Hilbert Matrix
A Hilbert Matrix is a square matrix whose each element is a unit fraction.
Properties:
- It is a symmetric matrix.
- Its determinant value is always positive.
Examples:
Input : N = 2
Output : 1 0.5
0.5 0.33
Input : N = 3
Output : 1.0000 0.5000 0.3333
0.5000 0.3333 0.2500
0.3333 0.2500 0.2000
Mathematically, Hilbert Matrix can be formed by the given formula:
Let H be a Hilbert Matrix of NxN. Then H(i, j) = 1/(i+j-1)
Below is the basic implementation of the above formula.
C++
// C++ program for Hilbert Matrix #include <bits/stdc++.h> using namespace std; // Function that generates a Hilbert matrix void printMatrix(int n) { float H[n][n]; for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { // using the formula to generate // hilbert matrix H[i][j] = (float)1.0 / ((i + 1) + (j + 1) - 1.0); } } for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) cout << H[i][j] << " "; cout << endl; } } // driver function int main() { int n = 3; printMatrix(n); return 0; } |
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Java
// Java program for // Hilbert Matrix import java.io.*; class GFG { // Function that generates // a Hilbert matrix static void printMatrix(int n) { float H[][] = new float[n][n]; for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { // using the formula // to generate // hilbert matrix H[i][j] = (float)1.0 / ((i + 1) + (j + 1) - (float)1.0); } } for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) System.out.print(H[i][j] + " "); System.out.println(); } } // Driver code public static void main (String[] args) { int n = 3; printMatrix(n); } } // This code is contributed // by anuj_67. |
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C#
// C# program for Hilbert Matrix
using System;
class GFG
{
// Function that generates
// a Hilbert matrix
static void printMatrix(int n)
{
float[,] H = new float[n, n];
for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { // using the formula to generate // hilbert matrix H[i, j] = (float)1.0 / ((i + 1) + (j + 1) - (float)1.0); } } for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) Console.Write(H[i, j] + " "); Console.WriteLine(""); } } // Driver code public static void Main() { int n = 3; printMatrix(n); } } // This code is contributed // by mits [tabbyending]
Output:
1 0.5 0.333333 0.5 0.333333 0.25 0.333333 0.25 0.2
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