Systems of Linear Algebraic Equations - Numerical Methods in Engineering with MATLAB - page 28

Systems of Linear Algebraic Equations

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Solve the simultaneousequations Ax = b

2.1 Introduction

In thischapterwe look at the solution of n linear, algebraicequations in n unknowns.
It is by far the longest and arguably the most importanttopic in the topic. There
is agoodreason for this—it is almost impossible to carry out numerical analysis
of any sort withoutencountering simultaneousequations. Moreover,equation sets
arising fromphysical problems are oftenvery large, consuming a lot of computa-
tional resources. It usuallypossible to reduce the storage requirements and the run
time by exploiting special properties of the coefficient matrix,such assparseness
(most elements of a sparse matrix arezero). Hence there are many algorithms ded-
icated to the solution of large sets of equations, each one being tailored to a partic-
ular form of the coefficient matrix (symmetric, banded,sparse, etc.).Awell-known
collection of these routines is LAPACK-Linear Algebra PACKage, originallywrittenin
Fortran77 1 .
Wecannot possiblydiscuss all the special algorithms in the limited space avail-
able. The best wecan do istopresent the basic methods of solution,supplemented
by a fewuseful algorithmsfor banded and sparse coefficient matrices.

Notation

A system of algebraicequationshas the form

1

LAPACK is the successor of LINPACK, a 1970s and 80s collection of Fortran subroutines.

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