Mathematica Solutions to the ISSAC' 97 Systems
Wolfram Research, Inc.
What is the 4-significant-digit approximation to the condition number
of the 256 by 256 Hilbert matrix?
Method 1: Calculate the ratio of the eigenvalues explicitly, using
brute force (the straightforward way).
The condition number for the matrix under consideration is the ratio
of the largest to the smallest eigenvalue.
The condition number of a Hilbert matrix of dimension is
roughly equal to (see ); for this is approximately . We
expect to multiply and subtract expressions in intermediate stages, so
to be on the safe side we'll use about twice as many digits.
Here is a plot of the 256 eigenvalues.
This is the answer.
Method 2: Use the power method for the matrix and its inverse (the
Fortunately, there is a closed-form expression for the matrix
elements of the inverse of a Hilbert matrix (see ). This allows us to iterate toward the
smallest and the largest eigenvalues using the power method.
As an aside, we note that the elements of this matrix have an
Again, about 800 digits should be enough.
We define two functions, N to normalize an eigenvector and
RayleighQuotient to calculate an eigenvalue.
We iterate until the ratio of the two eigenvalues remains constant up
to the required precision.
Here are plots of the components of these two eigenvectors.
This is the ratio of the two eigenvalues.
Method 3: Use LUDecomposition (the general way).
This method works for arbitrary matrices; we don't have to know the
An LU decomposition is more stable than an eigenvalue calculation, so
we don't have to use 800 digits. It is enough to have somewhat more
digits than the expected size of the condition number.