Numerical Solution of Systems of Equations

This chapter covers the numerical solution of linear and nonlinear systems of equations. Linear systems are discussed first, followed by more specialized methods to efficiently handle large linear systems. Ill-conditioning symptoms, as well as pertinent remedies are also introduced. The chapter ends with iterative solution of nonlinear systems of equations.

A linear system of n algebraic equations in n unknowns x1, x2, …, xn is in the form

$$\{\begin{array}{c}{a}_{11}{x}_{1}+{a}_{12}{x}_{2}+\cdots +{a}_{1n}{x}_{n}={b}_{1}\\ {a}_{21}{x}_{1}+{a}_{22}{x}_{2}+\cdots +{a}_{2n}{x}_{n}={b}_{2}\\ \cdots \\ {a}_{n1}{x}_{1}+{a}_{n2}{x}_{2}+\cdots +{a}_{nn}{x}_{n}={b}_{n}\end{array}\left(4.1\right)$$

where aij (i, j = 1, 2, …, n) and bk (k = 1, 2, …, n) are known constants, and aij’s are the coefficients. If every bk is zero, the system is homogeneous, ...