Truncation error refers to the discrepancy that occurs when an infinite process is approximated by a finite one. This is a common concept in numerical analysis and computational methods, where exact solutions are often impractical to obtain analytically. ### Key Points about Truncation Error: 1. **Origin**: It arises when a mathematical procedure is truncated or simplified.
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Truncation error is introduced by replacing an exact operation with a finite approximation. It remains even when all arithmetic is exact. For , the forward finite difference for its derivative at givesThus the truncation error is exactly : gives and gives . In finite precision, however, the numerator subtracts nearly equal values, and their roundoff error can be amplified by division by . For smooth data with bounded evaluation errors of scale , a typical bound has competing terms . Smaller reduces the first term and increases the second; with fixed positive constants the balancing scale is . This is why blindly refining a finite difference eventually stops improving it.