| hilbert {fBasics} | R Documentation |
Hilbert matrix
Description
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