The rational numbers are exact. ([:+/\|.)^:(i.5)1 1 1 1 1 2 2 3 3 5 5 8 fr=: 13 :'%/"1([:+/\|.)^:(i.y)1 1' (,.0j25":"0 fr 20);' ';x:,.fr 20 ┌───────────────────────────┬─┬──────────┐ │1.0000000000000000000000000│ │ 1│ │0.5000000000000000000000000│ │ 1r2│ │0.6666666666666666300000000│ │ 2r3│ │0.5999999999999999800000000│ │ 3r5│ │0.6250000000000000000000000│ │ 5r8│ │0.6153846153846154200000000│ │ 8r13│ │0.6190476190476190700000000│ │ 13r21│ │0.6176470588235294400000000│ │ 21r34│ │0.6181818181818181700000000│ │ 34r55│ │0.6179775280898876000000000│ │ 55r89│ │0.6180555555555555800000000│ │ 89r144│ │0.6180257510729614300000000│ │ 144r233│ │0.6180371352785145600000000│ │ 233r377│ │0.6180327868852458800000000│ │ 377r610│ │0.6180344478216818200000000│ │ 610r987│ │0.6180338134001252000000000│ │ 987r1597│ │0.6180340557275542100000000│ │ 1597r2584│ │0.6180339631667065600000000│ │ 2584r4181│ │0.6180339985218034100000000│ │ 4181r6765│ │0.6180339850173579600000000│ │6765r10946│ └───────────────────────────┴─┴──────────┘ Linda
-----Original Message----- From: Programming <programming-boun...@forums.jsoftware.com> On Behalf Of William Tanksley, Jr Sent: Friday, March 29, 2019 12:23 PM To: Programming forum <programm...@jsoftware.com> Subject: Re: [Jprogramming] converting from 'floating' to 'rational' Ian Clark <earthspo...@gmail.com> wrote: > But why should I feel obliged to carry on using lossy methods when > I've just discovered I don't need to? Methods such as floating point > arithmetic, plus truncation of infinite series at some arbitrary > point. The fact that few practical measurements are made to an > accuracy greater than 0.01% doesn't actually justify lossy methods in > the calculating machine. It merely condones them, which is something else > entirely. There will be a cost, of course. Supporting arbitrarily small and large numbers changes the time characteristics of the computations in ways that will depend on the log-size of the numbers -- and of course will blow the CPU's caching. Also, because the intermediate values are being stored with unlimited precision, you may find some surprises, such as values close to 1 which have enormous numerators and denominators. IMO it's a worthy experiment, especially if you wind up gathering data about the cost and benefit. There's some interesting reflections going on about this on the "unums" mailing list. The trouble with indefinite precision rationals is that they are overkill for all of the problems where they're actually needed, since the inputs and the solution will normally need to be expressed to only finite digits. Now, I don't think this makes doing experiments with them worthless; far from it. By tracking things like the smallest expected input (for example the smallest triangle side, or the largest ratio between sides) and the largest integer generated as intermediate value (perhaps also tracking the ratio in which this integer appeared), we can wind up answering how bad things can get (of course, this is the task of numerical analysis). Ulrich Kulisch developed technology called the "super-accumulator", which was supposed to function alongside the usual group of floating-point registers. It stored an overkill number of bits to permit it to accumulate multiple additions of products of arbitrary floats, the sort of operations you need to evaluate polynomials and linear algebra. Using this, he was able to show that a large number of operations which were considered unstable were possible to stabilize by providing this unrounded accumulator. In the end this wasn't made part of the IEEE standard, but it's being included in some of the numerical systems being developed in response to the need for more flexible floating point formats from the machine-learning world, where smaller-bitwidth floating point numbers both make stability a serious concern and also make the required size of the superaccumulator much smaller. > Ian Clark -Wm ---------------------------------------------------------------------- For information about J forums see https://eur01.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.jsoftware.com%2Fforums.htm&data=02%7C01%7C%7Cdd9b991445bd4aee1e7a08d6b462d9c2%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C636894733979469996&sdata=0uEWVVEQ3RUboB2xRtJgdxL%2FUSZGkeff1L9HAEGmhYM%3D&reserved=0 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
