Lecture 156:- Addition of Sparse Matrices

Adding sparse matrices involves combining their elements, taking care to preserve their sparsity. A sparse matrix is a matrix that contains mostly zero elements. Performing addition on sparse matrices can be computationally efficient if implemented correctly. Below, I'll outline the general steps to add two sparse matrices:

Let's say we have two sparse matrices A and B, and we want to compute their sum C = A + B.

1. Initialize an empty result matrix C with the same dimensions as A and B.
2. Iterate over the non-zero elements of matrices A and B:
   • For each non-zero element in A, retrieve its value and position (row, column).
   • For each non-zero element in B, retrieve its value and position (row, column).
3. Sum the corresponding non-zero elements from A and B and store the result in the corresponding position of matrix C.
4. If there are overlapping non-zero elements at the same position (row, column), add their values together in matrix C.
5. Continue iterating until all non-zero elements of both matrices are processed.
6. The resulting matrix C will now contain the sum of matrices A and B, while maintaining its sparsity.

Here's a Python code example to perform addition on two sparse matrices using dictionaries to store the non-zero elements:

pythonCopy code

def add_sparse_matrices(A, B):    # Initialize the result matrix C    C = {}    # Iterate over the non-zero elements of matrix A    for (row, col), value in A.items():        C[(row, col)] = C.get((row, col), 0) + value    # Iterate over the non-zero elements of matrix B    for (row, col), value in B.items():        C[(row, col)] = C.get((row, col), 0) + value    return C # Example usage A = {(0, 1): 5, (1, 2): 10, (2, 0): 3} B = {(0, 1): 7, (1, 1): 2, (2, 0): 1} result = add_sparse_matrices(A, B) print(result)  # Output: {(0, 1): 12, (1, 1): 2, (1, 2): 10, (2, 0): 4}

In this example, the matrices A and B are represented as dictionaries, where the keys are tuples (row, column), and the values are the corresponding non-zero elements in the matrices. The result matrix C is also represented as a dictionary with the same format. The function add_sparse_matrices combines the non-zero elements of A and B, taking care of sparsity, and returns the resulting sparse matrix C.