1 <HTML> 2 <HEAD> 3 <TITLE>Constructive Real Calculator and Library Implementation Notes</title> 4 </head> 5 <BODY BGCOLOR="#FFFFFF"> 6 <H1>Constructive Real Calculator and Library Implementation Notes</h1> 7 </body> 8 The calculator is based on the constructive real library consisting 9 mainly of <TT>com.sgi.math.CR</tt> and <TT>com.sgi.math.UnaryCRFunction</tt>. 10 The former provides basic arithmetic operations on constructive reals. 11 The latter provides some basic operations on unary functions over the 12 constructive reals. 13 <H2>General approach</h2> 14 The library was not designed to use the absolute best known algorithms 15 and to provide the best possible performance. To do so would have 16 significantly complicated the code, and lengthened start-up times for 17 the calculator and similar applications. Instead the goals were to: 18 <OL> 19 <LI> Rely on the standard library to the greatest possible extent. 20 The implementation relies heavily on <TT>java.math.BigInteger</tt>, 21 in spite of the fact that it may not provide asymptotically optimal 22 performance for all operations. 23 <LI> Use algorithms with asymptotically reasonable performance. 24 <LI> Keep the code, and especially the library code, simple. 25 This was done both to make it more easily understandable, and to 26 keep down class loading time. 27 <LI> Avoid heuristics. The code accepts that there is no practical way 28 to avoid diverging computations. The user interface is designed to 29 deal with that. There is no attempt to try to prove that a computation 30 will diverge ahead of time, not even in cases in which such a proof is 31 trivial. 32 </ol> 33 A constructive real number <I>x</i> is represented abstractly as a function 34 <I>f<SUB>x</sub></i>, such that <I>f<SUB>x</sub>(n)</i> produces a scaled 35 integer approximation to <I>x</i>, with an error of strictly less than 36 2<SUP><I>n</i></sup>. More precisely: 37 <P> 38 |<I>f<SUB>x</sub></i>(<I>n</i>) * 2<SUP><I>n</i></sup> - x| < 2<SUP><I>n</i></sup> 39 <P> 40 Since Java does not support higher order functions, these functions 41 are actually represented as objects with an <TT>approximate</tt> 42 function. In order to obtain reasonable performance, each object 43 caches the best previous approximation computed so far. 44 <P> 45 This is very similar to earlier work by Boehm, Lee, Cartwright, Riggle, and 46 O'Donnell. 47 The implementation borrows many ideas from the 48 <A HREF="http://reality.sgi.com/boehm/calc">calculator</a> 49 developed earlier by Boehm and Lee. The major differences are the 50 user interface, the portability of the implementation, the emphasis 51 on simplicity, and the reliance on a general implementation of inverse 52 functions. 53 <P> 54 Several alternate and functionally equivalent representations of 55 constructive real numbers are possible. 56 Gosper and Vuillemin proposed representations based on continued 57 fractions. 58 A representation as functions 59 producing variable precision intervals is probably more efficient 60 for larger scale computation. 61 We chose this representation because it can be implemented compactly 62 layered on large integer arithmetic. 63 <H2>Transcendental functions</h2> 64 The exponential and natural logarithm functions are implemented as Taylor 65 series expansions. There is also a specialized function that computes 66 the Taylor series expansion of atan(1/n), where n is a small integer. 67 This allows moderately efficient computation of pi using 68 <P> 69 pi/4 = 4*atan(1/5) - atan(1/239) 70 <P> 71 All of the remaining trigonometric functions are implemented in terms 72 of the cosine function, which again uses a Taylor series expansion. 73 <P> 74 The inverse trigonometric functions are implemented using a general 75 inverse function operation in <TT>UnaryCRFunction</tt>. This is 76 more expensive than a direct implementation, since each time an approximation 77 to the result is computed, several evaluations of the underlying 78 trigonometric function are needed. Nonetheless, it appears to be 79 surprisingly practical, at least for something as undemanding as a desk 80 calculator. 81 <H2>Prior work</h2> 82 There has been much prior research on the constructive/recursive/computable 83 real numbers and constructive analysis. Relatively little of this 84 has been concerned with issues related to practical implementations. 85 <P> 86 Several implementation efforts examined representations based on 87 either infinite, lazily-evaluated decimal expansions (Schwartz), 88 or continued fractions (Gosper, Vuillemin). So far, these appear 89 less practical than what is implemented here. 90 <P> 91 Probably the most practical approach to constructive real arithmetic 92 is one based on interval arithmetic. A variant that is close to, 93 but not quite, constructive real arithmetic is described in 94 <P> 95 Oliver Aberth, <I>Precise Numerical Analysis</i>, Wm. C. Brown Publishers, 96 Dubuque, Iowa, 1988. 97 <P> 98 The issues related to using this in a higher performance implementation 99 of constructive real arithmetic are explored in 100 <P> 101 Lee and Boehm, "Optimizing Programs over the Constructive Reals", 102 <I>ACM SIGPLAN 90 Conference on Programming Language Design and 103 Implementation, SIGPLAN Notices 25</i>, 6, pp. 102-111. 104 <P> 105 The particular implementation strategy used n this calculator was previously 106 described in 107 <P> 108 Boehm, Cartwright, Riggle, and O'Donnell, "Exact Real Arithmetic: 109 A Case Study in Higher Order Programming", <I>Proceedings of the 110 1986 ACM Lisp and Functional Programming Conference</i>, pp. 162-173, 1986. 111 </body> 112 </html> 113
