Problem in Understanding an Illustrative Example concerning the "Effect of Error in a Tabular Value".

Question

I am relatively new to the topic of "Numerical Methods and Analysis ". I was reading a book named, "Numerical Mathematical Analysis " by J Scarborough.

I was studying about the effect of an error in the tabular value. The book had an example stating that,

The law according to which an error is propagated in a difference table enables us to trace such an error to its source and correct it. As an illustration of the process of detecting and correcting an error in a tabulated function, let us consider the following table:

The book then went on as,

Here the third differences are quite irregular near the middle of the column, and the fourth differences are still more irregular. The irregu- larity begins in each column on the horizontal line corresponding to $x=0.40.$

Since the algebraic sum of the fourth differences is $1,$ the average value of the fourth differences is only about $0.1$ of a unit in the fifth decimal place. Hence the fourth differences found in this example are mostly accumulated errors.

Referring now, to Table $6$ (i.e the table below), we now have, $$-4\epsilon =-11,6\epsilon=17,\text{etc}$$ Hence, $\epsilon=3$ to the nearest unit. The true value of $y$ corresponding to $x=0.40$ is therefore $0.38945-0.00003=0.38942,$ since $(y_k+\epsilon)-\epsilon =y_k.$ The columns of differences can now be corrected, and it will be found that the third differences are practically constant.

I have a $4$ basic issues regarding the example illustrated.

• When they say, "Here the third differences are quite irregular near the middle of the column, and the fourth differences are still more irregular" , I don't get how exactly do they conclude it? I think they essentially, use the observation, that there is a sudden jump in the consecutive numerical values, in the middle of the third column. In other words, the numerical differences between two consecutive elements in the middle of the third column shows a sudden change or spike. If this is the case, then my question is, "How are they so sure, that the function value is not like that. I mean, it might very well happen that the function is defined in such a way, that the third differences have a non-uniform increase in numerical differences (between two consecutive elements)?"

• In the portion, where they say, "Since the algebraic sum of the fourth differences is $1,$ the average value of the fourth differences is only about $0.1$ of a unit in the fifth decimal place. Hence the fourth differences found in this example are mostly accumulated errors" , I dont understand what do they mean by, "Hence the fourth differences found in this example are mostly accumulated errors." Honestly, the term, "accumulated error" seems to make no sense to me. Does it have a definition?

Finally, how do they say,

Referring now, to Table $6$ (i.e the table below), we now have, $$-4\epsilon =-11,6\epsilon=17,\text{etc}$$

i.e to be more specific, how do they obtain the equations $$-4\epsilon =-11,6\epsilon=17,\text{etc}$$ from Table $6.$

• Lastly, even if I assume, that $\epsilon=3$ then, the original error magnitude is, $3\times 10^{-5}.$ Now, when they write,

The true value of $y$ corresponding to $x=0.40$ is therefore $0.38945-0.00003=0.38942,$ since $(y_k+\epsilon)-\epsilon =y_k.$

aren't they essentially assuming that the error resulted a value greater than the error corresponding to the correct value ? But it might have been the case, that the original value was, say $y_k$ and after the error it was $y_k-\epsilon$ and then accordingly we have to add $\epsilon$ to $y_k-\epsilon$ to obtain the correct value, $y_k.$

Any help regarding this will be greatly appreciated.

Answer 1

The original table of function values is presumably from a smooth function at equidistant steps at a small step size. Thus in the section under observation it can be assumed to be polynomial with rapidly falling coefficients. Applying iterated differences will eliminate these polynomial terms from the lowest on (reducing the degrees and slightly increasing the coefficients of the remaining terms). This will reduce the resulting columns towards zero.

On the other side, a perturbation in a single term propagates like a Pascal triangle, as demonstrated in the second table. At some point, the perturbation becomes clearly dominant over the iterated differences, so that the original error can be detected and reconstructed. The $\Delta_4$ column is the first where the divided differences are random and close to zero. The expected column in the Pascal triangle is $(1,-4,6,-4,1)$. The only nearly symmetric bump that has this structure is $(4,-11,17,-11,3)$. The factor 3 between these sequences is a first reasonable guess, one could apply regression or some weighted averaging to get a better value that will probably be slightly smaller.

Then follow the structure of the perturbation triangle back to its origin and correct the value at the correct decimal place.