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Java Forum / General / November 2005BigDecimal and trigonometrics
Does anyone know an easy way, or know of freely any available code / packages, to use sine and cosine on BigDecimal with full precision? /Magnus > Does anyone know an easy way, or know of freely any available code / > packages, to use sine and cosine on BigDecimal with full precision? AFAIK, the sine of a value may be an irrational number, which means it may take an infinite number of digits to represent. Therefore, what you are asking for is impossible on a computer with finite memory, nevermind whether it's running Java or not. - Oliver >>Does anyone know an easy way, or know of freely any available code / >>packages, to use sine and cosine on BigDecimal with full precision? [quoted text clipped - 3 lines] > asking for is impossible on a computer with finite memory, nevermind whether > it's running Java or not. That's only true if one assumes a traditional floating-point representation. There are plenty of techniques (continued fractions, symbolic algebra, etc.) for representing irrational numbers with infinite precision in finite memory. >>>Does anyone know an easy way, or know of freely any available code / >>>packages, to use sine and cosine on BigDecimal with full precision? [quoted text clipped - 8 lines] > symbolic algebra, etc.) for representing irrational numbers with infinite > precision in finite memory. You are, of course, correct. My bad. That being said, if the OP wishes to use BigDecimal, I think we can safely assume (s)he wants traditional floating-point representation. - Oliver On Mon, 14 Nov 2005 21:10:08 GMT, Jeffrey Schwab <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who said : >That's only true if one assumes a traditional floating-point >representation. There are plenty of techniques (continued fractions, >symbolic algebra, etc.) for representing irrational numbers with >infinite precision in finite memory. I think you are confusing rational repeaters (which can be accurately represented by a rational fraction) with irrationals that at best could be described as a limit of a series, e.g. pi, e. Remember the proof that there are non-rational numbers, discovered, IIRC by the ancient Greeks. There is no way you can map all rationals, much less all irrationals with perfect accuracy into a finite address space. There an infinite but countably many rationals and uncountably many irrationals. (One of my pet peeves is newscasters who use the term "uncountable" to mean "a large number" or "more than I can count on my fingers".) "Countable" to mathematicians means there exists a 1-1 mapping between the set and the set of integers.  SignatureCanadian Mind Products, Roedy Green. http://mindprod.com Java custom programming, consulting and coaching. Thank you for all replies. Unfortunatelly I expressed my self badly. What I meant was "arbitary precision". I would for example like to be able to calculate sin(0.2 and cosin(0.2) with 30 decimals. It's possible to do with Taylor expansion, but I hoped there would be some ready package avaible. Sorry for my bad English. /Magnus > Thank you for all replies. Unfortunatelly I expressed my self badly. > What I meant was "arbitary precision". I would for example like to be > able to calculate sin(0.2 and cosin(0.2) with 30 decimals. Are you forced to use the BigDecimal class ? If not then there are other arbitrary precision packages around which might do what you want. One example (which I have never used so I cannot speak for its quality or appropriateness): http://www.apfloat.org/apfloat_java/ -- chris > On Mon, 14 Nov 2005 21:10:08 GMT, Jeffrey Schwab > <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who [quoted text clipped - 19 lines] > "Countable" to mathematicians means there exists a 1-1 mapping between > the set and the set of integers. You are, of course, also correct. But in the specific case of the OP's problem, assuming (s)he wants the sine (or cosine) of a rational number, this can always be expressed in finite memory as the string "SIN(x)" where x is the rational number represented in some finite form (e.g. a finite decimal expansion, which is sometimes possible, or a fraction of two finite whole numbers, which is always possible). - Oliver >But in the specific case of the OP's >problem, assuming (s)he wants the sine (or cosine) of a rational number, >this can always be expressed in finite memory as the string "SIN(x)" where x >is the rational number represented in some finite form (e.g. a finite >decimal expansion, which is sometimes possible, or a fraction of two finite >whole numbers, which is always possible). There are an infinite number of rational numbers. Therefore you cannot represent them with a finite string much less all possible transcendental functions of them. If you limit yourself to all rational numbers expressible as the division of two 1,000,000-bit integers then the "rationals" can be represented by finite strings, but that is just the merest wisp of the total number of rationals. The problem here is we are wearing two hats -- mathematician and computer programmer. Computer programmers are always talking about finite subset approximations of the numbers that mathematicians think about. Computer programmers sometimes forget that, and that the mathematical rules don't perfectly apply on that limited set. SignatureCanadian Mind Products, Roedy Green. http://mindprod.com Java custom programming, consulting and coaching. >>But in the specific case of the OP's >>problem, assuming (s)he wants the sine (or cosine) of a rational number, [quoted text clipped - 8 lines] > represent them with a finite string much less all possible > transcendental functions of them. Yes, there are an infinite number of rational numbers, but every one of them can be represented with finitely many ASCII characters. And you only need a finite amount of ASCII characters to represent "SIN(" and ")". That is why you only need finite memory to represent the sine (or cosine) of any particular rational number. And if the OP wants to represent the sine (or cosine) or 10 rational numbers, this will also require "merely" finite memory. Same with a billion. In fact, as long as the OP wishes to represent the sine (or cosine) of a finite number of rational numbers, (s)he will only need finite memory. If (s)he wants an infinite number of rational numbers, that's a different story, with its own problems (where would these infinite number of rational numbers come from? user input? some sort of iterator?) - Oliver > If (s)he wants an infinite number of rational numbers, that's a >different story, with its own problems (where would these infinite number of >rational numbers come from? user input? some sort of iterator?) There is no upper bound on the RAM needed to represent a single rational as a string. There is also the practical manner of the size of the address space and the limit bigDecimal or bigInteger imposes on the length. You can only deal with a very small subset of the rationals in Java. One place you run into these super large numbers is in calculating combinatorics. SignatureCanadian Mind Products, Roedy Green. http://mindprod.com Java custom programming, consulting and coaching. > On Mon, 14 Nov 2005 21:10:08 GMT, Jeffrey Schwab > <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who [quoted text clipped - 19 lines] > "Countable" to mathematicians means there exists a 1-1 mapping between > the set and the set of integers. Can we assume that each rational number we need to represent must have be arrived at by some sequence of mathematical operations? If so, each such number can be represented by the sequence of operations that invokes it. > Can we assume that each rational number we need to represent must have be > arrived at by some sequence of mathematical operations? If so, each such > number can be represented by the sequence of operations that invokes it. For any number X (rational or otherwise), we can arrive at it by starting with X and doing zero operations on it. This doesn't nescessarily prove that X can be represented with finite memory though. If X is rational, then we sort of apriori have to assume that the step of "start with X" requires finite memory, to which we perform 0 operations (thus adding a overhead of 0 memory) and end up with finite memory requirements again. - Oliver >>Can we assume that each rational number we need to represent must have be >>arrived at by some sequence of mathematical operations? If so, each such [quoted text clipped - 3 lines] > starting with X and doing zero operations on it. This doesn't nescessarily > prove that X can be represented with finite memory though. Of course not. That would be cheating. > If X is rational, then we sort of apriori have to assume that the step > of "start with X" requires finite memory, to which we perform 0 operations > (thus adding a overhead of 0 memory) and end up with finite memory > requirements again. Yes, that's true. But can you please point me to an irrational number that cannot be derived by a sequence of mathematical operations on rational numbers? > Yes, that's true. But can you please point me to an irrational number > that cannot be derived by a sequence of mathematical operations on > rational numbers? No, but I think I can prove their existence if I can prove that all sequences of mathematical operations on rational numbers is countable (since the there are uncountably many irrational numbers). Every sequence of mathematical operations on rational numbers can be represented by some ASCII string (e.g. "1+1") You can order them by using Java's standard string sorting algorithm. Associate the first such legal string with the integer 1. Associate the second such legal string with the integer 2. And so on. You now have a 1 to 1 mapping between sequences of mathematical operations on rational numbers and the set of natural numbers, thus showing that there are only countably many sequences of mathematical operations on rational numbers. Note that I'm assuming the ASCII representation is finite, which I think is true as long as the number of operators and the number of arguments to each operator is finite in the sequence (and as long as each operator and each term can be represented by a finite number of characters, which is true for rational numbers). - Oliver >>Yes, that's true. But can you please point me to an irrational number >>that cannot be derived by a sequence of mathematical operations on >>rational numbers? > > No, but I think I can prove their existence Yes, of course they exist. Name one. > if I can prove that all > sequences of mathematical operations on rational numbers is countable (since [quoted text clipped - 18 lines] > each term can be represented by a finite number of characters, which is true > for rational numbers). Nicely done!!! So there certainly are unrepresentable irrational numbers. Let me "clarify" my position: Any irrational number that can be clearly represented in a traditional mathematical formula or proof, can also be represented in a computer program. On Wed, 16 Nov 2005 22:48:52 GMT, Jeffrey Schwab <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who said : >Any irrational number that can >be clearly represented in a traditional mathematical formula or proof, >can also be represented in a computer program. That is like saying that every describable mathematical problem has a solution that can be represented in Java, clear grandiosity. And clearly false -- look at the halting problems. You may be attempting to say something much weaker -- that anything that can be expressed in mathematical notation could be formulated in a Java String, e.g. as a TeX or PostScript program to typeset it. SignatureCanadian Mind Products, Roedy Green. http://mindprod.com Java custom programming, consulting and coaching. > On Wed, 16 Nov 2005 22:48:52 GMT, Jeffrey Schwab > <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who [quoted text clipped - 3 lines] >>be clearly represented in a traditional mathematical formula or proof, >>can also be represented in a computer program. [snip] > You may be attempting to say something much weaker -- that anything > that can be expressed in mathematical notation could be formulated in > a Java String, e.g. as a TeX or PostScript program to typeset it. I think the two are equivalent. Jeffrey is merely saying that you can represent the number in a computer program (e.g. as a String), not nescessarily that you could compute it's value. For example, you can represent Chaitin's constant as the string "Chaitain's constant" (or by its one character symbol if you have unicode support) on a computer with finite memory, even though actually computing the value of Chaitin's constant would be equivalent to solving the halting problem. http://en.wikipedia.org/wiki/Chaitin%27s_constant - Oliver >>On Wed, 16 Nov 2005 22:48:52 GMT, Jeffrey Schwab >><jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who [quoted text clipped - 19 lines] > > http://en.wikipedia.org/wiki/Chaitin%27s_constant Thanks, that's what I was trying to say. On Wed, 16 Nov 2005 16:56:02 GMT, Jeffrey Schwab <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who said : > But can you please point me to an irrational number >that cannot be derived by a sequence of mathematical operations on >rational numbers? The time until George Bush's death in microseconds from noon UTC today. SignatureCanadian Mind Products, Roedy Green. http://mindprod.com Java custom programming, consulting and coaching. > On Wed, 16 Nov 2005 16:56:02 GMT, Jeffrey Schwab > <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who [quoted text clipped - 6 lines] > The time until George Bush's death in microseconds from noon UTC > today. Nice one. :) Choose a random number uniformly between 0.0 and 1.0. It will almost surely be uncomputable. >>On Wed, 16 Nov 2005 16:56:02 GMT, Jeffrey Schwab >><jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who [quoted text clipped - 11 lines] > Choose a random number uniformly between 0.0 and 1.0. > It will almost surely be uncomputable. Nice job. I think you've actually got me there. :) We could certainly use symbols to stand for these numbers, but I don't think the symbols could really be said to "represent" the irrationals; for example, I don't think symbolic representations could be used to order such random numbers. Just for giggles, here's a way to assign symbols to truly random numbers: In order to be observed, such numbers must be generated by some process, at some finite rate. (It is understood that other irrational numbers exist, but it should also be understood that only finite number may ever be observed by people or computers, since there is a finite number of people and computers, each having finite faculties of observation.) We can then refer to the first number generated by the process as R1, the second as R2, etc. Randoms generated at the same instant may be refered to as RnA, RnB, etc., where n is an ordinal integer. On Wed, 16 Nov 2005 16:56:02 GMT, Jeffrey Schwab <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who said : >Yes, that's true. But can you please point me to an irrational number >that cannot be derived by a sequence of mathematical operations on >rational numbers? It has already been proved there are more irrational numbers that can be put into 1-1 corresponding with integers, hence Strings. "Most" (in the Lebesgue sense) irrationals cannot be described by a String. You are dabbling in measure theory, which is pretty far removed from computer science which is the study of finite numbers. SignatureCanadian Mind Products, Roedy Green. http://mindprod.com Java custom programming, consulting and coaching. > On Wed, 16 Nov 2005 16:56:02 GMT, Jeffrey Schwab > <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who [quoted text clipped - 6 lines] > It has already been proved there are more irrational numbers that can > be put into 1-1 corresponding with integers, hence Strings. Right, we agree on that. I'm just saying that a program needing to represent irrational numbers with "infinite" precision can do so by representing them in an abstract, symbolic way. > "Most" (in > the Lebesgue sense) irrationals cannot be described by a String. You > are dabbling in measure theory, which is pretty far removed from > computer science which is the study of finite numbers. I'm not familiar with Lebesgue, or "measure theory." If you have any good links on them, I'd be interested in learning. Computer science is NOT strictly the study of finite numbers. That's like saying Astronomy is the study of telescopes. Finite numbers are just some of the fundamental tools of computer science. On Wed, 16 Nov 2005 22:44:27 GMT, Jeffrey Schwab <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who said : >I'm not familiar with Lebesgue, or "measure theory." If you have any >good links on them, I'd be interested in learning. You are dancing around an area of advanced mathematics that you might find fascinating. Things you might google for are Aleph, countable, uncountable, 1-1 mapping, advanced probability theory, Lebesgue measure theory, the different kinds of infinity, Georg Kantor, transfinite numbers. rationals are like thin strips of celery fibre in a thick soup of irrationals. Even though you can always find an irrational between two rationals and a rational between two irrationals, in a very strong sense, irrationals are vastly more numerous, and most definitely can't be enumerated by a set of strings. How do you quantify that? You do it by integrating over all the rationals or over all the irrationals. The set of rationals is called a set of measure 0, because when you do that, their contribution comes out 0. Just how you do those integrals is called Lebesgue measure theory. The way you usually tackle the domain of knowledge is with first calculation of finite probablities and combinatorics, then probability, then the various types of infinity, then probability theory over these various infinite sets. You then wander around in Markovian processes (the things that turned me on most since they are so much like finite state automata.) and the astounding 0-1 law. This is stuff I did not learn until after I had my BSc back when I was studying it, so the books you may find won't necessarily be that accessible. Mathematicians tend to go for brevity, and elegance, as if they were constructing puzzles. They are not big on handing you any sort of intuitive understanding. SignatureCanadian Mind Products, Roedy Green. http://mindprod.com Java custom programming, consulting and coaching. > This is stuff I did not learn until after I had my BSc back when I was > studying it, so the books you may find won't necessarily be that > accessible. Mathematicians tend to go for brevity, and elegance, as if > they were constructing puzzles. They are not big on handing you any > sort of intuitive understanding. I learned this stuff by reading Roger Penrose's "The Emperor's New Mind". It's a long, thick book, but all the math stuff lies in the first 1/3rd of it. The rest of the book is equally fascinating as well, of course. http://www.amazon.com/gp/product/0140145346/103-2566881-0495862?v=glance&n=28315 5&v=glance - Oliver > On Wed, 16 Nov 2005 22:44:27 GMT, Jeffrey Schwab > <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who [quoted text clipped - 9 lines] > mapping, advanced probability theory, Lebesgue measure theory, the > different kinds of infinity, Georg Kantor, ITYM Cantor. > transfinite numbers. I'm familiar with all of those things except Lebesgue and "measure theory." > rationals are like thin strips of celery fibre in a thick soup of > irrationals. Even though you can always find an irrational between two > rationals and a rational between two irrationals, in a very strong > sense, irrationals are vastly more numerous, and most definitely can't > be enumerated by a set of strings. Right on. "Celery fibre" is an interesting analogy. :) > How do you quantify that? You do it by integrating over all the > rationals or over all the irrationals. The set of rationals is called > a set of measure 0, because when you do that, their contribution comes > out 0. Just how you do those integrals is called Lebesgue measure > theory. Not sure I follow, but I'll investigate. The proof I've seen before is this one: http://en.wikipedia.org/wiki/Cantor's_diagonal_argument > The way you usually tackle the domain of knowledge is with first > calculation of finite probablities and combinatorics, then > probability, then the various types of infinity, then probability > theory over these various infinite sets. Probability theory as I studied it in school was approached a very different way. We never saw irrational outcomes. It seems strange now, but this never struck me as odd. Anyway, probability is usually based on the randomness of some physical process, which means some random point is being chosen from a continuum of space or time; but whether the infinity of the continuum is the same as the infinity of irrationals is provably unprovable. Btw, have you read this? http://www.amazon.com/gp/product/019514743X/104-2580866-1835953?v=glance&n=28315 5&v=glance It's very good. > You then wander around in Markovian processes (the things that turned > me on most since they are so much like finite state automata.) and the > astounding 0-1 law. I don't know about that either, although I gather this is the same Markov of "Markov chain" fame. Looks like I've got some reading to do! > This is stuff I did not learn until after I had my BSc back when I was > studying it, so the books you may find won't necessarily be that > accessible. Mathematicians tend to go for brevity, and elegance, as if > they were constructing puzzles. They are not big on handing you any > sort of intuitive understanding. Some great books about Mathematics were not written by mathematicians. My personal favorite was written by a dentist: http://www.wwnorton.com/catalog/fall96/math.htm I've had it for three or four years now, and I'm still not half-way through it. On Thu, 17 Nov 2005 17:16:57 GMT, Jeffrey Schwab <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who said : >Georg Kantor, > >ITYM Cantor. I think he spelled it Kantor, but it is often transliterated into Cantor, singer in a synagogue, in English. Sort of like how Joe Green wrote Italian operas. SignatureCanadian Mind Products, Roedy Green. http://mindprod.com Java custom programming, consulting and coaching. On Thu, 17 Nov 2005 17:16:57 GMT, Jeffrey Schwab <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who said : >Probability theory as I studied it in school was approached a very >different way. Advanced probability theory you would hardly recognise as related to the probability of card games. It is all about integrating infinities. I have no idea what practical use it might ever have. SignatureCanadian Mind Products, Roedy Green. http://mindprod.com Java custom programming, consulting and coaching. On Thu, 17 Nov 2005 17:16:57 GMT, Jeffrey Schwab <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who said : >I don't know about that either, although I gather this is the same >Markov of "Markov chain" fame. Looks like I've got some reading to do! Markov processes are ones that have state but no memory of history. The are like generalisations of finite state automata. There is a probability function that maps the previous state onto the next. One of the strange results is that under some not that stringent conditions, if the probability of a state occurring is non-zero, no matter how vanishingly small that probability, if you wait long enough it will happen infinitely often. SignatureCanadian Mind Products, Roedy Green. http://mindprod.com Java custom programming, consulting and coaching. >> On Wed, 16 Nov 2005 16:56:02 GMT, Jeffrey Schwab >> <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who >> said : [snip] > I'm not familiar with Lebesgue, or "measure theory." If you have any > good links on them, I'd be interested in learning. http://en.wikipedia.org/wiki/Measure_theory SignatureThanks in Advance... IchBin, Pocono Lake, Pa, USA http://weconsultants.servebeer.com/JHackerAppManager __________________________________________________________________________ 'If there is one, Knowledge is the "Fountain of Youth"' -William E. Taylor, Regular Guy (1952-) > On Mon, 14 Nov 2005 21:10:08 GMT, Jeffrey Schwab > <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who [quoted text clipped - 7 lines] > I think you are confusing rational repeaters (which can be accurately > represented by a rational fraction) with irrationals I think I am not. > that at best > could be described as a limit of a series, e.g. pi, e. > > Remember the proof that there are non-rational numbers, discovered, > IIRC by the ancient Greeks. Proven geometrically, in fact, by trying to determine the length of the hypoteneuse of a right triangle. This sort of irrational number can be represented in finite space by a continued fraction. http://www.reference.com/browse/wiki/Continued_fraction > There is no way you can map all > rationals, much less all irrationals with perfect accuracy into a > finite address space. There an infinite but countably many rationals > and uncountably many irrationals. As proven by Georg Cantor, by a method I still find mind-blowingly simple and graceful. > (One of my pet peeves is > newscasters who use the term "uncountable" to mean "a large number" or > "more than I can count on my fingers".) I hear ya. :) It may be of comfort to know that "uncountable" has a non-mathematical meaning in the English language: http://dictionary.reference.com/search?q=uncountable > "Countable" to mathematicians means there exists a 1-1 mapping between > the set and the set of integers. You have a deep understanding! However, you don't need to instantiate all possible instances of a class in order to represent a subset of those instances, and similarly, you don't need to represent all irrational numbers in order to represent a (finite) subset of interest, as the OP requested. On Wed, 16 Nov 2005 07:20:48 GMT, Jeffrey Schwab <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who said : >You have a deep understanding! However, you don't need to instantiate >all possible instances of a class in order to represent a subset of >those instances, and similarly, you don't need to represent all >irrational numbers in order to represent a (finite) subset of interest, >as the OP requested. Your orgininal claim was much more extravagant: >That's only true if one assumes a traditional floating-point >representation. There are plenty of techniques (continued fractions, >symbolic algebra, etc.) for representing irrational numbers with >infinite precision in finite memory. Perhaps the problem is I interpreted your statement to mean "representing all irrational numbers with infinite precision in finite memory" where you meant "a few irrational numbers such as e, pi, pi, sin(a/b), sqrt(2) that commonly occur in calculus" SignatureCanadian Mind Products, Roedy Green. http://mindprod.com Java custom programming, consulting and coaching. > On Wed, 16 Nov 2005 07:20:48 GMT, Jeffrey Schwab > <jeff@schwabcenter.com> wrote, quoted or indirectly quoted someone who [quoted text clipped - 17 lines] > memory" where you meant "a few irrational numbers such as e, pi, pi, > sin(a/b), sqrt(2) that commonly occur in calculus" Yes, by "irrational numbers," I meant "those irrational numbers you derive." I did not mean that a computer program could enumerate all real numbers. Sorry for being unclear! You need a MacLaurin series expansion (from calculus) around a known reference point; then you can get arbitrary precision. I have a hard time imagining any practical application that would need more than the precision allowed by a double, though. I've mainly found BigDecimals valuable in currency calculations. Try this reference: http://www.tc.umn.edu/~ringx004/sidebar.html > Does anyone know an easy way, or know of freely any available code / > packages, to use sine and cosine on BigDecimal with full precision? > > /Magnus Free MagazinesGet these publications absolutely FREE for up to 12 months. There are no hidden fees and no obligation. Simply choose a title, complete the application form and submit it. Read more ...
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