Multiplying Vectors or Matrices by Sparse Matrices

These functions allow the multiplication of a normal (dense-representation) vector by a sparse matrix.

v * A: ModTupRng, MtrxSprs -> ModTupRng

V * A: Mtrx, MtrxSprs -> Mtrx

Given a dense-representation vector \(v\) or dense-representation matrix \(V\) with \(c\) columns, together with a sparse \(c\times n\) matrix \(A\), both over a ring \(R\), return the product \(v\cdot A\) or \(V\cdot A\).

This is generally fast if \(A\) is sparse and uses minimal memory.

MultiplyByTranspose(v, A): ModTupRng, MtrxSprs -> ModTupRng

MultiplyByTranspose(V, A): Mtrx, MtrxSprs -> Mtrx

Given a dense-representation vector \(v\) or dense-representation matrix \(V\) with \(c\) columns, together with a sparse \(n\times c\) matrix \(A\), both over a ring \(R\), return the product of \(v\) or \(V\) by the transpose of \(A\).

This is generally fast if \(A\) is sparse, and is much faster than computing the transpose of \(A\) first. For example, if the vector-matrix product \(v\cdot A\cdot A^{tr}\) is required, then the function call MultiplyByTranspose(v*A, A) should be used to avoid forming the matrix \(A\cdot A^{tr}\) which is usually dense. This product occurs in iterative algorithms such as Lanzcos.