function $c_{1}x+c_2,$ set up an overdetermined sys$-$

tem $Vc=T.$ The coefficient matrix $V$ can be

generated in MATLAB by setting

$V=[p,$ ones $(10,1)]$

or$,$ alternatively, by setting

$A=$vander$(p)$;$,V=A($;$,9$:$10)$

Note For any vector $x=(x_1,x_2,$dots,$x_{n+1}{)}^T,$

the MATLAB command vander$($x$)$ generates

a full Vandermonde matrix of the form

$([x_1^n,x_1^{n-1},$cdots,$x_1,1],[x_2^n,x_2^{n-1},$cdots,$x_2,1],[vdots,],[x_{n+1}^n,x_{n+1}^{n-1},$cdots,$x_{n+1},1])$

For a linear fit, only the last two columns of

the full Vandermonde matrix are used. More

information on the vander function can be

obtained by typing help vander. Once $V$

has been constructed, the least squares solu$-$

tion $c$ of the system can be calculated using the

MATLAB $""$ operation.

$($b$)$ To see how well the linear function fits the data,

define a range of pressure values by setting

$q=1$:$0\mathrm{.}1$:$10$;

The corresponding function values can be de$-$

termined by setting

$z=$polyval$(c,q)$;

We can plot the function and the data points with

the command

$($c$)$ Let us now try to obtain a better fit by us$-$

ing a cubic polynomial approximation. Again

we can calculate the coefficients of the cubic

polynomial

$c_1{x}^3+c_2{x}^2+c_{3}x+c_4$

that gives the best least squares fit to the data

by finding the least squares solution of an over$-$

determined system $Vc=T.$ The coefficient

matrix $V$ is determined by taking the last four

columns of the matrix $A=$ vander$($p$).$ To see

the results graphically, again set

$z=$polyral$(c,q)$

and plot the cubic function and data points, us$-$

ing the same plot command as before. Where do

you get the better fit, at the top or bottom of the

atmosphere?