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HilbertMatrix

HilbertMatrix[n]
gives the n×n Hilbert matrix with elements of the form .
HilbertMatrix
gives the m×n Hilbert matrix.
3×3 Hilbert matrix:
3×5 Hilbert matrix:
3×3 Hilbert matrix:
In[1]:=
Click for copyable input
Out[1]//MatrixForm=
 
3×5 Hilbert matrix:
In[1]:=
Click for copyable input
Out[1]//MatrixForm=
Hilbert matrix with machine-number entries:
Hilbert matrix with 20-digit precision entries:
A Hilbert matrix with machine-number entries:
A Hilbert matrix with 24-digit precision entries:
Find the exact inverse of the 3×3 Hilbert matrix:
Hilbert matrices are often used to compare numerical algorithms:
Compare methods for solving for known :
Solve using :
Solve using LinearSolve with Gaussian elimination:
Solve using LinearSolve using a Cholesky decomposition:
Solve using LeastSquares:
Compare errors:
Square Hilbert matrices are real symmetric:
The smallest eigenvalue of a square Hilbert matrix decreases exponentially with n:
The model is a reasonable predictor of magnitude for larger values of n:
The condition number increases exponentially with n:
The 2-norm condition number is the ratio of largest to smallest eigenvalue due to symmetry:
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