← Computational Physics
Derivatives Without Calculus
How do you differentiate a function you can only sample?
Finite differences are the bridge from continuous mathematics to something a machine can execute, and they carry the tension that runs through the whole path: the same h that reduces truncation error amplifies roundoff. This is where order of accuracy stops being notation and starts being a slope you can measure.
01Differentiating a function you can only samplecore25 min
Taylor gives you the formula and the error term in one line. Then floating point takes the error term back, and the result is the most important curve in the subject.
02Buying an order for freeadvanced25 min
Richardson extrapolation cancels the leading error term algebraically. Complex-step differentiation removes the subtraction entirely, and with it the whole roundoff branch.