Dynamic Multiple-Precision Arithmetic

Besides the classes real and interval, the dynamic classes long real (l_real) and long interval (l_interval) as well as the corresponding dynamic vectors and matrices are implemented including all arithmetic and relational operators and multiple-precision standard functions. The computing precision may be controlled by the user at run time. By replacing the real and interval declarations by l_real and l_interval, the user's application program turns into a multiple-precision program. This concept provides the user with a powerful and easy-to-use tool for error analysis. Furthermore, it is possible to write programs delivering numerical results with a user-specified accuracy by internally modifying the computing precision at run time in response to the error bounds for intermediate results within the algorithm.

All predefined operators for real and interval types are also available for l_real and l_interval. Additionally, all possible operator combinations between single and multiple-precision types are included. The following example shows a single-precision program and its multiple-precision version:

main()
{
  interval a, b;            /* Standard intervals           */
  a = 1.0;                  /* a   = [1.0,1.0]              */
  b = 3.0;                  /* b   = [3.0,3.0]              */
  cout << "a/b = " << a/b;  /* a/b = [0.333333333333, 
                                      0.333333333334]       */
}

main() 
{
  l_interval a, b;          /* Multiple-precision intervals */
  a        = 1.0;
  b        = 3.0;
  stagprec = 2;             /* global integer variable      */
  cout << "a/b = " << a/b; 
                     /* a/b = [0.333333333333333333333333,
                               0.3333333333333333333333334] */
}

At run time, the predefined global integer variable stagprec (staggered precision) controls the computing precision of the multiple-precision arithmetic in steps of a single real (64 bit words). The precision of a multiple-precision number is defined as the number of real s used to store the long number's value. An object of type l_real or l_interval may change its precision at run time. Components of a vector or a matrix may be of different precision. All multiple-precision arithmetic routines and standard functions compute a numerical result possessing a precision specified by the actual value of stagprec. Allocation, resize, and subarray access of multiple-precision vectors and matrices are similar to the corresponding single-precision data types.