20.7 Gauss elimination for the solution of linear equations 581
Gaussian quadrature formulas are used in applications in which very high accuracy is
required; for example, in quantum chemistry.
20.6 Methods in linear algebra
The problems of linear algebra are those associated with systems of linear equations,
as described in Chapters 17, 18, and 19:
(a) Solution of sets of simultaneous linear equations, or of the equivalent matrix
equation Ax 1 = 1 bfor the unknown vector x, where A is a given matrix of
coefficients and bis a known vector.
(b) Calculation of the determinant of a square matrix A.
(c) Calculation of the inverse matrixA
− 1of a square matrix A.
(d) Solution of the eigenvalue problemAx 1 = 1 λxfor a square matrix A.
There exists a vast and growing body of numerical methods for these problems driven
by, and driving, developments of computer technology. Many of these methods are
based on, or are related to, the elimination methodsdiscussed in the following
sections for problems of types (a), (b), and (c).
20.7 Gauss elimination for the solution of linear equations
The Gauss elimination method is the formalization of the elementary method of
solving systems of simultaneous linear equations. We consider the three equations
in three unknowns (see also Example 2.34).
8(1) x 1 + 12 y 1 + 13 z 1 = 126
(2) 2x 1 + 13 y 1 + 1 z 1 = 134 (20.42)
(3) 3x 1 + 12 y 1 + 1 z 1 = 139
The method proceeds in a sequence of steps.
Step 1. Elimination ofx
Equation (1) is called the pivot equationand the term in xis called the pivotof the
step. The elimination of xfrom all subsequent equations is achieved by subtraction
of appropriate multiples of (1). Thus, subtraction of 2 1 × 1 (1) from (2) and of 3 1 × 1 (1)
from (3) gives the new set of equations
(1) x 1 + 12 y 1 + 13 z 1 = 1 26
(2′) −y− 15 z 1 = 1 − 18 (20.43)
(3′) − 4 y− 18 z 1 = 1 − 39
8These particular equations arise in a problem on measures of grain in Chapter 8 of the Jiuzhang suanshu, and
they are solved there by a method essentially identical to Gauss elimination. Gauss described his method in the
Theoria motus corporum celestium(Theory of motion of the heavenly bodies) of 1809.