Write a matlab code in class for Gibbs$-$Poole$-$Stockmeyer algorithm and then call the functions like this in script file and then do step wise explaination of how does the GPS bandwidth reduction of sparse matrix work. then do flop count analysis where you explain what is the total flop count

THis is for reverse cuthill mckee algorithm

classdef ReorderingSSM

properties

initial$\_$matrix

degree $\%$ Store degrees of nodes

end

methods

function obj $=$ ReorderingSSM$($m$)$

obj.initial$\_$matrix $=$ m;

obj.degree $=$ sum$($m$,2)$; $\%$ Precompute degrees

end

function R $=$ CuthillMckee$($obj$)$

n $=$ size$($obj$.$initial$\_$matrix, $1)$;

degrees $=$ obj.degree;

Q $=[]$;

R $=$ zeros$(1,$ n$)$;

notVisited $=$ true$(1,$ n$)$;

position $=0$;

while any$($notVisited$)$

i $=$ find$($notVisited$,1)$;

Q$($end $+1)=$ i;

notVisited$($i$)=$ false;

while ~isempty$($Q$)$

u $=$ Q$(1)$;

Q$(1)=[]$;

position $=$ position $+1$;

R$($position$)=$ u;

adj$\_$nodes $=$ find$($obj$.$initial$\_$matrix$($u$,$ :$))$;

$[$~$,$ idx$]=$ sort$($degrees$($adj$\_$nodes$))$;

adj$\_$nodes $=$ adj$\_$nodes$($idx$)$;

for v $=$ adj$\_$nodes

if notVisited$($v$)$

Q$($end $+1)=$ v;

notVisited$($v$)=$ false;

end

end

end

end

R $=$ R$(1$:position$)$; $\%$ Truncate R to the actual size

end

function R $=$ ReverseCuthillMckee$($obj$)$

cuthill $=$ obj.CuthillMckee$()$;

R $=$ flip$($cuthill$)$;

end

function visualizeMatrix$($obj$,$ reordered$)$

$\%$ Visualize using improved graphics and better colormap

subplot$(1,3,1)$;

spy$($obj$.$initial$\_$matrix$)$;

title$(\text{'}$Original Matrix'$)$;

axis square;

subplot$(1,3,2)$;

spy$($reordered$)$;

title$(\text{'}$Custom RCM Reordered Matrix'$)$;

axis square;

$\%$ Visualizing the built$-$in reordered matrix

subplot$(1,3,3)$;

r$\_$matlab $=$ symrcm$($obj$.$initial$\_$matrix$)$;

reordered$\_$matlab $=$ obj.initial$\_$matrix$($r$\_$matlab, r$\_$matlab$)$;

spy$($reordered$\_$matlab$)$;

title$(\text{'}$MATLAB symrcm Reordered Matrix'$)$;

axis square;

end

end

end

script:

n $=500$;

density $=0\mathrm{.}001$;

matrix $=$ sprand$($n$,$ n$,$ density$)$;

$\%$ Print the original order before reordering

fprintf$(\text{'}$Original order before reordering:

$\text{'})$;

disp$(1$:n$)$;

$\%$ Creating an instance of ReorderingSSM with the generated matrix

m $=$ ReorderingSSM$($matrix$)$;

$\%$ Timing and applying the custom Reverse Cuthill$-$Mckee algorithm

tic;

r$\_$custom $=$ m$.$ReverseCuthillMckee$()$;

customTime $=$ toc;

$\%$ Print the order after custom RCM reordering

fprintf$(\text{'}$Order after custom RCM reordering:

$\text{'})$;

disp$($r$\_$custom$)$;

$\%$ Timing and applying MATLAB's built$-$in symrcm function

tic;

r$\_$matlab $=$ symrcm$($matrix$)$;

matlabTime $=$ toc;

$\%$ Print the order after MATLAB's symrcm reordering

fprintf$(\text{'}$Order after MATLAB symrcm reordering:

$\text{'})$;

disp$($r$\_$matlab$)$;

$\%$ Reordering the matrix according to the custom RCM

reordered$\_$matrix$\_$custom $=$ matrix$($r$\_$custom, r$\_$custom$)$;

$\%$ Print execution times

fprintf$(\text{'}$Time taken by custom RCM method: $\%\mathrm{.}4$f seconds

$\text{'},$ customTime$)$;

fprintf$(\text{'}$Time taken by MATLAB built$-$in symrcm method: $\%\mathrm{.}4$f seconds

$\text{'},$ matlabTime$)$;

$\%$ Visualize the matrices

m$.$visualizeMatrix$($reordered$\_$matrix$\_$custom$)$;

keep the template of the class same just change the agorithm