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  The main sources of error

The major Sources of Error:

Truncation Error:

Truncation error is defined as the error cause straight by an approximation method. For illustration all numerical integration methods are approximations and thus there is error even if the calculations are performed exactly. Numerical differentiation as well has a truncation error, as will the differential equations methods we will study in Part IV which are base on numerical differentiation formulas.

There are two methods to minimize truncation error (1) utilize a higher order method and (2) use a finer grid so that points are closer together. Except the grid is very small truncation errors are usually much larger than round-off errors. The clear trade-off is that a smaller grid requires more calculations which in turn produces more round-off errors as well as requires more running time.

Round-off Error:

Round-off error forever occurs when a finite number of digits are recorded after an operation. luckily this error is extremely small. The standard measure of how little is called machine epsilon. It is defined as the smallest number that is able to be added to 1 to produce another number on the machine that is if a smaller number is added the result will be rounded down to 1. In IEEE standard double accuracy (used by Mat lab and most serious software) machine epsilon is 2−52 or about 2.2 × 10−16. A different however equivalent way of thinking about this is that the machine records 52 floating binary digits or about 15 floating decimal digits. Therefore there are never more than 15 significant digits in any calculation. This of course is more than sufficient for any application.

Nevertheless there are ways in which this very small error can cause problems.

To see an unexpected incidence of round-off try the following commands:

> (2^52+1) - 2^52
> (2^53+1) - 2^53

Loss of Precision (as well called Loss of Significance):

One way in which little round-off errors are able to become more significant is when significant digits cancel.

For illustration if you were to calculate:

1234567(1 − .9999987654321)

Then the result must be:

1.52415678859930.

However if you input

> 1234567 * ( 1 - .9999987654321)

you will get:

> 1.52415678862573

In the correct calculation there are 15 significant digits however the Mat lab calculation has only 11 significant digits.

Even though this seems like a silly example this type of loss of precision can happen by accident if you aren’t careful. For illustration in f′(x) ≈ (f(x + h) −f(x))/h you will lose precision when h gets too small. Try:

>format long
>format compact
> f=inline(’x^2’,’x’)
>for i = 1:30
> h=10^(-i)
>df=(f(1+h)-f(1))/h
>relerr=(2-df)/2
>end

At first the relative error reduces since truncation error is reduced. Then loss of precision takes over as well as the relative error increases to 1.

Bad Conditioning:

We encountered awful conditioning in Part II when we talked about solving linear systems. Bad conditioning signifies that the problem is unstable in the sense that small input errors can produce large output errors. This is able to be a problem in a couple of ways. First the measurements utilized to get the inputs can’t be completely accurate. Second the computations beside the way have round-off errors. Errors in the calculations near the beginning especially can be magnified by a factor close to the condition number of the matrix. Therefore what was a very small problem with round-off can become a very big problem.

It turns out that matrix equations aren’t the only place where condition numbers occur. In any problem one is able to define the condition number as the maximum ratio of the relative errors in the output versus input that is condition # of a problem = max{Relative error of output/Relative error of inputs}. An simple example is solving a simple equation

f(x) = 0.

Presume that f′ is close to zero at the solution x*. Then a very little change in f (caused perhaps by an inaccurate measurement of some of the coefficients in f) be able to cause a large change in x*. It can be exposed that the condition number of this problem is 1/f′(x*).

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