Why do algebraic calculations?

 

• in many research fields, one often needs to process very large algebraic expressions (of perhaps hundreds of thousands or more terms, e.g., collision integrals, computations from the general theory of relativity, celestial mechanics calculations) or perform long analytical operations

• compared to a human, a computer does not make errors (assuming the programming is correct! This unfortunately is not always the case ...)

• there exist algorithms which cannot easily be performed by a human with pencil and paper (e.g., factorization , quantifier elimination )

• many others:
  • algebraic solutions are usually more compact than a set of numerical solutions; algebraic solution gives more direct information about the relationship between the variables than figures
  • algebraic solutions are always exact, numerical solutions will normally be approximations; this can arise from rounding and truncation errors, further errors can creep in when the user interpolates data given in tabular form
  • computer algebra can save both time and effort in solving a wide range of problems; much larger problems can be investigated than by using traditional methods
  • computer algebra reduces the need for tables of functions, series and integrals; symbolic computation using computers have highlited many errors in such materials
  • traditional teaching of applied mathematics has to involve much time in teaching techniques of solution; computer algebra systems tend to produce solutions quickly and without errors, so they enable more time to be devoted to studying the properties of the solution
  • using of computer algebra allows also very effective construction of numerical algorithms and their semiautomatic programming by code generation, the effectiveness of the work and reliability of the results can be strongly increased