LAB ACTIVITY $8\mathrm{.}5\mathrm{.}1$: MATLAB: Least Squares Approximation This tool is provided by a third party. Though your activity may be recorded, a page refresh may be needed to fill the banner. $0/1$ MATLAB: Least Squares Approximation In this activity you will use a least squares approximation to find a curve of best fit for a data set. Consider the four points in the plane: $(-2,3),(-1,1),(1,0),$ and $(2,1).$ Use the least squares approximation to find the best$-$fit line for this data. Note Y is the vector b in the inconsistent system Ax$=$b$.\%$Enter the data as two column vectors. X $=(-2-112].\text{'}$ Y $=[3101].\text{'}\%$Use the length$()$ command to determine the size of the column vector. m $=$ length$($X$)\%$Set up the appropriate matrix A to find the best$-$fit line of the form y$=$C$+$Dx$.$ The first column Sof A will contain all l$\text{'}$s$.$ This is achieved here using the ones $()$ command to create a column $\%$vector of length m of all i$\text{'}$s$.$ The second column of A contains X$.$ A $=[$ones $($m$,1)$ X$]\%$Calculate the matrix products. A$\_$transposeA $=$ A$.\text{'}*$ A A$\_$transposeY $=$ A$.\text{'}*$ Y$\%$Use the backslash operation to solve the overdetermined system. Soln$1=$ A$\_$transposeA$\backslash$A$\_$transposeY The points will range from $-4$ to $4$ in increments of $0\mathrm{.}1.\%$Generate points to plot the best$-$fit line. X$=-4$: $0\mathrm{.}1$ : $4$ ylinear $=$ Soln$1(1)+$ Soln$1(2)*$X $\%$The following sequence of commands plots the data and the line of best fit. plot$($x$,$ ylinear, X$,$ Y$,\text{'}$k$*\text{'})$;grid;shg The MATLAB code produces the following image. $3.2.21\mathrm{.}510\mathrm{.}50-0\mathrm{.}543-2.01234$ A line may not be the best fit for this data. For this activity, use the same data and find the best$-$fit parabola.Script e Save C Reset MATLAB Documentation These are provided for you. $1\%$The same data is used for the activity. $2$ X $=[-2-112].\text{'}3$ Y $=[3101].\text{'}45\%$Use the length$()$ command to determine the size of the column vector X$.$ Store this value in m$.6$ m $=$ length$($X$)7\%$Set up the appropriate matrix A to find the best$-$fit parabola of the form y$=$C$+$Dx$+$Ex$^2.$ The $8\%$first column of A will contain all l$\text{'}$s$,$ using the ones$)$ command. The second column of A $9\%$contains x values that are stored in X$.$ The third column of A contains the squared x values $10\%$that are stored in X$.$ Elementwise multiplication of x by itself, using $.*$ operator, will $11\%$produce the desired values for the third column. $12$ A $=[$ones $($m$,1)$ X$]13\%$Calculate the matrix products. These are provided for you. $14$ A$\_$transposeA $=$ A$.\text{'}*$ A $15$ A$\_$transpose Y $=$ A$.\text{'}*$ Y $16|17$ Use the backslash operation to solve the overdetermined system. Store this in Soln$2.18$ Soln$2=$ A$\_$transposeA$\backslash$A$\_$transposeY $19$ Define the x values to use for plotting the best$-$fit parabola. This creates a vector x$.20\%$This is provided for you. $21$ x$=-4$: $0\mathrm{.}1$ : $422\%$Define the best$-$fit parabola, storing it in yquadratic. Elementwise multiplication of the $23\%$x values times themselves to square them is achieved by using $.*$ operator $($because x is a vector$).24$ yquadratic $=$ Soln$2(1)+$ Soln$2(2)*$x $25\%$The following sequence of commands plots the data and the best$-$fit parabola. The command is $26\%$provided for you. $27$ plot$($x$,$ yquadratic, X$,$ Y$,\text{'}$k$*\text{'})$;grid;shg $28293031$