Well known waveforms or signal equations for filter testing
My question is whether there is a set of well known waveforms with algorithms that can be programmatically generated?


Yes and no. There are many wave forms that can be created programatically. However these may or may not be useful for you test. 

Your test should focus on determining whether the algorithm meets the stated requirements. Hence the best way to build a test is to run down the list of specifications and requirements and build test that clearly establish whether the implementation meets the requirement or not . Once you know what exactly a specific test needs to assess and what the pass/fair criteria are, you can start thinking about how to build the test. That includes what wave forms but also what test equipment, what test flow, what data acquisition, what metrics etc. 

Wave forms are a means to an end. Once you have a clear definition of what the end is, the wave form question will most likely become obvious.