Numerical Computation with Guaranteed Accuracy
This page provides information on numerical computation with guaranteed accuracy.
Topic
1. Kepler's conjecture is proved recently using numerical computation with guaranteed accuracy in
Thomas C. Hales:``Cannonballs and Honeycombs'', Notices of the AMS, Vol. 47, No.4 (2000/4) pp.440-449
For fast verification, fast numerical computation tools are needed. The following is the state of art of level of optimization of Matlab, Octave and Scilab.
(1) MATLAB6 is optimized by ATLAS.
(2) Octave2.1.xx can be optimized by ATLAS. In the first place, compile ATLAS. Then, copy generated all .a files to /usr/local/lib. Finally, do ./configure, make and make install. By this process, we can get an optimized Octave.
Remark: 1. Although A*B, lu(A) etc. are optimized by this procedure, A\b may not be optimized. Thus, I have made a m-files lusol.m . This program calculates lu-decomposition by the optimized lu(A) of Octave, then calculates forward substitution and backward elimination by for-loops of Octave. Let us consider Ax=b, where A is square matrix and x and b vectors. Via PentiumIII 600MHz machine, LAPACK+ATLAS solves 1000 times 1000 matrix A problem by 2.5 second. The lusol(A,b) solves the same problem within 4 second. Since Octave's A\b solves the same problem by 15 seconds, lusol is much faster.
(3) Optimization of Scilab through ATLAS seems difficult.
Instructions for changing rounding mode of IEEE754.1. Rounding Instructions in C.(Table)
2. Rounding instructions of Octave on Linux (Octfile)
3. Rounding instructions of Scilab on Windows (Dll)
Matlab program for numerical computation with guaranteed accuracy.
(1) Matrix Equation(m-files).