matrix-hilbert: Hilbert Matrix

Description Usage Arguments Details Value References Examples

Description

Creates a Hilbert matrix.

Usage

1

Arguments

n

an integer value, the dimension of the square matrix.

Details

In linear algebra, a Hilbert matrix is a matrix with the unit fraction elements.

The Hilbert matrices 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.

The Hilbert matrix is symmetric and positive definite.

Value

hilbert generates a Hilbert matrix of order n.

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

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## Create a Hilbert Matrix:
   H = hilbert(5)
   H                              

Example output

Loading required package: timeDate
Loading required package: timeSeries
          [,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

fBasics documentation built on March 13, 2020, 9:09 a.m.