Hilbert matrix

Creates a Hilbert matrix.

Usage

hilbert(n)

Arguments

n

an integer value, the dimension of the square matrix.

Details

An \(n,n\) matrix with \((i,j)\)th element equal to \(1/(i+j-1)\) is said to be a Hilbert matrix of order \(n\). Hilbert matrices are symmetric and positive definite.

They are canonical examples of ill-conditioned matrices, making them notoriously difficult to use in numerical computation. For example, the 2-norm condition number of a 5x5 Hilbert matrix above is about 4.8e5.

Value

a matrix

References

Hilbert D., Collected papers, vol. II, article 21.

Beckermann B, (2000); The condition number of real Vandermonde, Krylov and positive definite Hankel matrices, Numerische Mathematik 85, 553--577, 2000.

Choi, M.D., (1983); Tricks or Treats with the Hilbert Matrix, American Mathematical Monthly 90, 301--312, 1983.

Todd, J., (1954); The Condition Number of the Finite Segment of the Hilbert Matrix, National Bureau of Standards, Applied Mathematics Series 39, 109--116.

Wilf, H.S., (1970); Finite Sections of Some Classical Inequalities, Heidelberg, Springer.

Examples

## Create a Hilbert Matrix:
   H = hilbert(5)
   H                              
#>           [,1]      [,2]      [,3]      [,4]      [,5]
#> [1,] 1.0000000 0.5000000 0.3333333 0.2500000 0.2000000
#> [2,] 0.5000000 0.3333333 0.2500000 0.2000000 0.1666667
#> [3,] 0.3333333 0.2500000 0.2000000 0.1666667 0.1428571
#> [4,] 0.2500000 0.2000000 0.1666667 0.1428571 0.1250000
#> [5,] 0.2000000 0.1666667 0.1428571 0.1250000 0.1111111