In this thesis we study three
aspects of symbolic methods for numerical applications. One is the computing of
symbolic differentiation. Specifically, we describe SEMT, a software package
based on C++ templates, which allows the programmer to create symbolic
expressions, substitute variables in them, and compute their derivatives. The
unique aspect of SEMT is that these manipulations are all done at compile time.
In other words, SEMT effectively coerces the compiler to do symbolic
computation as part of the compilation process. Beyond the theoretical
interest, SEMT has application in an efficient, generic and easy to use
implementation of many numerical algorithms. In the second part of this thesis
we show that the problem of compiling C++ programs is Turing complete: for
every Turing machine $U$, there exists a C++ program $P$, such that a C++
compiler will halt compiling $P$ if and only if $U$ halts. Therefore, you
should not be searching for a different vendor if your C++ compiler gets into
an endless loop, or if it sets arbitrary limits on your program. The flaw is
with the language. On the positive side, we have that arbitrarily complex
computations can be carried out at compile time. The consequence is that a
skilled programmer can employ the compiler to do whatever partial evaluation is
required for optimizing the runtime. A detailed transition for any Turing
Machine into C++ compile time processing using templates is provided. The third
aspect is an approach for transforming systems of partial differential
equations in order to obtain new formulations which are more accessible to
numerical solution. An algorithm is developed for generating such
transformations automatically, using symbolic computations employing Gröbner
bases. The algorithm is implemented using freely available software.
