Many iterative numerical algorithms can be considered as dynamical systems. Since
control theory deals with the study of dynamical systems, it has the potential to provide
new insight, analysis tools and even new algorithms to the field of numerical analysis.
Although the early knowledge of this observation, the research in this application area
is scarce. In this thesis we use control theoretic ideas to study and design numerical
algorithms. We will focus on developing analysis tools that could help speedup existing
algorithms and design new algorithms that are fast or have a low computational time.
Search for faster algorithms is an active research area in the fields of numerical analysis
and computing, which is driven by the requirements of many applications. One of the
ways to speedup an algorithm is to design a custom computational hardware for it. For
several algorithms it has been shown that a customized computational hardware can give
significant speedups over general processors. One of the parameters in custom hardware
design is the choice of number representation. While in general purpose computational
hardware this is fixed to a 64 bit floating point with a precision of 53 bits, in custom
hardware design any number of bits can be chosen. A circuit in lower precision utilizes less
hardware resources on a chip, allowing more parallel computational blocks to be built on
the chip that could speedup calculations in an algorithm. However, reducing the precision
increases the errors in computations. There is a trade-off between speed and accuracy of
computations and one would like to know what precision is just enough for a given algorithm
to guarantee solutions of a desired accuracy. A bound on the error in the solution obtained
by an algorithm as a function of precision would be useful in the decision of precision.
Such a bound is called forward error bound and the process of finding it is called forward
error analysis. In this thesis we use control theoretic ideas to present forward error analysis
schemes for algorithms in finite precision. While forward error analysis is a standard
technique in numerical analysis, the objective is usually to just show that such a bound
exists. Whereas, for our application we would like to find a tight bound. In this thesis
we present three control theoretic forward analysis schemes. The first schemes is based on
the control tools of l8-stability and input-to-state stability. The other two schemes are
based on results from the study of quantization effects in control systems. The proposed
schemes are applied on the successive iteration method and the classical iterative method
for solving a system of linear equation. The obtained bounds are shown to be tighter than
the existing bounds in the numerical analysis literature.
While most of the efforts to speedup an algorithm are to customize the hardware for it,
we should also look for new algorithms that are more suitable for hardware, i.e. the design
of algorithms should be aware of what type of computations can be easily accelerated in
hardware. In this thesis we use control theoretic ideas to present a new algorithm for
solving a positive definite system of linear equations. The proposed algorithm in highly
parallelizable and can effectively take advantage of parallel computational hardware. It is
shown that the proposed algorithm has lower time complexity than other state of the art
algorithms for the case of parallel computing.