When performing computations on sparse matrices, latency in the memory hierarchy becomes much more of a factor. Sparse matrices lack the spatial locality in the data stream typically found in matrix operations. As a result, new matrix representations have been proposed.One the earliest sparse matrix representations is the Yale Sparse Matrix Format. It stores an initial sparse m × n matrix, M in row form using three one-dimensional arrays. Let R be the number of nonzero entries in M. We construct an array A of length R that contains all nonzero entries of M (in left -to-right top-to-bottom order). We also construct a second array IA of length m + 1 (i.e., one entry per row, plus one). IA(i) contains the index in A of the fi rst nonzero element of row i. Row i of the original matrix extends from A(IA(i)) to A(IA(i+1)−1). Th e third array, JA, contains the column index of each element of A, so it also is of length R.Consider the sparse matrix X below and write C code that would store this code in Yale Sparse Matrix Format.Row 1 [1, 2, 0, 0, 0, 0]Row 2 [0, 0, 1, 1, 0, 0]Row 3 [0, 0, 0, 0, 9, 0]Row 4 [2, 0, 0, 0, 0, 2]Row 5 [0, 0, 3, 3, 0, 7]Row 6 [1, 3, 0, 0, 0, 1]> In terms of storage space, assuming that each element in matrix X is single precision fl oating point, compute the amount of storage used to store the Matrix above in Yale Sparse Matrix Format.> Perform matrix multiplication of Matrix X by Matrix Y shown below.[2, 4, 1, 99, 7, 2]Put this computation in a loop, and time its execution. Make sure to increase the number of times this loop is executed to get good resolution in your timing measurement. Compare the runtime of using a naïve representation of the matrix, and the Yale Sparse Matrix Format.> Can you fi nd a more effi cient sparse matrix representation (in terms of space and computational overhead)?
When performing computations on sparse matrices, latency in the memory hierarchy becomes much more of a factor. Sparse matrices lack the spatial locality in the data stream typically found in matrix operations. As a result, new matrix representations have been proposed.
One the earliest sparse matrix representations is the Yale Sparse Matrix Format. It stores an initial sparse m × n matrix, M in row form using three one-dimensional arrays. Let R be the number of nonzero entries in M. We construct an array A of length R that contains all nonzero entries of M (in left -to-right top-to-bottom order). We also construct a second array IA of length m + 1 (i.e., one entry per row, plus one). IA(i) contains the index in A of the fi rst nonzero element of row i. Row i of the original matrix extends from A(IA(i)) to A(IA(i+1)−1). Th e third array, JA, contains the column index of each element of A, so it also is of length R.
Consider the sparse matrix X below and write C code that would store this code in Yale Sparse Matrix Format.
Row 1 [1, 2, 0, 0, 0, 0]
Row 2 [0, 0, 1, 1, 0, 0]
Row 3 [0, 0, 0, 0, 9, 0]
Row 4 [2, 0, 0, 0, 0, 2]
Row 5 [0, 0, 3, 3, 0, 7]
Row 6 [1, 3, 0, 0, 0, 1]
> In terms of storage space, assuming that each element in matrix X is single precision fl oating point, compute the amount of storage used to store the Matrix above in Yale Sparse Matrix Format.
> Perform matrix multiplication of Matrix X by Matrix Y shown below.
[2, 4, 1, 99, 7, 2]
Put this computation in a loop, and time its execution. Make sure to increase the number of times this loop is executed to get good resolution in your timing measurement. Compare the runtime of using a naïve representation of the matrix, and the Yale Sparse Matrix Format.
> Can you fi nd a more effi cient sparse matrix representation (in terms of space and computational overhead)?
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