Solve this C language problem $($c$)(5$ points$)$ Suppose the recursive calls on lines $20-23$ are replaced with the following

code:float M$2[]=$ matmul$($A$21+$ A$22,$ B$11,$ N$/2)$;

float M$3[]=$ matmul$($A$11,$ B$12-$ B$22,$ N$/2)$;

float M$4[]=$ matmul$($A$22,$ B$21-$ B$11,$ N$/2)$;

float M$5[]=$ matmul$($A$11+$ A$12,$ B$22,$ N$/2)$;

float M$6[]=$ matmul$($A$21-$ A$11,$ B$11+$ B$12,$ N$/2)$;

float M$7[]=$ matmul$($A$12-$ A$22,$ B$21+$ B$22,$ N$/2)$;and the return statement on line $25$ is replaced with:

return $[[$M$1+$ M$4-$ M$5+$ M$7,$ M$3+$ M$5],$

$[M2+M4,M1-M2+M3+M6]$

Note that this computes the same result! Use big$-$ notation to express the new

number of float$/$float multiplies in terms of the size $N.($Hint: write a new recurrence

relation$).$

Consider the following pseudocode that recursively multiplies two $NN$ matrices by

breaking them into eight $\frac{N}{2}\frac{N}{2}$ matrix multiplies:$//$ base case return $[$A$[0,0]*$ B $[0,0]]$;$//$ get A submatricesfloat A$12[]=$ A$[$:N$/2,$ N$/2$:$]$;float A$22[]=$ A$[$N$/2$:$,$N$/2$:$]$;float B$11[]=$ B$[$:N$/2,$ :N$/2]$;float B$21[]=$ B$[$N$/2$:$,$ :N$/2]$;$//$ make recursive callsfloat C$12[]=$ matmul$($A$12,$ B$22,$ N$/2)+$ matmul$($A$11,$ B$12,$ N$/2)$;float C$22[]=$ matmul$($A$21,$ B$12,$ N$/2)+$ matmul$($A$22,$ B$22,$ N$/2)$;$\}$For simplicity, you may assume that N is a power of two.

$($a$)(3$ points$)$ Write a recurrence relation that describes the number of times the above

algorithm multiplies two floats $($line $4)$ as a function of $N.$

$($b$)(2$ points$)$ Use the Master Theorem to compute a big$-$ closed form of your recur$-$

rence relation from part $($a$).$ Show all intermediate steps.