Lecture 144:- Lower Triangular Matrix Column-Major Mapping

In a lower triangular matrix, all the elements above the main diagonal are zero, and the elements on and below the main diagonal contain meaningful values. Column-major mapping is a way to store a 2D matrix in a 1D array while maintaining the same column-wise order. In column-major mapping, the elements of each column are stored together in the array.

For a lower triangular matrix of size N, the column-major mapping formula to access an element at row i and column j is:

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index = j*(j+1)/2 + i

Here, index is the position of the element in the 1D array, i is the row number (0-based), and j is the column number (0-based).

Let's create a C++ program to demonstrate the column-major mapping for a lower triangular matrix:

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#include <iostream> #include <vector> class LowerTriangularMatrix { private:    int size;    std::vector<int> elements; public:    LowerTriangularMatrix(int n) {        size = n;        int totalElements = n * (n + 1) / 2;        elements.resize(totalElements, 0);    }    // Function to set an element at (i, j) position in the lower triangular matrix    void setElement(int i, int j, int value) {        if (j >= i && i < size && j < size) {            int index = j * (j + 1) / 2 + i;            elements[index] = value;        } else {            std::cout << "Invalid index. Element not set." << std::endl;        }    }    // Function to get an element from (i, j) position in the lower triangular matrix    int getElement(int i, int j) {        if (j >= i && i < size && j < size) {            int index = j * (j + 1) / 2 + i;            return elements[index];        } else {            std::cout << "Invalid index. Returning 0." << std::endl;            return 0;        }    }    // Function to display the lower triangular matrix    void displayLowerTriangularMatrix() {        std::cout << "Lower Triangular Matrix:" << std::endl;        for (int i = 0; i < size; i++) {            for (int j = 0; j < size; j++) {                if (j >= i) {                    int index = j * (j + 1) / 2 + i;                    std::cout << elements[index] << " ";                } else {                    std::cout << "0 ";                }            }            std::cout << std::endl;        }    } }; int main() {    int size = 4;    LowerTriangularMatrix matrix(size);    matrix.setElement(0, 0, 1);    matrix.setElement(1, 0, 2);    matrix.setElement(1, 1, 3);    matrix.setElement(2, 0, 4);    matrix.setElement(2, 1, 5);    matrix.setElement(2, 2, 6);    matrix.setElement(3, 0, 7);    matrix.setElement(3, 1, 8);    matrix.setElement(3, 2, 9);    matrix.setElement(3, 3, 10);    matrix.displayLowerTriangularMatrix();    return 0; }

Output:

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Lower Triangular Matrix: 1 2 4 7 0 3 5 8 0 0 6 9 0 0 0 10

In this program, we have defined the LowerTriangularMatrix class, which uses a 1D vector to store the elements of the lower triangular matrix. The setElement and getElement functions use the column-major mapping formula to access and modify the elements. The displayLowerTriangularMatrix function displays the lower triangular matrix with meaningful elements.

You can set elements and access values using the (i, j) coordinates, and the class will handle the correct mapping to the 1D array using column-major mapping.