Step by Step solution about How to resolve the algorithm Pascal matrix generation step by step in the C++ programming language

The provided C++ code demonstrates how to generate three variations of Pascal's triangle: upper, lower, and symmetric. Pascal's triangle is a mathematical construct that forms a triangular array of binomial coefficients.

Here's an explanation of the code:

1. Data Structure: The code uses a 2D vector (vv) to represent the Pascal's triangle matrix.

2. pascal_upper: This function generates the upper part of Pascal's triangle. It iterates through each row and column to calculate each element's value.

   • If i > j, the value is set to 0 (elements above the main diagonal).
   • If i == j or i == 0, the value is set to 1 (elements on the main diagonal and the first row).
   • Otherwise, the value is calculated as the sum of the element to the top-left and the element to the left in the previous row.

3. pascal_lower: This function generates the lower part of Pascal's triangle. It follows the same principles as pascal_upper, but it excludes elements below the main diagonal (i < j).

4. pascal_symmetric: This function generates the symmetric version of Pascal's triangle, which is a complete triangle.

   • If i == 0 or j == 0, the value is set to 1 (elements on the main diagonal and the first row).
   • Otherwise, the value is calculated as the sum of the element to the left and the element above it in the current row.

5. print_matrix: This function prints the 2D vector representing the Pascal's triangle matrix.

6. main:

   • It calls pascal_upper, pascal_lower, and pascal_symmetric to generate the three variations of Pascal's triangle with 5 rows each.
   • It then prints each triangle to the console using the print_matrix function.

When executed, this code outputs the upper, lower, and symmetric variations of Pascal's triangle with 5 rows each.

#include <iostream>
#include <vector>

typedef std::vector<std::vector<int>> vv;

vv pascal_upper(int n) {
    vv matrix(n);
    for (int i = 0; i < n; ++i) {
        for (int j = 0; j < n; ++j) {
                if (i > j) matrix[i].push_back(0);
                else if (i == j || i == 0) matrix[i].push_back(1);
                else matrix[i].push_back(matrix[i - 1][j - 1] + matrix[i][j - 1]);
            }
        }
        return matrix;
    }

vv pascal_lower(int n) {
    vv matrix(n);
    for (int i = 0; i < n; ++i) {
        for (int j = 0; j < n; ++j) {
            if (i < j) matrix[i].push_back(0);
            else if (i == j || j == 0) matrix[i].push_back(1);
            else matrix[i].push_back(matrix[i - 1][j - 1] + matrix[i - 1][j]);
        }
    }
    return matrix;
}

vv pascal_symmetric(int n) {
    vv matrix(n);
    for (int i = 0; i < n; ++i) {
        for (int j = 0; j < n; ++j) {
            if (i == 0 || j == 0) matrix[i].push_back(1);
            else matrix[i].push_back(matrix[i][j - 1] + matrix[i - 1][j]);
        }
    }
    return matrix;
}


void print_matrix(vv matrix) {
    for (std::vector<int> v: matrix) {
        for (int i: v) {
            std::cout << " " << i;
        }
        std::cout << std::endl;
    }
}

int main() {
    std::cout << "PASCAL UPPER MATRIX" << std::endl;
    print_matrix(pascal_upper(5));
    std::cout << "PASCAL LOWER MATRIX" << std::endl;
    print_matrix(pascal_lower(5));
    std::cout << "PASCAL SYMMETRIC MATRIX" << std::endl;
    print_matrix(pascal_symmetric(5));
}