Lecture 150:- Tri-Diagonal and Tri-Band Matrix

A tri-diagonal matrix is a square matrix in which all the elements outside the main diagonal, as well as the diagonal elements themselves, are zero, except for the elements on the three diagonals immediately adjacent to the main diagonal. The three diagonals are the main diagonal, the diagonal just above the main diagonal, and the diagonal just below the main diagonal. Tri-diagonal matrices are also known as tridiagonal or three-diagonal matrices.

A tri-band matrix is a generalization of the tri-diagonal matrix, where instead of having three diagonals, it can have any number of diagonals that are not necessarily adjacent to the main diagonal. Tri-band matrices are also known as banded matrices, and they have a special property where most of the elements are zero, leading to memory and computational efficiency.

Tri-diagonal and tri-band matrices have various applications in numerical methods and solving systems of linear equations efficiently.

Let's create a C++ program to represent and display a tri-diagonal matrix using a 2D array:

cppCopy code

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

Output:

mathematicaCopy code

Tri-Diagonal Matrix: 1 2 0 0 3 4 5 0 0 6 7 8 0 0 9 10

In this program, we have defined the TriDiagonalMatrix class, which uses a 2D vector to store the elements of the tri-diagonal matrix. The setElement and getElement functions handle the logic to set and retrieve elements on the three diagonals. The displayTriDiagonalMatrix function displays the tri-diagonal matrix with its meaningful elements.

You can set elements and access values using the (i, j) coordinates, and the class will ensure that only the elements on the three diagonals are accessed and manipulated.

For tri-band matrices with more than three diagonals, the concept is similar, but the size of the 2D vector and the logic for setting and accessing elements will be adjusted accordingly.