如何测试特殊多项式的正确实现？

Question

I need to implement the calculation of some special polynomials in Java (the language is not really important). These are calculated as a weighted sum of a number of base polynomials with fixed coefficients.

Each base polynomial has 2 to 10 coefficients and there are typically 10 base polynomials considered, giving a total of, say 20-50 coefficients.

Basically the calculation is no big deal but I am worried about typos. I only have a printed document as a template. So i would like to implement unit tests for the calculations. The issue is: How do I get reliable testing data. I do have another software that is supposed to calculate these functions but the process is complicated and also error prone - I would have to scale the input values, go through a number of menu selections in the software to produce the output and then paste it to my testing code.

I guess that there is no way around using the external software to generate some testing data, but maybe you have some recommendations for making this type of testing procedure safer or minimize the required number of test cases .

I am also worried about providing suitable input values: Depending on the value of the independent variable, certain terms will only have a tiny contribution to the output, while for other values they might dominate.

The types of errors I expect (and need to avoid) are:

• Typos in coefficients
• Coefficients applied to wrong power (ie a_7*x^6 instead of a_7*x^7 - just for demonstration, I am not calculating this way but am using Horner's scheme)
• Off-by one errors (ie missing zero order or highest order term)

Answer 1

Since you have a polynomial of degree 10, testing at 11 distinct points should give certainty.

However, already a test at one well-randomized point, x=1.23004 to give an idea (away from small fractions like 2/3, 4/5), will with high probability show a difference if there is an error, because it is unlikely that the difference between the wrong and the true polynomial has a root at exactly this place.