mathjs
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Math.js is an extensive math library for JavaScript and Node.js. It features a flexible expression parser and offers an integrated solution to work with numbers, big numbers, complex numbers, units, and matrices.
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###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)**
````