Don't you become even a bit suspicious when the answers don't agree? Consider the number on the lower right hand corner:
PLJsAPL: 2815820646.25489 Dyalog: 2815825985 NARS2000: 2815820182 J Rational: 2815827300 (true answer). If you are proud of implementation X, I can bring it down to earth real quick by asking it to invert H 2*n, or by working in rational numbers. H=: %@>:@(+/~)@i. (+/ .* %.) H 20x 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 On Mon, Jun 18, 2012 at 6:32 PM, Paul Jackson <plj...@gmail.com> wrote: > I only raised it because it is so bad compared to everyone else I've > looked at, including mine. An original copy of my result and my > implementation code are available on request. Please note, I cannot > speak for others but I didn't take a Hilbert matrix into account in > writing mine. > > Paul > > PLJsAPL=: 0 : 0 > 80.9999531610822 ¯3239.99665986097 41579.9420943319 > ¯249479.578965133 810808.433035166 ¯1513508.76245454 > 1621616.2451529 ¯926637.713155974 218789.430959011 > ¯3239.99667368818 172799.763174579 ¯2494795.89900529 > 15966690.2071222 ¯54053889.1945398 103783451.186115 > ¯113513134.744105 65894238.5094722 ¯15752839.8286812 > 41579.9425036479 ¯2494795.91108115 38419849.2522369 > ¯256132286.350815 891889090.567644 ¯1748102418.55488 > 1942335832.29382 ¯1141617699.88458 275674708.595751 > ¯249479.582883114 15966690.3618727 ¯256132287.510015 > 1756335480.92061 ¯6243223179.83275 12430950040.9097 > ¯13984817837.5535 8302674292.97357 ¯2021614600.3302 > 810808.450415449 ¯54053889.9691135 891889098.317587 > ¯6243223204.79328 22544971251.5605 ¯45450659629.1162 > 51648474522.865 ¯30918813482.0781 7581054973.44456 > ¯1513508.80311881 103783453.123491 ¯1748102440.39083 > 12430950134.7588 ¯45450659786.3692 92554068055.4922 > ¯106051533619.168 63930593442.9569 ¯15768594736.8895 > 1621616.2969454 ¯113513137.31893 1942335863.28028 > ¯13984817987.4734 51648474857.9506 ¯106051533934.55 > 122367153478.811 ¯74205153752.1709 18396694252.0436 > ¯926637.747107004 65894240.2479428 ¯1141617721.69509 > 8302674405.58929 ¯30918813765.3446 63930593798.5238 > ¯74205153941.7251 45229808347.2382 ¯11263282403.1867 > 218789.439925196 ¯15752840.2978892 275674714.654419 > ¯2021614632.92992 7581055060.93416 ¯15768594860.4611 > 18396694339.8914 ¯11263282427.9574 2815820646.25489 > ) > > DialogAPL=: 0 : 0 > 81.00000837 ¯3240.000441 41580.00571 ¯2.494800309E5 > 8.108100843E5 ¯1.513512123E6 1.621620094E6 ¯9.266400324E5 > 2.187900028E5 > ¯3240.000352 172800.0149 ¯2494800.132 1.596672026E7 > ¯5.405399896E7 1.037836745E8 ¯1.135133904E8 6.589439254E7 > ¯1.575287780E7 > 41580.0033 ¯2494800.068 38419919.12 ¯2.561327823E8 > 8.918909013E8 ¯1.748106101E9 1.942340048E9 ¯1.141620239E9 > 2.756753345E8 > ¯249480.0094 15966719.51 ¯256132777.2 1.756338956E9 > ¯6.243235864E9 1.243097584E10 ¯1.398484736E10 8.302692071E9 > ¯2.021618982E9 > 810809.9941 ¯54053995.43 891890869.5 ¯6.243235770E9 > 2.254501711E10 ¯4.545075286E10 5.164858121E10 ¯3.091887772E10 > 7.581070803E9 > ¯1513511.922 103783666.1 ¯1748106017 1.243097550E10 > ¯4.545075234E10 9.255425620E10 ¯1.060517489E11 6.393072306E10 > ¯1.576862667E10 > 1621619.848 ¯113513379.8 1942339933 ¯1.398484685E10 > 5.164858015E10 ¯1.060517480E11 1.223673983E11 ¯7.420530117E10 > 1.839673057E10 > ¯926639.8769 65894385.62 ¯1141620161 8.302691705E9 > ¯3.091887687E10 6.393072205E10 ¯7.420530065E10 4.522989667E10 > ¯1.126330416E10 > 218789.963 ¯15752875.99 275675313.6 ¯2.021618879E9 > 7.581070548E9 ¯1.576862633E10 1.839673034E10 ¯1.126330410E10 > 2.815825985E9 > ) > > NARS2000=: 0 : 0 > 80.99994931 ¯3239.996392 41579.93755 > ¯249479.5464 810808.3137 ¯1513508.519 > 1621615.966 ¯926637.5448 218789.3894 > ¯3239.996409 172799.7447 ¯2494795.584 15966687.95 > ¯54053880.89 103783434.2 > ¯113513115.2 65894226.72 ¯15752836.92 > 41579.93807 ¯2494795.6 38419843.93 ¯256132248.1 > 891888949.5 ¯1748102130 1942335500 > ¯1141617499 275674658.9 > ¯249479.5515 15966688.15 ¯256132249.6 1756335208 > ¯6243222172 1.243094797E10 > ¯1.398481546E10 8302672851 ¯2021614244 > 810808.3362 ¯54053881.9 891888959.6 ¯6243222204 > 2.254496755E10 ¯4.545065203E10 > 5.164846577E10 ¯3.091880818E10 7581053660 > ¯1513508.571 103783436.7 ¯1748102158 > 1.243094809E10 ¯4.545065223E10 9.255405253E10 > ¯1.060515157E11 6.393058258E10 ¯1.576859205E10 > 1621616.032 ¯113513118.6 1942335540 > ¯1.398481565E10 5.164846618E10 ¯1.060515161E11 > 1.223671329E11 ¯7.420514126E10 1.839669115E10 > ¯926637.5882 65894228.96 ¯1141617527 8302672992 > ¯3.091880853E10 6.393058301E10 > ¯7.420514148E10 4.522980079E10 ¯1.126328053E10 > 218789.4008 ¯15752837.52 275674666.5 ¯2021614284 > 7581053766 ¯1.576859219E10 > 1.839669126E10 ¯1.126328056E10 2815820182 > ) > > > On Mon, Jun 18, 2012 at 4:11 PM, Roger Hui <rogerhui.can...@gmail.com> > wrote: > > Do not be alarmed! :-) The Hilbert matrix is notoriously ill-conditioned > > and attempts to invert it does go very bad very fast. This is not due to > > any deficiency in the algorithm or implementation but to the fact that > > 64-bit floats do not have sufficient precision to represent the answer. > > > > http://en.wikipedia.org/wiki/Hilbert_matrix > > http://www.jsoftware.com/papers/APLQA.htm#Hilbert_matrix > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
