Transcription of Square Roots via Newton’s Method
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Square Roots via Newton’s Method
Square Roots via newton s methods . G. Johnson, MIT Course 4, 20151 OverviewNumerical methodscan be distinguished from other branches of analysis and computer scienceby three characteristics: They work with arbitraryrealnumbers (and vector spaces/extensions thereof): the desiredresults are not restricted to integers or exact rationals (although in practice we only evercomputerational approximationsof irrational results). Like in computer science (= math + time = math + money), we are concerned not onlywith existence and correctness of the solutions (as in analysis), but with thetime(and othercomputational resources, memory) required to compute the result. We are also concerned withaccuracyof the results, because in practice we only ever haveapproximateanswers: Some algorithms may be intrinsically approximate like the newton s- Method exampleshown below, they convergetowardsthe desired result but neverreachit in a finite numberof fast they convergeis a key question.
be equivalent to Newton’s method to find a root of f(x) = x2 a. Recall that Newton’s method finds an approximate root of f(x) = 0 from a guess x n by approximating f(x) as its tangent line f(x n)+f0(x n)(x x n),leadingtoanimprovedguessx n+1 fromtherootofthetangent: x n+1 = x n f(x n) f0(x n); andforf(x) = x2 ...
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