• Author(s): Johan Brinch
• Supervisor(s): Jakob Grue Simonsen
• Link: brinchj-2009_roots.pdf [ preview ]
• AMS publication: Volume 82, Number 283, July 2013, Pages 1837–1858

Abstract:

I investigate the normality of algebraic irrational numbers, specifically by analyzing their chi^2, nabla^2chi^2 and Kolmogorov-Smirnov distributions.

I analyze 39 Pisot numbers, 47 Salem numbers and the roots of 15 randomly generated polynomials with integer coefficients, all computed to 5 bases with a precision of 2 * 3^18 approx 2^29.5 bits (about 233 million decimals).

I show that the frequencies of the digit sequences tested are equidistributed with a confidence value of 5%. Specifically, digit sequences of size 10 in bases 2 and 3, size 7 in base 5, size 6 in base 7 and size 5 in base 10 are equally frequent. This result is true for all 101 numbers. I show that no pattern can be found in the statistical anomalies and that only few anomalies survive a prefix size of 2 * 3^18 approx 2^29.5 .

The results lend credence to Borel's conjecture that all algebraic irrationals are normal in all bases.

Furthermore, popular root finding algorithms are discussed and their per- formance when computing roots to arbitrary precision compared. Specifically, I give a comparison of Newton's, Laguerre's and Gupta-Mittal's methods using MPSolve as reference. The results show that Newton's and Laguerre's methods are much faster than the competition. Gupta-Mittal is shown to reduce to the classical Power Method, faster than MPsolve, but much slower than Newton's and Laguerre's methods. Gupta-Mittal appears to be unsuited, while Laguerre's method shows great performance.

Keywords

algebraic, normal, polynomial, roots, power method, laguerre, statistics, chi square, pisot numbers, salem numbers, monic polynomials, multiple roots