Mathematical Methods for Physics and Engineering : A Comprehensive Guide

27.3 SIMULTANEOUS LINEAR EQUATIONS

contain non-zero entries. Such matrices are known as tridiagonal matrices. They

may also be used in numerical approximations to the solutions of certain types

of differential equation.

A typical matrix equation involving a tridiagonal matrix is as follows:

0

b 1 c (^10)
b 2 c 2
b 3 c 3
bN
a 2
a 3
aN–1
aN

...

x 1

x 2

x 3

...

xN–1

xN

y 1

y 2

y 3

...

yN–1

yN

=

bN–1cN–1

......

(27.30)

So as to keep the entries in the matrix as free from subscripts as possible, we

have useda,bandcto indicate subdiagonal, leading diagonal and superdiagonal

elements, respectively. As a consequence, we have had to change the notation for

the column matrix on the RHS frombto (say)y.

In such an equation the first and last rows involvex 1 andxN, respectively, and

so the solution could be found by lettingx 1 be unknown and then solving in turn

each row of the equation in terms ofx 1 , and finally determiningx 1 by requiring

the next-to-last line to generate forxNan equation compatible with that given

by the last line. However, if the matrix is large this becomes a very cumbersome

operation, and a simpler method is to assume a form of solution

xi− 1 =θi− 1 xi+φi− 1. (27.31)

Since theith line of the matrix equation is

aixi− 1 +bixi+cixi+1=yi,

we must have, by substituting forxi− 1 ,that

(aiθi− 1 +bi)xi+cixi+1=yi−aiφi− 1.

This is also in the form of (27.31), but withireplaced byi+1. Thus the recurrence

formulae forθiandφiare

θi=

−ci

aiθi− 1 +bi

,φi=

yi−aiφi− 1

aiθi− 1 +bi

, (27.32)

provided the denominator does not vanish for anyi. From the first of the matrix

equations it follows thatθ 1 =−c 1 /b 1 andφ 1 =y 1 /b 1. The equations may now

be solved for thexiin two stages without carrying through an unknown quantity.

First, all theθiandφiare generated using (27.32) and the values ofθ 1 andφ 1 ,

then, as a second stage, (27.31) is used to evaluate thexi, starting withxN(=φN)

and working backwards.