CSRFn

CSRFn is a term occasionally used in sparse linear algebra to denote an operator implemented on a matrix stored in the compressed sparse row (CSR) format. In this usage, CSRFn(A, x) returns the matrix–vector product y = A x, where A is the sparse matrix represented by CSR arrays. The designation can be encountered as either a concrete callable in code or as an abstract operator in theoretical discussions, with variations in exact interpretation across sources.

Definition and representation: A CSR matrix stores three arrays: values of the nonzero entries, column indices of those entries, and a row pointer indicating where each row starts in the value and index arrays. CSRFn uses this structure to compute y_i by summing over the nonzeros in row i: y_i = sum_k val[k] * x[col_ind[k]] for k from row_ptr[i] to row_ptr[i+1] − 1. The standard CSR layout enables efficient row-wise traversal, which is well suited to the matrix-vector product.

Applications and implementations: CSRFn is commonly used to represent and apply sparse matrices in iterative solvers (such as GMRES, CG) and in preconditioning routines. It appears in numerical libraries and research code as a practical abstraction for performing SpMV (sparse matrix–vector multiplication) with CSR data, and may be implemented across CPUs and accelerators (CPU, GPU) with attention to memory access patterns. Transpose or operations requiring column-wise access often require conversion to a different format (e.g., CSC) or explicit handling within the CSRFn framework.

Limitations and context: The term CSRFn is not universally standardized and may vary in definition by author or library. Its usefulness hinges on efficient CSR storage, which favors row-wise computations but may require format conversions for certain operations.

See also: Compressed Sparse Row, sparse matrices, SpMV, iterative methods, linear algebra libraries.