Validated numerics with Python: the ValidiPy package

Abstract—We introduce the ValidiPy package for validated numerics in Python. This suite of tools, which includes interval arithmetic and automatic differenti-ation, enables rigorous and guaranteed results using floating-point arithmetic. We apply the ValidiPy package to two classic problems in dynamical systems, calculating periodic points of the logistic map, and simulating the dynamics of a chaotic billiard model. Index Terms—validated numerics, Newton method, floating point, interval arith-metic Floating-point arithmetic Scientific computation usually requires the manipulation of real numbers. The standard method to represent real numbers internally in a computer is floating-point arithmetic, in which a real number a is represented as a=±2e×m. The usual double-precision (64-bit) representation is that of the IEEE 754 standard [IEEE754]: one bit is used for the sign, 11 bits for the exponent e, which ranges from −1022 to +1023, and the remaining 52 bits are used for the "mantissa" m, a binary string of 53 bits, starting with a 1 which is not explicitly stored. However, most real numbers are not explicitly representable in this form, for example 0.1, which in binary has the infinite periodic expansion 0.0 0011 0011 0011 0011..., in which the pattern 0011 repeats forever. Representing this in a computer with a finite number of digits, via truncation or rounding, gives a number that differs slightly from the true 0.1, and leads to the following kinds of problems. Summing 0.1 many times-- a common operation in, for example, a time-stepping code, gives the following unexpected behaviour. a = 0.1 total = 0.0 print("%20s %25s " % ("total", "error")) for i in xrange(1000)