This question is from the textbook, A First Course in Numerical Methods Uri M $$Ascher Chen Greif. Chapter $10,$ Question # $0.$ Review questions $0($a$,$d$,$e$,$f$,$h$)10\mathrm{.}8$ Exercises

$0.$ Review questions

$($a$)$ Distinguish between the terms data fitting, interpolation, and polynomial interpolation.

$($b$)$ Distinguish between $($discrete$)$ data fitting and approximating a given function.

$($c$)$ What are basis functions? Does an approximant $v(x)$ that is written as a linear combina$-$

tion of basis functions have to be linear in $x?$

$($d$)$ An interpolating polynomial is unique regardless of the choice of the basis. Explain why.

$($e$)$ State one advantage and two disadvantages of using the monomial basis for polynomial

interpolation.

$($f$)$ What are Lagrange polynomials? How are they used for polynomial interpolation?

$($g$)$ What are barycentric weights?

$($h$)$ State the main advantages and the main disadvantage for using the Lagrange represen$-$

tation.