Question

Problem Two: (Based on Chapra, Problems 12.9 Consider the simultaneous nonlinear equations: 2-5-y y+i- 1. Plot the equations and identify the solution graphically. Page 1 of 2 2. Solve the system of equations using successive substitution, starting with the initial guess xo-y-1.5. Show two complete iterations. Evaluate &s for the second iteration. 3. Redo Part 2 using Newton-Raphson method . Automate the solutions in Parts 2 and 3 using MATLAB scripts 5. Solve the system of nonlinear equations by calling the MATLAB function newtmult. Use the initial guess xo- 5. Your solution should achieve an accuracy of 10 significant figures. Report the solution, number of iterations and error.

Answer 1

1.Answers to the first four subparts

Plotting the two graphs.

The intersection point is (x,y) =(-1.6,1,56) and (1.6,1.56)

2.successive substitution:

$\\x^{2}= 5- y^{2}\Rightarrow \ x=\sqrt{5-y^{2}} \\y_{0}=1.5 \ , x_{0}=1.5 \\ x_{1} = \sqrt{5-y_{0}^{2}} =1.658 \\ y+1 = x^{2} \Rightarrow \ y = x^{2}-1 \\y_{1} = 1.25 \\x_{2} = \sqrt{5-1.25^{2}} =1.854 \\y_{2} =\Rightarrow \ y = 1.658^{2}-1=1.748$

3. Newton Raphson:

$\\x^{2}= 5- y^{2}\Rightarrow \ x=\sqrt{5-y^{2}} \\f'(y) = \frac{-2y}{2\sqrt{5-y^{2}}} \\ y+1 = x^{2} \Rightarrow \ y = x^{2}-1 \\ f'(x) =2x\\ \\y_{0}=1.5 \ , x_{0}=1.5 \\ Newton \ raphson \ method\ \\x_{n+1} =x_{n} - \frac{f(x)}{f'(x)}\\ \\y_{n+1} =y_{n} - \frac{f(y)}{f'(y)}\\ \\ x_{1} = 3.33 \ , \ y_{1} =1.083 \ , \ x_{2} =6.86 \ , \ y_{2} = -0.433$

4.

xsucccesive=1.5;ysuccessive=1.5;

for i= 1:2

xtemp=xsucccesive;

xsucccesive = sqrt(5- ysuccessive.*ysuccessive)

ysuccessive=xtemp.*xtemp -1

end

xnr=1.5;ynr=1.5;

for i= 1:2

xtemp=xnr;

dify1= -ynr/(sqrt(5- ynr.*ynr));

fy1=sqrt(5- ynr.*ynr);

xnr = xnr - (fy1/dify1)

fx2=xtemp.*xtemp -1;

diffx2 = 2*xtemp;

ynr = ynr - (fx2/diffx2)

end