Lecture 147:- Upper Triangular Matrix Row-Major Mapping

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

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

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

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 row-major mapping for an upper triangular matrix:

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#include <iostream> #include <vector> class UpperTriangularMatrix { private:    int size;        // Size of the matrix    std::vector<int> elements; // 1D vector to store diagonal elements public:    // Constructor to create an upper triangular matrix of given size    UpperTriangularMatrix(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 upper triangular matrix    void setElement(int i, int j, int value) {        if (i <= j && i < size && j < size) {            int index = i * (i + 1) / 2 + j;            elements[index] = value;        } else {            std::cout << "Invalid index. Element not set." << std::endl;        }    }    // Function to get an element from (i, j) position in the upper triangular matrix    int getElement(int i, int j) {        if (i <= j && i < size && j < size) {            int index = i * (i + 1) / 2 + j;            return elements[index];        } else {            std::cout << "Invalid index. Returning 0." << std::endl;            return 0;        }    }    // Function to display the upper triangular matrix    void displayUpperTriangularMatrix() {        std::cout << "Upper Triangular Matrix:" << std::endl;        for (int i = 0; i < size; i++) {            for (int j = 0; j < size; j++) {                if (i <= j) {                    int index = i * (i + 1) / 2 + j;                    std::cout << elements[index] << " ";                } else {                    std::cout << "0 ";                }            }            std::cout << std::endl;        }    } }; int main() {    int size = 4;    UpperTriangularMatrix matrix(size);    matrix.setElement(0, 0, 1);    matrix.setElement(0, 1, 2);    matrix.setElement(0, 2, 3);    matrix.setElement(0, 3, 4);    matrix.setElement(1, 1, 5);    matrix.setElement(1, 2, 6);    matrix.setElement(1, 3, 7);    matrix.setElement(2, 2, 8);    matrix.setElement(2, 3, 9);    matrix.setElement(3, 3, 10);    matrix.displayUpperTriangularMatrix();    return 0; }

Output:

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

In this program, we have defined the UpperTriangularMatrix class, which uses a 1D vector to store the elements of the upper triangular matrix. The setElement and getElement functions use the row-major mapping formula to access and modify the elements. The displayUpperTriangularMatrix function displays the upper 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 row-major mapping.