An Experimental Interval Arithmetic Package in Maple. Экспериментальный пакет интервальной арифметики для системы Maple - PDF

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Interval Computations No 2, 1993 An Experimental Interval Arithmetic Package in Maple Amanda E. Connell and Robert M. Corless We describe an experimental arbitrary-precision interval arithmetic package

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Interval Computations No 2, 1993 An Experimental Interval Arithmetic Package in Maple Amanda E. Connell and Robert M. Corless We describe an experimental arbitrary-precision interval arithmetic package written for the computer algebra system Maple. The functions implemented are those of the proposed Basic Interval Arithmetic Subroutine (BIAS) library. We give here an overview of the package design, some examples of the usage and code, and a report on our experiences. The package will be made available under the name INTPAK from the Maple share library (anonymous ftp to daisy.uwaterloo.ca in a subdirectory of the directory maple) at a future date. Экспериментальный пакет интервальной арифметики для системы Maple А. Е. Коннелл, Р. М. Корлесс Описан экспериментальный пакет интервальной арифметики произвольной точности, написанный для системы компьютерной алгебры Maple. Функции реализованы согласно спецификациям библиотеки базовых подпрограмм интервальной арифметики (BIAS). Статья содержит обзор структуры пакета, несколько примеров использования и программирования, а также описывает опыт его применения авторами. В будущем пакет будет доступен всем желающим под именем INTPAK в составе библиотеки разделенного пользования Maple по anonymous ftp, по адресу daisy.uwaterloo.ca, в одном из подкаталогов каталога maple. * This work was supported by NSERC. Amanda Connell s summer work term was financed by an NSERC Undergraduate scholarship, and her presentation of this work at the Numerical Analysis with Automatic Result Verification conference in Lafayette, LA in 1993 was supported by NSERC, by NSF, and by Waterloo Maple Software, Inc. We would also like to thank George Corliss, Stephen Merrill, and Jim Phillips for many informative discussions, pointers, and bug reports. c A. E. Connell, R. M. Corless, 1994 An Experimental Interval Arithmetic Package in Maple Introduction The computer algebra system Maple [3] has high-quality arbitrary-precision point arithmetic. If the environment variable Digits is set to a value high enough that the hardware or C floating-point arithmetic cannot be used, then Maple s software arbitrary-precision arithmetic is used. In this case, arithmetic operations and single function evaluations are claimed to be accurate to within 0.6 ulp (units in the last place). [The basis of this claim is that functions are evaluated using as many guard digits as necessary: for example, the routine evalf/w uses a complicated formula involving Digits and the magnitude of 1 + e x to conservatively determine how many guard digits are required to guarantee accuracy of W (x) to 0.6 ulp. See [4] for a discussion of W (x).] However, this level of accuracy is claimed only for single operations, and no facilities exist in Maple (as distributed) for studying the effects of the propagation of errors, either of roundoff errors or of data uncertainties. Maple also has very efficient facilities for exact integer and rational arithmetic, which avoids roundoff error altogether, but this comes with the inherent storage penalty of possibly rapid growth of the integers involved as the computation proceeds. Maple allows very large integers (limited essentially only by the amount of memory on the machine), but some problems exhibit exponential growth in the storage requirements, which makes this approach impractical in these cases. In other cases, though, where the growth of the integers is not too severe, Maple s existing arithmetic is already satisfactory: one can do a long computation over the rationals and then convert to a decimal expansion as the final step and incur no more error than 0.6 ulp. Examples include solving linear systems of equations with rational coefficients or evaluating a determinant of a matrix of rational numbers. However, no computation with expressions containing transcendental functions or requiring root extraction can be done in this way, and doing the computation symbolically instead can incur a tremendous penalty in the form of intermediate expression swell, which can cause super-exponential growth in the storage requirements as the computation proceeds. Further, the numerical stability of the final expression may be in doubt (Maple makes no claim to evaluate arbitrary expressions to 0.6 ulp accuracy). For example, though Maple can quite easily exactly invert rational matrices of moderately large size, such as Hilbert matrices, asking Maple to 122 A. E. Connell, R. M. Corless compute the (well-conditioned) eigenvalues of the Hilbert matrix of, say, order 25 by 1) computing the characteristic polynomial exactly and then 2) attempting to find the roots of this polynomial by using a numerical method such as is implemented in the Maple command fsolve is a bad idea and leads to unnecessary expense and large errors. The roots of this polynomial are ill-conditioned, as many are, and this leads to trouble. Maple can successfully use this approach to find the eigenvalues of the Hilbert matrix of order 15, using the very precise routine realroot, written by Bruce Char, to isolate all fifteen positive roots of the characteristic polynomial (which, incidentally, needs 125-digit arithmetic to represent the coefficients accurately). The routine realroot, which comes packaged with Maple, provides exact dyadic rational intervals of arbitrary tightness containing each real root of the input univariate polynomial. For this family of examples, however, the extremely large size of the polynomial coefficients for higher-order Hilbert matrices soon defeats Maple by exhausting our patience. Arithmetic with large rational numbers not only takes storage space, it takes time. It is better not to introduce the ill-conditioning in the first place and instead work directly with the eigenvalue problem. Another example is the evaluation of functions defined by expressions automatically produced by Maple. For example, consider the expressions for the roots of a quadratic generated by the quadratic formula: solve(x^2 + 2*b*x + c, x); 2 1/2 2 1/2 - b + (b - c), - b - (b - c) r := unapply( [1],b,c); 2 1/2 r := (b,c) - - b + (b - c) In this example, the evaluation of r(b, c) is well-known (to numerical analysts) to be numerically unstable for b/c 1: but Maple does not know that, and indeed neither would an inexperienced user. More realistically, the algebraic solutions to cubics and quartics are also known to be unstable for certain values of the coefficients, but this is even less widely understood. Indeed, the expressions Maple can generate as the solution to more complicated problems can be much too large for human analysis and can also be of very dubious numerical stability. Some numerical analysts of our acquaintance refer to such output as wallpaper expressions because wallpaper An Experimental Interval Arithmetic Package in Maple 123 is about all they re good for 1. It would be useful if Maple could warn the user when the result of evaluating the expression is unreliable. Fixed-precision interval arithmetic does not do this instead, it provides the complementary tool of telling the user that when the interval bounds are tight, the computation was reliable. Of course, sometimes the bounds are not tight, which may be due to an unstable calculation or perhaps due to pessimism on the part of interval arithmetic, which is caused by insufficient use of the correlation of different parts of the expression. This problem might be curable in two ways: add more intelligence to the interval expression evaluation, as is done in [5], or we could instead apply more brute force in the form of increasing the number of digits carried in the calculation from the beginning. Both approaches seem very feasible in a high-level computational environment such as Maple. Thus the construction of an arbitrary-precision interval arithmetic package in Maple seems of interest. Other reasons for the construction of this package include possible applications to honest plotting [7], to code development (to provide numerical analyses of proposed code for e.g. sequence acceleration or for evaluation of the Jacobian elliptic functions), and to education we will see an example later, of numerical integration, that can profitably be used in either a first-year calculus class or in a numerical analysis class. 2 Summary of the proposed BIAS standard George Corliss has a draft proposal [6] for a standard common core interval arithmetic subroutine library, which will help enable users to write portable and efficient programs using interval arithmetic. We implemented our package using this proposed standard for two reasons: one, we were novices in the field of interval arithmetic and were grateful for such a simple way to take advantage of the many years experience of others, and, two, by adhering to a standard we could expect that others would be able to quickly use the package. At this writing the only users of this package are a small seminar group run by George Corliss at Marquette University. From the comments of that group, who implemented such routines as the solution of 1 This expression was originally used by Morven Gentleman, to our knowledge. It graphically and memorably describes the very real problem of expression swell, which is to symbolic computation what exponential amplification of roundoff error is to floating-point computation. 124 A. E. Connell, R. M. Corless linear systems by Gaussian elimination and by the preconditioned interval Gauss-Seidel algorithm of [8] using the package described here, it is clear that the extent of our adhering to the proposed standard made their work easier. The BIAS specifications are presented in [6] in a language independent form along with Fortran, Ada, and C bindings. The language-independent specifications address these areas: Declare a variable to be of interval type Arithmetic Operations Elementary functions Utility functions Membership functions Commonly used constants Error handling Rudimentary I/O We have implemented the type declarations, the arithmetic operations, the elementary functions, some of the utility functions and membership functions, and some of the rudimentary I/O. We have not strictly adhered to the standard, using our own names for some of the functions (strict adherence would be better, and in the next version of the package we will attempt to ensure true compliance). Arithmetic operations (required) The binary operations implemented are &+, &-, &*, &/, and &inv which all take interval or numeric inputs and return intervals on output. The interval [, ] may be returned by &inv or &/ if a division by zero occurs. Elementary functions (optional) The unary elementary functions implemented are &sqr, &sqrt, &ln, &exp, &** = &^, &intpower, &sin, &cos, &tan, &arcsin, &arccos, &arctan, &sinh, &cosh, and &tanh. These functions all compute inclusions of the An Experimental Interval Arithmetic Package in Maple 125 range of these functions for interval arguments, including degenerate interval arguments. Utility functions (required) The functions construct, ru, and rd, to construct an interval (with optional outward rounding), round up by one ulp, and round down by one ulp, are basic to the package. (The real name of, for example, ru is Interval_round_up but the short alias is used throughout the package for convenience). Utility functions (optional) The functions &midpoint, &width, &intersect, &union, and &is_in are also present. George Corliss contributed another utility in this class, the routine is_disjoint. All of these routines perform the natural functions indicated by their names. Remark The names of the arithmetic operations had to be different from that of the proposed standard since Maple s built-in operations +,, etc., cannot be overloaded and indeed suffer from some very undesirable automatic simplifications that cannot be turned off (for example, [0, 1] [0, 1] is automatically simplified to 0 before any user routine would have a chance to even look at it). So we were forced to use Maple s so-called inert operators &+, &-, &*, etc. These have two drawbacks: one is that they are slightly ugly (and take more than twice as long to type because & is not often used!) and, more seriously, Maple has the precedence rules for these operators hard-wired in, incorrectly. This bug is a historical one the Maple group implemented inert operators, then realized that &* could be used for matrix (non-commutative) multiplication (the ordinary * operator has hardwired commutativity built-in), and so the precedence for the inert operator &* was lowered from that of all the other inert operators to that of ordinary multiplication. This leaves us with a situation where the precedence of these operators is exactly the opposite of what you would expect: a &+ b &* c is parsed as (a + b) c. The only cure for this is to explicitly use parentheses to force correct evaluation. For convenience, we implemented a routine to 126 A. E. Connell, R. M. Corless convert from the use of ordinary Maple operators to the use of the interval operators here, but it would be more convenient still not to have to use it. A request that this precedence bug be fixed has been forwarded to Maple. 3 Sample code and usage We exhibit the Maple code for the exponential function as an example. # PROCEDURE &EXP # expinfinity is only called from &exp. # It deals with FAIL and +/- infinity. # Like most of the other subroutines &exp # takes floating point intervals # or numerics (which are converted into intervals). expinfinity:=proc(x); if x=fail then FAIL elif x=infinity then infinity elif x=-infinity then 0 else evalf(exp(x)) fi; end: Interval_exp:=proc(x); if type(x,interval) then if x=[] then [] elif x[1]=fail or x[2]=fail then [FAIL,FAIL] else [Interval_Round_Down(expinfinity(x[1])), Interval_Round_Up(expinfinity(x[2]))] fi; elif type(x,num_or_fail) then Interval_exp(construct(x)) else # Want to return unevaluated here. Interval_exp(x) fi; end: alias( &exp =Interval_exp): This routine uses the built-in Maple exponential function to provide the basic results and uses monotonicity of the exponential function to allow exponentials of intervals. We also allow arithmetic with the symbols FAIL and infinity which are the Maple equivalents of NaN (Not a Number); though of course they indicate also that the result is not a symbolic answer either. If &exp is called with a symbolic argument, it is simply returned unevaluated, that is as a sequence of symbols which can be processed and perhaps evaluated later. For example, if x has no interval or numerical value, then the following sequence results: &exp(x); &exp(x) An Experimental Interval Arithmetic Package in Maple 127 We have written a utility routine called convert/interval and another called inapply which together convert symbolic expressions to a form using the interval operators. The first utility recursively walks an expression, replacing the ordinary Maple operators with the appropriate interval ones. The second utility converts an expression into an operator (i.e. procedure) which evaluates its arguments using the interval operators. We give some examples of the usage of the interval arithmetic package in the following sample Maple session. load( intpak.mpl ); f := sin(x)/(1+ cos(x))^2; sin(x) f := (1 + cos(x)) f_interval := inapply(f,x); f_interval := x - (&sin x) &* inv((1 &+ (&cos x)) &intpower 2) Digits := 20; Digits := 20 f_interval( [0,0.5] ); [0, ] # plotting shows the function is monotonic # and that above bound is tight. asympt(ei(x),x); # The exponential integral function. / \ 1/x O(----) exp(x) \ x x x x x / # The following trick for removing the Landau O-symbol # from a Maple expression is # regrettably still occasionally necessary. subs(o=0, ): convert( ,polynom); / \ 1/x exp(x) \ x x x x / p := inapply( ,x); p := x - (inv(x &intpower 1) &+ (inv(x &intpower 2) &+ ((2 &* inv(x &intpower 3)) &+ ((6 &* inv(x &intpower 4)) &+ (24 &* inv(x &intpower 5)))))) &* (&exp x) 128 A. E. Connell, R. M. Corless p([100.,200.]); [ *10, *10 ] 4 Numerical integration: a didactic exercise In the beautifully written article [9], professor Kahan exhibits several interesting integrals. He uses them to make several points, among which are that numerical integration is provably impossible, numerical integration is better than symbolic integration, and that the integration algorithm incorporated into the HP34C calculator (now inherited by the HP48 family) is a good one. Following a tradition exemplified by much of Kahan s own work, we will use one of his own examples to support exactly the opposite points. We first examine his proof that numerical integration is impossible. Of course, he is correct, if point algorithms are used. His proof strongly uses the fact that point algorithms may not look at the program that evaluates the integrand: in a certain sense this disallows interval arithmetic, which in effect analyzes the integrand by replacing point arithmetic with interval arithmetic. But if we are permitted to cut the ground from under this proof by modifying the interpreter or compiler so the program for the integrand uses interval arithmetic instead, then the conclusion becomes more doubtful. Certainly the rest of Kahan s proof no longer goes through, since it essentially consists of spying on the numerical integrator by feeding a zero integrand to it and reporting back to the malicious user as to where the integrand was probed; then, one constructs a second function, zero precisely at those points but positive everywhere else, with a nonzero integral. This second, maliciously designed, function cannot be accurately evaluated with that quadrature method. Of course, this does not apply to interval techniques: one cannot design a malicious function in that way as no intermediate information is overlooked. However, it may be that numerical integration is still impossible, due to some reason that we ourselves have not noticed. In particular, one might be able to defeat any interval techniques by using integrands with integrable singularities. Second, Kahan uses the example An Experimental Interval Arithmetic Package in Maple 129 I(n, x) = x 0 dt 1 + t n with n = 64 to show that numerical integration is superior to symbolic integration. We agree with Kahan that this example is atypically modest out of consideration for the typesetter, because the symbolic answer has only 32 terms, and so the point we make below on the utility of symbolic techniques is not really general. However, we believe that point to be interesting and useful. Kahan s numerical algorithm, implemented on the HP34C, makes short work of this problem for n = 64 and x = 1 and gives a much more satisfactory answer than the 32-term symbolic answer in terms of arctans and logs. But what about larger n? The first such n we tried on the HP48SX was n = The calculator returns the answer I(1024, 1) = 1 in under a second, and claims this answer is correct to twelve decimal places. Alas, that can t be right. We may show that I(1024, 1) by computing the first three nonzero terms in the Taylor series for 1/(1 + t n ) = 1 t n + t 2n and integrating term by term (we will get tighter bounds later). So we may suspect that the algorithm of the HP34C (and hence the HP48SX) has fallen afoul of the impossibility proof mentioned earlier, and indeed this is the case. When we spy on the integrator, we find that it never evaluates the
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