`hilb`

Hilbert matrix, with entries 1/(i+j-1).

Category: Mathematics > Elementary Math

Syntax

H = hilb(n)

Returns the n-by-n Hilbert matrix.

Description

The Hilbert matrix is the classical example of an ill-conditioned matrix. Its condition number grows exponentially with n (about 1.6e13 already for n = 10), so solving H*x = b in floating point loses most digits for modest sizes. It comes from least-squares fitting by polynomials on the interval from 0 to 1.

Its exact inverse has integer entries and is returned by invhilb.

Input arguments

n: Nonnegative integer, the order of the matrix.

Output arguments

H: Symmetric positive definite matrix with H(i, j) = 1/(i + j - 1).

Example

A 3-by-3 Hilbert matrix

H = hilb(3)
cond(hilb(8))

Output

H = 3×3

    1.0000    0.5000    0.3333
    0.5000    0.3333    0.2500
    0.3333    0.2500    0.2000
ans = 1.5258e+10

See also

invhilb magic hankel cond gallery