Generate Hilbert matrix

stm_hilbert() creates an n x n Hilbert matrix, where each element is defined as Hij = 1 / (i + j - 1), where i and j are the row and column indices.

Usage

stm_hilbert(n)

Arguments

n

A positive integer specifying the size of the Hilbert matrix. It represents both the number of rows and columns.

Value

An n x n matrix with elements Hij = 1 / (i + j - 1).

Details

The Hilbert matrix is used in numerical analysis and is known for its ill-conditioning as n increases. This function uses the outer() function to generate the matrix efficiently.

Examples

stm_hilbert(3)
#>           [,1]      [,2]      [,3]
#> [1,] 1.0000000 0.5000000 0.3333333
#> [2,] 0.5000000 0.3333333 0.2500000
#> [3,] 0.3333333 0.2500000 0.2000000
stm_hilbert(5)
#>           [,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