Numerical Analysis Assignment 2 — Root Finding
 Question #1
Consider the function $f=\sin(x)$ with $x$ in radians. Find the root $f=0$ for the initial interval $\frac{1}{2}\pi \le x \le \frac{3}{2} \pi$ using the bisection method. Do so in two different ways:
 (a) By hand, with enough iterations to yield a root accurate to at least 4 significant digits. How many iterations are needed to find a root accurate to at least 4 significant digits? (b) With a C code that starts as follows: #include #include #include #include double f(double x){  double ret;  ret=sin(x);  return(ret);}int main(void){
 08.30.16
 Question #2
Consider the function $f=\sin(x)$ with $x$ in radians. Find the root $f=0$ for the initial condition $x_0=2.8$ using the secant method. Do so in two different ways:
 (a) By hand, with enough iterations to yield a root accurate to at least 4 significant digits. How many iterations are needed to find a root accurate to at least 4 significant digits? (b) With a C code that starts as follows: #include #include #include #include double f(double x){  double ret;  ret=sin(x);  return(ret);}int main(void){
 Question #3
Consider the function $f=\sin(x)$ with $x$ in radians. Find the root $f=0$ for the initial condition $x_0=2.8$ using the Newton method. Do so in two different ways:
 (a) By hand, with enough iterations to yield a root accurate to at least 4 significant digits. How many iterations are needed to find a root accurate to at least 4 significant digits? (b) With a C code that starts as follows: #include #include #include #include double f(double x){  double ret;  ret=sin(x);  return(ret);}double dfdx(double x){  double ret;  ret=cos(x);  return(ret);}int main(void){
 Question #4
 (a) Prove that the order of convergence $p$ of the secant method is 1.618. (b) Verify that the results obtained in Question #2 with the secant method do exhibit an order of convergence of 1.618. Explain why there are discrepancies if applicable.
 Question #5
Knowing that $$|\epsilon_{n+1}|_{\rm Newton}=\left|\frac{1}{2} \frac{f^{\prime\prime}(r)}{f'(r)} \right| |\epsilon_n|^2$$ and $$|\epsilon_{n+1}|_{\rm secant}=\left|\frac{1}{2} \frac{f^{\prime\prime}(r)}{f'(r)} \right|^{1/1.618} |\epsilon_n|^{1.618}$$ Do the following:
 1. Prove that $$\frac{|\epsilon_n|_{\rm Newton}}{|\epsilon_n|_{\rm secant}}=\left( \left|\frac{1}{2} \frac{f^{\prime\prime}(r)}{f'(r)} \right| |\epsilon_0|\right)^{\left(2^n-1.618^n \right)}$$ Note: the latter should be proven fully without skipping steps or making an assumption/simplification. 2. For $|0.5f^{\prime\prime}(r)/f'(r)||\epsilon_0|$ set to 0.3 and 0.03, tabulate the error for $n$ set to 2, 4, and 6. What do you deduce from this?
 10.02.17
 $\pi$