**(*Mathematica Codes for Newton-Raphson Approach*)**

**(*Write Your Own Function Below*)**

**Sigma = 100000*(Sqrt[epsilon] + 1/80*Sin[Pi*epsilon/0.002]) - 4000;**

**SigmaDerivative = D[Sigma, epsilon];**

**epsilontable = {0.0002};**

**Error = {1};**

**SetTolerance = 0.0005;**

**MaximumIteration = 100;**

**i = 1;**

**(*Newton-Raphson Algorithm Implementation*)**

**While[And[i <= MaximumIteration, Abs[Error[[i]]] > SetTolerance],**

**epsilonnew =**

**epsilontable[[i]] - (Sigma /.**

**epsilon -> epsilontable[[i]])/(SigmaDerivative /.**

**epsilon -> epsilontable[[i]]);**

**epsilontable = Append[epsilontable, epsilonnew];**

**Errornew = (epsilonnew - epsilontable[[i]])/epsilonnew;**

**Error = Append[Error, Errornew];**

**i++];**

**L = Length[epsilontable];**

**SolutionTable =**

**Table[{i - 1, epsilontable[[i]], Error[[i]]}, {i, 1, L}];**

**SolutionTable1 = {"Iteration Number", "Epsilon", "Error"};**

**L = Prepend[SolutionTable, SolutionTable1];**

**Print["Showing Results for the Newton-Raphson Approach when Sigma is \**

**4,000 and Initial Guess is 0.0002"]**

**ScientificForm[L // MatrixForm, 4]**

__Program Output:__

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