Taylor & Padé
Compare Taylor and Padé approximations and their numerical errors.
About this tool
Compare a Taylor sum at 0 with a diagonal Padé entry[n/n]. Choose an order and evaluate a free x by typing, clicking or dragging. The table shows actual function/approximation values and absolute floating-point differences. The error view makes differences easier to inspect.
Padé uses a numerator and denominator whose formal product relation matches the power series through the requested order. Entries can simplify. For 1/(1+x²), orders n≥2 reproduce the rational function exactly as[0/2]. The defective[1/1] entry cancels a common x and becomes 1, with fewer matching terms after cancellation; details show the effective degrees and matching order.
Taylor for 1/(1+x²) has convergence radius 1 because of complex poles±i, although the function is finite on the whole real line. Higher degree need not help outside that radius. exp and sin have entire Taylor series; finite sums still have truncation error. Padé can introduce real poles that the original function does not have. No finite pole neighborhood or large-value band is silently removed.
Linear or sign(y)log 10(1+|y|) axes are explicit; transformed ticks show original values. Large linear magnitudes carry a labeled common factor. Roots/poles and values are numerical approximations in double precision, not error certificates. A displayed zero difference may mean identical floating-point results, not exact mathematical equality. Available Taylor orders 0–12 and Padé orders 0–8, x in[−100,100], plot bounds[−20,20].
Function, view and scale choices update directly; numeric edits follow after a short pause or Enter. Switching functions adapts the standard plot range; a custom range is retained. The compact legend names the selected orders. ±log denotes the signed logarithmic axis explained above.
Browser storage and JSON preserve unfinished fields, function, plot choices, open sections and completed results. Restoring a finished session cheaply reconstructs the bounded comparison with the same method; unfinished work stays unfinished. Numeric drafts are limited to 128 characters.
Only input values are remembered in this browser. Calculated results and experiment histories are not saved.