Hilbert Matrix

A Hilbert Matrix is a square matrix whose each element is a unit fraction.
Properties:

  1. It is a symmetric matrix.
  2. Its determinant value is always positive.
  3. 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++

filter_none

edit
close

play_arrow

link
brightness_4
code

// 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;
}

chevron_right


Java

filter_none

edit
close

play_arrow

link
brightness_4
code

// 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.

chevron_right


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


My Personal Notes arrow_drop_up

Maths is the language of nature

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.



Improved By : vt_m, Mithun Kumar



Article Tags :
Practice Tags :


Be the First to upvote.


Please write to us at contribute@geeksforgeeks.org to report any issue with the above content.