Problem $1(15$ points$)($Core Course Outcome $5)()/()$ Mat.labGrader

Write a function mySecondDerivativeOrder$3$ that calculates the second derivative d$^(2)($y$)/($d$)$x$^(2)$x$,$y x with spacing h$.$

The function input shall be

y: one$-$dimensional column vector of y values

h: scalar value of spacing h

The function output shall be

d $2$ y : column vector of the same size as y containing d$^(2)($y$)/($d$)$x$^(2)$

Calculate d$^(2)($y$)/($d$)$x$^(2)$ using only the following formulas that use the nomenclature of Table $8-1$ and the lecture notes, but are different

from the formulas listed there:

f$^(\text{'}\text{'})($x$\_($i$))=(164$f$($x$\_($i$))-465$f$($x$\_($i$|1))+460$f$($x$\_(($i$)/(2)))-170$f$($x$\_($i$|3))+11$f$($x$\_(($i$)/(5))))/(60$h$^(2))+$O$($h$^(3))$

f$^(\text{'}\text{'})($x$\_($i$))=(56$f$($x$\_($i$-1))-105$f$($x$\_($i$))+40$f$($x$\_($i$|1))+10$f$($x$\_(($i$)/(2)))-$f$($x$\_($i$|4)))/(60$h$^(2))+$O$($h$^(3))$

f$^(\text{'}\text{'})($x$\_($i$))=(-5$f$($x$\_($i$)2)+80$f$($x$\_($i$-1))-150$f$($x$\_($i$))+80$f$($x$\_($i$+1))-5$f$($x$\_($i$|2)))/(60$h$^(2))+$O$($h$^(4))$

f$^(\text{'}\text{'})($x$\_($i$))=(-2$f$($x$\_($i$-4))+5$f$($x$\_($i$-3))+50$f$($x$\_($i$-1))-110$f$($x$\_($i$))+57$f$($x$\_($i$|1)))/(60$h$^(2))+$O$($h$^(3))$

f$^(\text{'}\text{'})($x$\_($i$))=(57$f$($x$\_($i$)-5)-230$f$($x$\_($i$)-4)+290$f$($x$\_($i$)-3)-235$f$($x$\_($i$-1))+118$f$($x$\_($i$)))/(60$h$^(2))+$O$($h$^(3))$

Pay special attention to the sub$-$indices in the formulas, since some indices are skipped! For every data point, use the most accurate

formula, if Eq$.(3)$ is the most accurate, followed by Eqs. $(2)$ and $(4),$ and followed by Eqs. $(1)$ and $(5).$