Real number arithmetic in a general purpose language?_问答_开发者

As (hopefully) most of you know, floating point 开发者_开发问答arithmetic is different from real number arithmetic. It's for starters imprecise. Many numbers, especially decimals (0.1, 0.3) cannot be represented, leading to problems like this. A more thorough list can be found here.

Are there any general purpose languages that have built-in support for something closer to real number arithmetic? If not, what are good libraries that support this?

EDIT: Arbitrary precision decimal datatypes are not what I am looking for. I want to be able to represent numbers like 1/3, sqrt(3), or 1 + 2i as well.

Though I hate to say it, Fortran. It has extensive support for arbitrary-precision arithmetic and tons of support for big-number calculations. It's ancient and gross, but it gets the job done.

All the numbers used in your examples are algebraic numbers, and can be represented finitely as roots of polynomials with integer coefficients.

The same cannot be said of real numbers in general, which is easily seen when one considers that the reals are uncountable, but the set of computer programs is countable. Therefore most reals will not have a finite representation in code.

What you are looking for is symbolic calculation (MATLAB and other tools used in math and engineering are good at it).

If you want a general purposed language, I think expression tree in C# is good point to start with. In the essence, the ability to store the expression (instead of evaluate the expression into real values) is the key to be able to perform symbolic calculation. Note that expression tree does not provide symbolic calculation, it just provides the data structure that supports symbolic calculation.

This question is interesting, but raises some issues. First, you will never be able to represent all the real numbers using a (even theoretically infinite) computer, for cardinality reasons.

What you are looking for is a "symbolic numbers" datatype. You can imagine some sort of expression tree, with predefined constants, arithmetical operations, and perhaps algebraic (roots of polynomials) and transcendantal (exp, sin, cos, log, etc) functions.

Now the fun part of the story: you cannot find an algorithm which tells whether two such trees are representing the same number (or equivalently, which test whether such a tree is zero). I won't state anything precise, but as a hint, this is similar to the Halting Problem (for computer scientists) or the Gödel Incompleteness Theorem (for mathematicians).

This renders such a class pretty useless.

For some subfields of the reals, you have canonical forms, like a/b for the rationals, or finite algebraic extensions of the rationals (a/b + ic/d for complex rationals, a/b + sqrt(2) * a/b for Q[sqrt(2)], etc). These can be used to represent some particular sets of algebraic numbers.

In practice, this is the most complicated thing you will need. If you have a particular necessity, like ranges of floating point numbers (to prove some result is whithin a specified interval, this is probably the closest you can get to real numbers), or arbitrary precision numbers, you have freely available classes everywhere. Google boost::range for the former, and gmp for the latter.

There are several languages with support for rational and complex numbers. Scheme, for instance, has support built in for arbitrarily precise rational numbers, and complex numbers with either rational, floating point, or integral coefficients:

> (+ 1/2 1/3)
5/6
> (* 3 1+1/2i)
3+3/2i
> (+ 1/2 .5)
1.0

If you want to go beyond rational numbers or complex numbers with rational coefficients, to algebraic numbers such as sqrt(2) or closed-form numbers like e, you will probably have to look beyond general purpose programming languages, and use a special purpose mathematical language like Mathematica or Maxima.

To cover the real numbers with any flair you'll need a symbolic package.

Boost, the C++ project, has a Rational library, but that's only part of the story.

You have irrational numbers in all sorts of forms (pi, base of the natural logarithm, square and cube roots, the Champernowne constant, to name only a few). The only way I know of to handle arithmetic operations is a symbolic package with smarts as to the relationship amongst all of these numbers. Assuming you could express e^pi, how would you add one to it? Or take the square root of it?

Mathematica might handle these cases.

Java: java.math.BigDecimal

C#: decimal

A lot of languages have support for that: Java has BigDecimal, Perl has Math::BigFloat and Math::BigRat, Haskell has Integer and a lot of libraries and languages are listed in the wikipedia.

Ada natively supports fixed-point math as well as floating-point. Fixed-point can be much more exact than floating-point, as long as the number's exponents remain in range.

If you need floating-points, but more precision than IEEE gives, there are bignum packages around for just about every language.

I think that's about the best you can do. Neither scheme can exactly represent repeating decimals (like 1/3). It would probably be possible to come up with a scheme that does, but I know of no language that supports such a thing with a built-in type. Even that won't help you with irrational numbers (like pi and e). I believe there's even a theorem that says there will always be unrepresentable numbers, no matter what scheme you come up with.

EDIT: Arbitrary precision decimal datatypes are not what I am looking for. I want to be able to represent numbers like 1/3, sqrt(3), or 1 + 2i as well.

Ruby has a Rational class, so 1/3 can be expressed exactly as Rational(1,3). It also has a Complex class.

Scheme defines rationals, bignums, floating point and complex numbers. An implementation is not required to support them all, but if they are present, you can mix them and they will to "the right thing".

While its not "built-in", I think C++ (maybe C#) is your best bet. There are classes out there that have been written for this purpose.

http://www.oonumerics.org/oon/