Matrix operations are fundamental in many areas of computer science and data analysis. One efficient way to implement matrix operations is by using recursive templates in C++. Recursive templates allow us to handle matrices of varying dimensions without sacrificing performance. In this article, we will explore how to implement recursive templates for common matrix operations such as addition, multiplication, and transposition.

Adding Matrices

To begin, let’s define a Matrix class that will hold the data and dimensions of a matrix. We will also define a recursive template function add for adding two matrices.

template <size_t Rows, size_t Cols>
struct Matrix {
    std::array<std::array<double, Cols>, Rows> data;
};

template <size_t Rows, size_t Cols, size_t Index = 0>
void add(const Matrix<Rows, Cols>& A, const Matrix<Rows, Cols>& B, Matrix<Rows, Cols>& result) {
    for (size_t i = 0; i < Cols; ++i) {
        result.data[Index][i] = A.data[Index][i] + B.data[Index][i];
    }

    if constexpr (Index + 1 < Rows) {
        add<Rows, Cols, Index + 1>(A, B, result);
    }
}

In the above code snippet, we use a recursive template function add to add corresponding elements of matrices A and B and store the result in the result matrix. The if constexpr statement ensures that the recursion terminates when Index reaches the number of rows.

Multiplying Matrices

Next, let’s implement a recursive template function multiply for multiplying two matrices.

template <size_t Rows, size_t Cols, size_t K, size_t IndexI = 0, size_t IndexJ = 0>
void multiply(const Matrix<Rows, K>& A, const Matrix<K, Cols>& B, Matrix<Rows, Cols>& result) {
    if constexpr (IndexJ == Cols) {
        IndexI++;
        IndexJ = 0;
    }

    if constexpr (IndexI == Rows) {
        return;
    }

    double sum = 0;
    for (size_t k = 0; k < K; ++k) {
        sum += A.data[IndexI][k] * B.data[k][IndexJ];
    }
    result.data[IndexI][IndexJ] = sum;

    multiply<Rows, Cols, K, IndexI, IndexJ + 1>(A, B, result);
}

In the code snippet above, the multiply function multiplies matrices A and B by calculating the dot product of each row of A with each column of B, and stores the result in the result matrix. The recursion is controlled by IndexI and IndexJ, which increment until the multiplication is complete.

Transposing a Matrix

Lastly, let’s implement a recursive template function transpose to transpose a matrix.

template <size_t Rows, size_t Cols, size_t Index = 0>
void transpose(const Matrix<Rows, Cols>& input, Matrix<Cols, Rows>& output) {
    for (size_t i = 0; i < Rows; ++i) {
        output.data[Index][i] = input.data[i][Index];
    }

    if constexpr (Index + 1 < Cols) {
        transpose<Rows, Cols, Index + 1>(input, output);
    }
}

The transpose function swaps the rows and columns of the input matrix and stores the result in the output matrix. The recursion continues until Index reaches the number of columns.

Conclusion

Recursive templates provide a flexible and efficient way to perform matrix operations in C++. They allow us to handle matrices of arbitrary dimensions without sacrificing performance. By implementing recursive template functions for matrix addition, multiplication, and transposition, we have demonstrated the power and elegance of this technique.

#matrix #recursive-templates