Problem $3$: You experiment with a GPU to compute the following multiplication C$=$A x B$,$ where $m=n=l=102$ the elements of $A,B,$ and $C$ are all with the same bit$-$width of $4$ Bytes. When implementing on GPU, each thread is in charge of computing for one element in C$.$ Please answer the following questions $($please show detailed steps.$)(25$ pts$).$

$A=\left[\begin{array}{ccc}{x}_{1,1}& \text{c}\text{d}\text{o}\text{t}\text{s}& x_{1,l}\\ \text{v}\text{d}\text{o}\text{t}\text{s}& \text{d}\text{d}\text{o}\text{t}\text{s}& \text{v}\text{d}\text{o}\text{t}\text{s}\\ x_{m,1}& \text{c}\text{d}\text{o}\text{t}\text{s}& x_{m,l}\end{array}\right]$ and $B=\left[\begin{array}{ccc}{y}_{1,1}& \text{c}\text{d}\text{o}\text{t}\text{s}& y_{1,n}\\ \text{v}\text{d}\text{o}\text{t}\text{s}& \text{d}\text{d}\text{o}\text{t}\text{s}& \text{v}\text{d}\text{o}\text{t}\text{s}\\ y_{l,1}& \text{c}\text{d}\text{o}\text{t}\text{s}& y_{l,n}\end{array}\right]$

a$)$ From the hardware standpoint, the GPU has $4$ Streaming Multiprocessors $($SMs$),$ each with exclusive L$1$ and L$2$ caches, and $8$ warp schedulers managing $4$ warps per scheduler. Each warp consists of $32$ cores. From the software perspective, threads within the same block can share data directly, while threads across different blocks cannot. What is the maximum block size that matches the GPU hardware architecture? $(5$ pts$)8432=1024$

b$)$ Following the result from a$),$ how many blocks should we have for computing the entire matrix C$?$ Please use dim$3$ we have learned to initialize the threading. $(5$ pts$)$ dim $3$block$(32,32)$ dim $3$ thread $(32,32)$

c$)$ After applying tiling, with the tile size of $3232,$ houmany tiles can cover the entire matrix? Given the block size calculated in a$),$ how many titles can cover thejentire block? $(pts1024\frac{1024}{256}=4096,\frac{1024}{256}=4$

d$)$ Following the block and tile sizes in c$),$ what is the proper size of shared memory that we need to request $($Hint: threads within the same block can share data directly, while threads across different blocks cannot$)?$ Please explain your answer. $(10$ pts$)$

Just c & d