𝜎 = 100000 (√𝜀 + (1/80) sin (𝜋𝜀/0.002))

Now, taking the first derivative of the above relationship, we may write,

𝜎' = 100000 ((1/2√𝜀) + 19.653 cos (1570.80𝜀))

The NR method implements an iterative approach by considering a tangent to a curve, for which the function under consideration becomes zero. If

*f(xⱼ)*is any function and*f′(xⱼ**)*is the derivative of that function at point*xⱼ,*then*according to NR approach, we may write as,**xⱼ₊₁ =*

*xⱼ -*

*f(xⱼ) /*

*f′(xⱼ*

*)*

The MNR works in a similar way, but is helpful for converging a solution efficiently for roots with multiplicity and may be represented as,

*xⱼ₊₁ =*

*xⱼ - ⟮*

*f(xⱼ)*

*f′(xⱼ*

*)⟯ /⟮*

*f′(xⱼ*

*)² -*

*f(xⱼ)*

*f′′(xⱼ*

*)⟯*

The following Mathematica program is developed to plot the function and its derivatives.

**(*Function Defined in the Problem*)**

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

**y1 = 4000;**

**y2 = 6000;**

**sigmap = D[sigma, epsilon];**

**sigmap2 = D[sigma, {epsilon, 2}];**

**(*Plotting the Function, its Derivatives, Upper and Lower Bounds*)**

**A1 = Plot[{sigma, y1, y2}, {epsilon, 0, 0.005}, ImageSize -> Large,**

**AxesLabel -> {"Epsilon", "Sigma"}, PlotStyle -> {Red, Green, Black, Thick}]**

**A2 = Plot[sigmap, {epsilon, 0, 0.005}, ImageSize -> Large,**

**AxesLabel -> {"Epsilon", "Sigma'"}, PlotStyle -> {Red, Thick}]**

**A3 = Plot[sigmap2, {epsilon, 0, 0.005}, ImageSize -> Large,**

**AxesLabel -> {"Epsilon", "Sigma''"}, PlotStyle -> {Red, Thick}]**

**(*Initial Guesses*)**

**sigma /. epsilon -> 7.928*10^-4**

**sigma /. epsilon -> 7.924*10^-4**

**sigma /. epsilon -> 3.882*10^-3**

**sigma /. epsilon -> 3.879*10^-3**

__Mathematica Program for the NR Method:__**(*Mathematica Codes for Newton-Raphson Approach*)**

**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]**

__Mathematica Program for the MNR Method:__

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

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

**epsilontable = {0.0002};**

**SigmaDerivative = D[Sigma, epsilon] /. epsilon -> epsilontable[[1]];**

**Error = {1};**

**SetTolerance = 0.0005;**

**MaximumIteration = 100;**

**i = 1;**

**(*Implementation of the Modified Newton-Raphson Algorithm*)**

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

**epsilonnew =**

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

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

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

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

**i++];**

**T = Length[epsilontable];**

**SolutionTable =**

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

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

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

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

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

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

__Results:__

The following table shows the comparison of results from NR and MNR methods. We see that at iteration number 17, the Modified Newton-Raphson approach converges, whereas the Newton-Raphson approach is still iterating. However, it is noticeable that the step is faster than the Modified approach. Nevertheless, Newton-Raphson approach fails to converge early while the Modified approach, although iterates slowly, but converges eventually.

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