We discuss some main points of computer-assisted proofs based
on reliable numerical computations. Such so-called self-validating numerical
methods in combination with exact symbolic manipulations result in very
powerful mathematical software tools. These tools allow proving mathematical
statements (existence of a fixed point, of a solution of an ODE, of
a zero of a continuous function, of a global minimum within a given range,
etc.) using a digital computer. To validate the assertions of the underlying
theorems...
The paper has been presented at the 12th International Conference on Applications of
Computer Algebra, Varna, Bulgaria, June, 2006
The computation of the exact solution set of an interval linear
system is a nontrivial task [2, 13]. Even in two and three dimensions a lot of
work has to be done. We demonstrate two different realizations. The first
approach (see [16]) is based on Java, Java3D, and the BigRational package
[21]. An applet allows modifications of the matrix coefficients and/or...
The paper has been presented at the 12th International Conference on Applications of
Computer Algebra, Varna, Bulgaria, June, 2006
The Maple Power Tool intpakX [24] de nes Maple types for
real intervals and complex disc intervals. On the level of basic operations,
intpakX includes the four basic arithmetic operators, including extended
interval division as an extra function. Furthermore, there are power, square,
square root, logarithm and exponential functions, a set of standard functions,
union,...
This work reports on a new software for solving linear systems
involving affine-linear dependencies between complex-valued interval parameters.
We discuss the implementation of a parametric residual iteration
for linear interval systems by advanced communication between the system
Mathematica and the library C-XSC supporting rigorous complex interval
arithmetic. An example of AC electrical circuit illustrates the use of the
presented software.
* This work was partly supported by the...
The C++ class library C-XSC for scientific computing has been
extended with the possibility to compute scalar products with selectable accuracy in version 2.3.0. In previous versions, scalar products have always
been computed exactly with the help of the so-called long accumulator. Additionally, optimized floating point computation of matrix and vector operations using BLAS-routines are added in C-XSC version 2.4.0. In this article
the algorithms used and their implementations, as well as some potential
pitfalls...
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