ) = x i ξ)x i 1 ξ) f[x i 1,x i,ξ]. 4) From error estimation of polynomial interpolation, we learned that = f η 1 ), η 1 I[x i 1,x i ]; f[x i 1,x i,ξ] = 1 f η ), η I[x i 1,x i,ξ], 4

5 where I[x i 1,x i ] is the smallest interval containing x i 1 and x i, and I[x i 1,x i,ξ] the smallest interval containing x i 1, x i, ξ { If ξ is a simple zero, that is, f ξ) 0, there exists a bound M and an interval J = x x ξ } ǫ for some ǫ > 0 such that 1 f η ) f η 1 ) M, 5) for any η 1,η J. Let e i = M x i ξ and e 0,e 1 < min{1,ǫm}. By induction and using 4) and 5) we can easily show that e i+1 = M x i+1 ξ M ei M ei 1 M M = e ie i 1, and e i min{1,ǫm}, for i = 1,,... Let q = 1 + 5)/ be the root of the equation z z 1 = 0. Then we have e i K qi, i = 0,1,,... where K = max{e 0, q e 1 } < 1. This is because by induction) e i+1 e i e i 1 K qi K qi 1 = K qi 1 q+1) = K qi 1 q = K qi+1. Thus the secant method converges at least as well as a method of order p = 1+ 5 = One-step secant requires one additional function evaluation. But one-step Newton requires two f and f ). Therefore two secant steps are as expensive as single Newton step. But two secant steps has a convergence order of 1.618).618. This explains why in practice the secant method always dominates Newton s method with numerical derivatives. References [1] J. Stoer and R. Bulirsch. Introduction to Numerical Analysis. Springer-Verlag New York, Inc., nd edition, [] M. Erdmann. Lecture notes for Mathematical Fundamentals for Robotics. The Robotics Institute, Carnegie Mellon University, [3] W. H. Press, et al. Numerical Recipes in C++: The Art of Scientific Computing. Cambridge University Press, nd edition, 00. 5