3. How Far Away Is Infinity?
Jörg Waldvogel · Society for Industrial and Applied Mathematics eBooks · 2004
O God! I could be bounded in a nutshell, and count myself a king of infinite space, were it not that I have bad dreams. —William Shakespeare (Hamlet, Act 2, Scene 2, Verse 226) Problem 3 The infinite matrix A with entries a11 = 1, a12 = 1/2, a21 = 1/3, a13 = 1/4, a22 = 1/5, a31 = 1/6, and so on, is a bounded operator on ℓ2. What is ∥ A ∥? In §3.1 we reduce the infinite-dimensional problem to the task of calculating a limit of finite-dimensional matrix norms. A simple estimate will be given that determines two digits. In §3.2 we calculate the matrix norms without the need of much background knowledge by using MATLAB's built-in norm command and approach the limit by extrapolation. This will bring us 12 digits, but without satisfactory evidence of correctness. A similar but somewhat more efficient algorithm based on the power method will help us produce 21 digits in §3.3. In §3.4 we use second-order perturbation theory to obtain a precise understanding of the error of approximation. Striving for higher accuracy turns out to be particularly difficult for Problem 3. In fact, Trefethen's first publication of the results on his web page in May 2002 reported only 15 digits, as opposed to 40 digits on the other problems. In a later version, thanks to a method of Rolf Strebel, the missing 25 digits were added. Strebel's method, which is the subject of §3.5, is based on the Euler-Maclaurin sum formula and is the most efficient method to get the full accuracy of 16 digits in IEEE double-precision arithmetic. Though the method performs with remarkable success, the convergence rate slows down for very high accuracies beyond, say, 100 digits. We will overcome this difficulty in §3.6 by capturing infinity via complex analysis and contour integration.