Transposition (telecommunications)

Abstract: Let A and B be two sparse matrices whose orders are p by q and q by r. Their product C -A B requires N nontrlvial multiplications where 0 <_ N <_ pqr. The operation count of our algorithm is usually proportional to N; however, its worse case is O(p, r, NA, N) where N A is the number of elements in A This algorithm can be used to assemble the sparse matrix arising from a finite element problem from the basic elements, using ~-1 [order (g)]2 operations where m is the total number of basic elements and order(g) is the order of the ~th element matrix. The concept of an unordered merge plays a key role m obtaining our fast multiplication algorithm It forces us to accept an unordered sparse row-wise format as output for the product C The permuted transposition algorithm computes ( R A ) T i n O(p, q, NA) operations where R is a permutation matrix It also orders an unordered sparse row-wise representation. We can combine these algorithms to produce an O(M) algorithm to solve A x = b where M is the number of multiplications needed to factor A into L U