3  A Singular Problem

The value sought is the largest singular value of the matrix. Approximate the matrix by the upper-left 2048 × 2048 matrix and use Mathematica's SingularValues function (which uses the QR algorithm) or use Matlab's norm function. Confidence in the answer is obtained by using extrapolation on the squence of values one gets by looking at all upper-left submatrices whose dimension is a power of 2. We got 13 digits. Bornemann obtained 20 digits. R. Strebel of ETH-Zurich obtained 40 digits:
1.274224152821228188212340639725078099472

RowBox[{RowBox[{f[i_, j_], , :=, , RowBox[{2., /, (2 - i + i^2 - 3 j + 2 i j + j^2)}]}], ;}] SingularValues[Array[f, {2048, 2048}]] 〚2, 1〛

1.27422

Here is the extrapolation method. We look at 13 powers of two and then extrapolate. There are ways (using higher precision) to learn that the extrapolated result is likely correct to 13 digits. In fact, the extrapolated result below turns out to be correct when rounded to 15 digits.

data = Table[{n, SingularValues[mat[2^n]] 〚2, 1〛}, {n, 0, 12}] ; TableForm[data]

RowBox[{      , RowBox[{1      1, ... bsp; , 1.27422}], , RowBox[{4096,       , 1.27422}]}]}]

SequenceLimit[Last /@ data]

1.2742241528212157

Created by Mathematica  (June 27, 2004)