Algorithms for the implementation of element wise operations between a Dense and Sparse matrices:

• Algorithm 1 x(dense, sparse)

• Algorithm should clone DenseMatrix and call the x(d(i,j), s(i,j)) operation for the items in the Dense and Sparse matrices (iterating on the Sparse matrix nonzero items), updating the cloned matrix.

• Output type is a DenseMatrix (the cloned matrix)

• x() operation invoked NZ times (number of nonzero items in SparseMatrix)

  Cij = x(Dij, Sij);    Sij != 0
  Cij = Dij        ;    otherwise
  

• Algorithm 2 x(dense, sparse)

• Algorithm should iterate SparseMatrix (nonzero items) and call the x(d(i,j),s(i,j)) operation for the items in the Sparse and Dense matrices (since zero & X == zero)

• Output type is a SparseMatrix since the number of nonzero items will be less or equal the number of nonzero elements in the Sparse Matrix.

• x() operation invoked NZ times (number of nonzero items in SparseMatrix)

  Cij = x(Dij, Sij);    Sij != 0
  Cij = 0          ;    otherwise
  

• Algorithm 3 x(dense, sparse)

• Algorithm should iterate SparseMatrix (nonzero and zero items) and call the x(s(i,j),d(i,j)) operation for the items in the Dense and Sparse matrices

• Output type is a DenseMatrix

• x() operation invoked M*N times

  Cij = x(Dij, Sij);    Sij != 0
  Cij = x(Dij, 0)  ;    otherwise
  

• Algorithm 4 x(sparse, sparse)

• Algorithm should iterate on the nonzero values of matrices A and B and call x(Aij, Bij) when both matrices contain value at (i,j)

• Output type is a SparseMatrix

• x() operation invoked NZ times (number of nonzero items at the same (i,j) for both matrices)

  Cij = x(Aij, Bij);    Aij != 0 && Bij != 0
  Cij = Aij        ;    Aij != 0
  Cij = Bij        ;    Bij != 0
  

Algorithms for the implementation of element wise operations between a Sparse matrices:

• Algorithm 5 x(sparse, sparse)

• Algorithm should iterate on the nonzero values of matrices A and B and call x(Aij, Bij) for every nonzero value.

• Output type is a SparseMatrix

• x() operation invoked NZ times (number of nonzero values in A only + number of nonzero values in B only + number of nonzero values in A and B)

  Cij = x(Aij, Bij);    Aij != 0 || Bij != 0
  Cij = 0          ;    otherwise
  

• Algorithm 6 x(sparse, sparse)

• Algorithm should iterate on the nonzero values of matrices A and B and call x(Aij, Bij) when both matrices contain value at (i,j).

• Output type is a SparseMatrix

• x() operation invoked NZ times (number of nonzero items at the same (i,j) for both matrices)

  Cij = x(Aij, Bij);    Aij != 0 && Bij != 0
  Cij = 0          ;    otherwise
  

• Algorithm 7 x(sparse, sparse)

• Algorithm should iterate on all values of matrices A and B and call x(Aij, Bij)

• Output type is a DenseMatrix

• x() operation invoked MxN times

  Cij = x(Aij, Bij);
  

• Algorithm 8 x(sparse, sparse)

• Algorithm should iterate on the nonzero values of matrices A and B and call x(Aij, Bij) when both matrices contain value at (i,j). Use the value from Aij when Bij is zero.

• Output type is a SparseMatrix

• x() operation invoked NZ times (number of nonzero items at the same (i,j) for both matrices)

  Cij = x(Aij, Bij);    Aij != 0 && Bij != 0
  Cij = Aij        ;    Aij != 0
  Cij = 0          ;    otherwise
  

• Algorithm 9 x(sparse, sparse)

• Algorithm should iterate on the nonzero values of matrices A x(Aij, Bij).

• Output type is a SparseMatrix

• x() operation invoked NZA times (number of nonzero items in A)

  Cij = x(Aij, Bij);    Aij != 0
  Cij = 0          ;    otherwise
  

Algorithms for the implementation of element wise operations between a Sparse and Scalar Value:

• Algorithm 10 x(sparse, scalar)

• Algorithm should iterate on the nonzero values of matrix A and call x(Aij, N).

• Output type is a DenseMatrix

• x() operation invoked NZ times (number of nonzero items)

  Cij = x(Aij, N);    Aij != 0
  Cij = N        ;    otherwise
  

• Algorithm 11 x(sparse, scalar)

• Algorithm should iterate on the nonzero values of matrix A and call x(Aij, N).

• Output type is a SparseMatrix

• x() operation invoked NZ times (number of nonzero items)

  Cij = x(Aij, N);    Aij != 0**
  Cij = 0        ;    otherwise**
  

• Algorithm 12 x(sparse, scalar)

• Algorithm should iterate on the zero and nonzero values of matrix A and call x(Aij, N).

• Output type is a DenseMatrix

• x() operation invoked MxN times.

  Cij = x(Aij, N);    Aij != 0
  Cij = x(0, N)  ;    otherwise
  

Algorithms for the implementation of element wise operations between a Dense and Dense matrices:

• **Algorithm 13 x(dense, dense)

• Algorithm should iterate on the values of matrix A and B for all dimensions and call x(Aij..z,Bij..z)

• Output type is a DenseMatrix

• x() operation invoked Z times, where Z is the number of elements in the matrix last dimension. For two dimensional matrix Z = MxN

  Cij..z = x(Aij..z, Bij..z)**
  

Algorithms for the implementation of element wise operations between a Dense Matrix and a Scalar Value:

• Algorithm 14 x(dense, scalar)

• Algorithm should iterate on the values of matrix A for all dimensions and call x(Aij..z, N)

• Output type is a DenseMatrix

• x() operation invoked Z times, where Z is the number of elements in the matrix last dimension.

  Cij..z = x(Aij..z, N)**