How can you represent and use sparse matrices in functional programming?

In functional programming, sparse matrices can be represented using data structures like lists of lists, maps or tuples.

A sparse matrix is a matrix in which most of the elements are zero. In a functional programming language, you can represent a sparse matrix in several ways. One common method is to use a list of lists, where each inner list represents a row of the matrix. However, this can be inefficient if the matrix is very large and contains many zeros, as you're using up memory to store those zeros.

A more efficient way to represent a sparse matrix is to use a map or a dictionary. In this case, you would store only the non-zero elements of the matrix, along with their positions. For example, you could use a map where the keys are tuples representing the positions of the non-zero elements, and the values are the non-zero elements themselves. This way, you're only using memory to store the non-zero elements and their positions, which can be a significant saving if the matrix is large and mostly zeros.

Another method is to use a list of tuples, where each tuple contains the row and column indices of a non-zero element and its value. This is similar to the map method, but instead of using a map to store the positions and values, you're using a list of tuples. This can be more efficient than the map method if the matrix is very large and the non-zero elements are spread out, as you're not using up memory to store the keys of a map.

In terms of using sparse matrices in functional programming, you would typically use higher-order functions like map, filter and reduce to manipulate them. For example, you could use the map function to apply a function to all non-zero elements of the matrix, the filter function to find all non-zero elements that meet a certain condition, and the reduce function to combine all non-zero elements in some way.