## Abstract

In considering the reliability of numerical programs, it is normal to \limit

our study to the semantics dealing with numerical precision" (Martel,

2005) [Mar05]. On the other hand, there is a great deal of work on the

reliability of programs that essentially ignores the numerics. The thesis

of this paper is that there is a class of problems that fall between these

two, which could be described as \does the low-level arithmetic imple-

ment the high-level mathematics". Many of these problems arise because

mathematics, particularly the mathematics of the complex numbers, is

more dicult than expected: for example the complex function log is

not continuous, writing down a program to compute an inverse function

is more complicated than just solving an equation, and many algebraic

simplication rules are not universally valid.

The good news is that these problems are theoretically capable of being

solved, and are practically close to being solved, but not yet solved, in

several real-world examples. However, there is still a long way to go

before implementations match the theoretical possibilities.

our study to the semantics dealing with numerical precision" (Martel,

2005) [Mar05]. On the other hand, there is a great deal of work on the

reliability of programs that essentially ignores the numerics. The thesis

of this paper is that there is a class of problems that fall between these

two, which could be described as \does the low-level arithmetic imple-

ment the high-level mathematics". Many of these problems arise because

mathematics, particularly the mathematics of the complex numbers, is

more dicult than expected: for example the complex function log is

not continuous, writing down a program to compute an inverse function

is more complicated than just solving an equation, and many algebraic

simplication rules are not universally valid.

The good news is that these problems are theoretically capable of being

solved, and are practically close to being solved, but not yet solved, in

several real-world examples. However, there is still a long way to go

before implementations match the theoretical possibilities.

Original language | English |
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Publication status | Published - 26 Sept 2012 |

Event | SYNASC 2012: 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing - Timisoara, Romania Duration: 25 Sept 2012 → 28 Sept 2012 |

### Conference

Conference | SYNASC 2012: 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing |
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Country/Territory | Romania |

City | Timisoara |

Period | 25/09/12 → 28/09/12 |