(no subject)
…as a technique fell by the wayside…interpolation was discussed…you’d crash and burn, because you’re making a mistake…if you made the slightest mistake it’d propagate through…did you ever hear of the phrase lookup table?...make a giant table of functions…how does your pocket calculator do sines and cosines…the problem is, you need to get analytic derivatives…anybody want to look it up? it’s not that hard to find…my point here is that it can be very useful…mechanism after mechanism…this is interpolating….a lot of the rules for integration are based on that…if I have ten data points what can I fit it with?...the idea is you can fit exactly an (n-1) polynomial…as you add more and more terms, the condition number gets worse and worse…on page six, I show you the different representations…you can find it under my name on the website…interpolation…the critical thing is on page 11…it’s right there, a figure… … … c is defined…you do this in two stages…okay, I did umm…okay…okay there are eight data points and they are in red…then here is the equivalent…the y values…okay here it is…I’m not interested in the details of how to calculate it…you calculate something that goes through the points, alright…we say…okay…xd equally spaced…then we do this Lagrange coefficient thing again….okay…this is twelve…this is twenty…you can stick them together again this way…this little thing just completely dominates…okay…you notice none of them work…none of them worked very good…this is the one thing I want you to remember…for certain types of things, the more points, the worse the fit…this is completely dependent on the Lagrange…it turns out, what happens is, the best way to do it is to space the data…this is uniformly spaced data…this is independent of the method…this explosion at the edge doesn’t occur everywhere, but it does occur…it turns out…it turns out that it’s going to be one or two things…this dominates the nature of interpolation…you don’t fit something with a twenty order polynomial…don’t ever fit anything with more than a quintic...it fits it by a partial cubic…you match the value…the first derivative and the second derivative…good point…r is red and o is circle…good point…it’s barely visible…that’s supposed to be green…now look what happened here…guess what…it fit….there are certain types of functions that blow up, some that don’t…and you can see that in the picture on page 11…it completely goes crazy there…you need to visualize it…there’s no bizarre things there…why do they act so bad?...notice what’s happening…as you go in it gets more and more…it concentrates the data points at the outer edges…then we can go through and we can do a fit on these points…let’s see what we get…now can you see this picture, or is it still too green?...or red…