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Muller’s Method Dan Heflin Sarah Hare

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Muller’s Method What problem does it solve? How is it represented? Where does it come from?

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What does Muller’s Method Solve? It finds the Roots of functions including Complex. This method allows the user to find roots for functions that do not have real roots.

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How is Muller’s Method Represented?

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Where does Muller’s Method Come From? x n-1 xnxn f(x n-1 ) f(x n ) Secant Method Muller’s Method

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Algorithm for Muller’s Method For[i=3,(i<=maxit)&&(Abs[xn1-xn2]<=err),i++, t=(xn-xn1)*(xn-xn2); u=(xn1-xn)*(xn1-xn2); v=(xn2-xn)*(xn2-xn1); If[Abs[t]>0 &&Abs[u]>0&&Abs[v]>0, a= (Y0/t)+(Y1/u)+(Y2/v); b=(-Y0(xn1+xn2)/t)+(-Y1(xn+xn2)/u)+ (-Y2(xn+xn1)/v); c=(Y0 (xn1*xn2)/t)+(Y1 (xn*xn2)/u)+ (Y2 (xn*xn1)/v); r1=(-b+Sqrt[b^2 -4a*c])/(2a); r2=(-b-Sqrt[b^2 -4a*c])/(2a); If[Abs[xn2-r1]<Abs[xn2-r2], xn = xn1; xn1 = xn2; xn2 = r1; Y0 = f[xn]; Y1= f[xn1]; Y2= f[xn2];,(*else*) xn = xn1; xn1 = xn2; xn2 = r2; Y0 = f[xn]; Y1= f[xn1]; Y2= f[xn2]; ];

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Representation Again

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Example

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Continued

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Since r1 is closer to the actual root of the function, we choose r1 instead of r2. The program continues until the limit of iterations has been met, or the actual root has been found. This is the first iteration of Muller’s Method.

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