Lecture 7 handout.pdf - Math 472 Lecture 7 Chapter 1...

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Math 472 Lecture 7Chapter 1Solving EquationsRuoyu WuU Michigan1 / 14
1.4 Newton’s methodThe Newton’s (or the Newton–Raphson) MethodOne step of Newton’s Methodx1=x0-f(x0)f0(x0)2 / 14
1.4 Newton’s methodNewton’s Method:x0initial guess;fori= 0,1,2, . . . , N-1doxi+1g(xi) :=xi-f(xi)f0(xi);endreturnxNSpecial case of FPI, withg(x) :=x-f(x)f0(x)3 / 14
1.4 Newton’s methodExample 1.11Use Newton’s Method to find a root ofx3+x-1 = 0Solution: (Recall the third choice for FPI in the previous lecture)4 / 14
1.4 Newton’s methodIn the previous example, once convergence takes hold, the number ofcorrect places in the approximation roughlydoubleson each iterationThis is the characteristic of“quadratic convergence”Definition 1.10 (Quadratic Convergence)Leteibe the error afteristep of an iterative algorithm. The iteration isquadratically convergentifM:= limi→∞ei+1e2i<5 / 14
1.4 Newton’s methodTheorem 1.11 (Quadratic Convergence of Newton’s Method)Letf(·)betwice continuously differentiableandf(r) = 0.Iff0(r)6= 0, thenNewton’s Method is locally and quadratically convergent tor. Furthermore,M:= limi→∞ei+1e2i=f00(r)2f0(r)(1)In the previous exampleM0.85which agrees with the tableProof of local convergence: a particular form of fixed-point iterationg(x) =x-f(x)f0(x). Checkg0(r) = 0.

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