Fit the following data using a Cubic spline using natural boundary conditions$$

$$

You will do this by setting up the matrix equation as

Ax$=$bAx$=$b

$.$ You can use numpy arrays to set up

AA

and

bb

$.$ Now$$

bb

$=$ np$.$array corrects a row vector.$$ To change this to a column vector you need to do the following$$

bb

$=$

bb

$[$:$,$ np$.$newaxis$].$ Now the matrix equation looks just like we set up in the lecture.$$

Next, you will use np$.$dot$($np$.$linalg.inv$($

AA

$),$

bb

$).$ I am using the short name np for numpy.$$ What this routine does is first find the inverse of

AA

$,$ and then take the dot product of each row of the inverse of

AA

with the column vector

bb

$.$ In essence, doing the matrix$-$vector multiply.$$ So$,$ this one command calls two functions.$$ Look this up so you get a better understanding of this routine.$$hjb$$

$$

Saturn $5$ Rocket Launch Data$$

x $($time in minutes$)$

y $($velocity in $1000$ ft per second$$

$0$

$1$

$1$

$2$

$2\mathrm{.}5$

$9$

$3$

$9\mathrm{.}2$

$4$

$10$

$5$

$12$

$6$

$14\mathrm{.}5$

$7$

$17$

$8$

$20$

$8\mathrm{.}75$

$23$

$9$

$23\mathrm{.}5$

$10$

$24$

$11$

$25\mathrm{.}5$

$11\mathrm{.}25$

$25\mathrm{.}9$

$11\mathrm{.}5$

$25\mathrm{.}9$

$$