Inverse of a Hilbert matrix
This functionality does not run in MATLAB.
linalg::invhilbert(n) returns the inverse of the n×n Hilbert matrix H. The n×n Hilbert matrix H = (hi, j)1 ≤ i ≤ m, 1 ≤ j ≤ n 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.
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 )
The dimension of the matrix: a positive integer
n×n matrix of the domain Dom::Matrix(R).
Hilbert matrices of large dimension are notoriously ill-conditioned, challenging any numerical inversion scheme.
linalg::invhilbert uses the formula
for the inverse of the n×n Hilbert matrix H. All entries of are integers.