MTH3051 Introduction to Computational Mathematics - User Web ...

School of Mathematical Sciences
Monash University
no reason to trust the answers from a 4-digit computer. This is better seen in the third
table which lists the relative errors. Here you can see that the 2 and 4 digit computers
yield answers that are 100% in error – they produce junk!
The first big question is What causes this problem? When λ ≪ 1, the numbers 1 and
√
1 − 4λ are almost equal. Thus when we compute their difference we introduce a large
relative error.
The second big question is Can we cure this problem? In this case, yes, we have at least
two options, both of which entail a re-working of the mathematics prior to handing
control over to the computer. That is we search for different algorithms to compute x
in the case where λ ≪ 1.
Option 1. Here we do some simple algebra
x =
=
1 − (1 − 4λ)(1/2)
2
2λ
1 + (1 − 4λ) (1/2)
=
1 − (1 − 4λ)(1/2)
2
(
)
1 + (1 − 4λ) (1/2)
1 + (1 − 4λ) (1/2)
We see that when λ ≪ 1 we do not have any problem with cancellation between nearly
equal numbers (i.e no loss of precision).
Relative errors for Option 1
λ Exact 2-digits 4-digits 6-digits 8-digits
1.00e-01 1.1270e-01 2.3972e-02 1.4777e-05 2.9691e-06 4.7730e-08
1.00e-02 1.0102e-02 1.0102e-02 2.0307e-04 5.0924e-06 4.3889e-08
1.00e-03 1.0010e-03 1.0010e-03 2.0030e-06 2.0030e-06 5.0090e-09
1.00e-04 1.0001e-04 1.0001e-04 1.0001e-04 2.0003e-08 2.0003e-08
Option 2. This time we use a Taylor series. For λ ≪ 1 we can expnad √ 1 − 4λ as
1 − 2λ + O (λ 2 ). Thus we find, for λ ≪ 1,
x = λ + O ( λ 2)
Relative errors for Option 2
λ Exact 2-digits 4-digits 6-digits 8-digits
1.00e-01 1.1270e-01 1.1270e-01 1.1270e-01 1.1270e-01 1.1270e-01
1.00e-02 1.0102e-02 1.0102e-02 1.0102e-02 1.0102e-02 1.0102e-02
1.00e-03 1.0010e-03 1.0010e-03 1.0010e-03 1.0010e-03 1.0010e-03
1.00e-04 1.0001e-04 1.0001e-04 1.0001e-04 1.0001e-04 1.0001e-04
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