# Documentation

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# `linalg`::`invhilbert`

Inverse of a Hilbert matrix

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## Syntax

```linalg::invhilbert(`n`, <`R`>)
```

## Description

`linalg::invhilbert(n)` returns the inverse of the n×n Hilbert matrix H. The n×n Hilbert matrix H = (hi, j)1 ≤ im, 1 ≤ jn is defined by .

`linalg::invhilbert` uses an explicit formula for the inverse.

Note that the entries of the inverse of a Hilbert matrix are integers. But the returned matrix is defined over the standard component domain `Dom::ExpressionField()` so that no conversion is necessary when working with other functions that expect or return matrices over that component domain.

`linalg::invhilbert(n,Dom::Integer)` returns the inverse of the n×n Hilbert matrix defined over the integers.

## Examples

### Example 1

We compute the inverse of the 3×3 Hilbert matrix:

`A := linalg::invhilbert(3)`

This is a matrix of the domain `Dom::Matrix()`.

If you prefer a different component ring, the matrix may be converted into the desired domain afterwards (see `coerce`, for example). Alternatively, one can specify the component ring when calling `linalg::invhilbert`, for example the domain `Dom::Float`:

`A := linalg::invhilbert(3, Dom::Float)`

`domtype( A )`

## Parameters

 `n` The dimension of the matrix: a positive integer `R` The component ring: a domain of category `Cat::Rng`; default: `Dom::ExpressionField``()`

## Return Values

n×n matrix of the domain `Dom::Matrix``(R)`.

## References

N.J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM 1996

## Algorithms

Hilbert matrices of large dimension are notoriously ill-conditioned, challenging any numerical inversion scheme.

`linalg::invhilbert` uses the formula

where

for the inverse of the n×n Hilbert matrix H. All entries of are integers.

#### Mathematical Modeling with Symbolic Math Toolbox

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