HW$1$

Due: $8/30/2024$ Friday $11$:$59$pm$.$

Submit scanned$/$electronic answer sheets to Canvas in ONE PDF$,$ Label your problems.

Problem $1$

In a $4-$digit decimal number system, tilde$(a)=1\mathrm{.}234$ and tilde$(b)=4\mathrm{.}567$ are two machine numbers.

What is relative rounding error?

$($don$\text{'}$t just write an answer, show necessary steps such as formula for relative rounding error$)$

Problem $2$

What is the rounding errors when computing the derivative of $f(x)=sinx$ at $x_0=2.($exact value $=cos(2))$ using the definition $f^{\text{'}}(x_0)$~~$\frac{f(x_0+h)-f(x_0)}{h},$ when $h=1e-8,1e-5?$

Use MATLAB to solve it$,$ print out the results and paste$/$write your MATLAB coding script on the answer sheet.

$($hint: may revise from the code posted on canvas$-$code$\_$demo: L$1\_$rounding$\_$deri.m$)$

Problem $3.$

$u=[1,2,3{]}^T,v=[4,5,6{]}^T,$ what is the flops of $2u*v+1?$

$($don$\text{'}$t just write an answer, give necessary explanation$/$show necessary steps$).$

Problem $4$

solve $f(x)=x^3-3x+1=0,$ with iteration:

$x_{k+1}=x_k-\frac{x_k^3-3x_k+1}{3x_k^2-3}$

$x_0=2.$

a$.$ use MATLAB to find smallest $k$ such that

b$.$ for the smallest $k$ required in Part a$,$ what is the value of $f(x_k)?$

Use MATLAB solve the problem, write out the required results, and paste$/$write your MATLAB script on answer sheets. $($hint: may refer StopCriteria$\_$successive$\_$approx.m and StopCriteria$\_$check$\_$residual.m on Canvas$-$files$\_$code demo$)$