NUMERICAL METHODS
Solve the following tridiagonal matrix equation, in which only non-zero elements are
shown:
12
−12 1
2 − 12
311
342
− 22
x 1
x 2
x 3
x 4
x 5
x 6
=
4
3
− 3
10
7
− 2
. (27.33)
The solution is set out in table 27.7, in which the arrows indicate the general flow of the
calculation. First, the columns ofai,bi,ciandyiare filled in from the original equation
(27.33) and then the recurrence relations (27.32) are used to fill in the successive rows
starting at the top; on each row we work from left to right as far as and including theφi
column. Finally, the bottom entry in the thexicolumn is set equal to the bottom entry in
the completedφicolumn and the rest of thexicolumn is completed by using (27.31) and
working up from the bottom. Thus the solution isx 1 =2;x 2 =1;x 3 =3;x 4 =−1;x 5 =
ai bi ci aiθi− 1 +bi θi yi aiφi− 1 φi xi
↓ 012 → 1 − 2 40 4 2 ↑
↓− 121 → 4 − 1 / 4 3 − 4 7/4 1 ↑
↓ 2 − 12 →− 3 / 2 4/3 − 3 7/2 13/3 3 ↑
↓ 311 → 5 − 1 / 5 10 13 − 3 / 5 − 1 ↑
↓ 342 → 17/5 − 10 / 17 7 − 9 / 5 44/17 2 ↑
↓− 220 → 54/17 0 − 2 − 88 / 17 1 → 1 ↑Table 27.7 The solution of tridiagonal matrix equation (27.33). The arrows
indicate the general flow of the calculation, as described in the text.2;x 6 =1.
27.4 Numerical integrationAs noted at the start of this chapter, with modern computers and computer
packages – some of which will present solutions in algebraic form, where that
is possible – the inability to find a closed-form expression for an integral no
longer presents a problem. But, just as for the solution of algebraic equations, it
is extremely important that scientists and engineers should have some idea of the
procedures on which such packages are based. In this section we discuss some of
the more elementary methods used to evaluate integrals numerically and at the
same time indicate the basis of more sophisticated procedures.
The standard integral evaluation has the formI=∫baf(x)dx, (27.34)where the integrandf(x) may be given in analytic or tabulated form, but for the
cases under consideration no closed-form expression forIcan be obtained. All