Advanced Truncation Error Calculator

Calculation Result

Choose a method, enter values, and calculate an error estimate.

Taylor and Maclaurin Series Error

Calculate the polynomial, actual error, next-term estimate, and Lagrange bound.

Function f(x)

Examples: sin(x), exp(x), log(1+x), sqrt(1+x).

Expansion point a

Evaluation point x

Polynomial degree n

Exact value

Target error tolerance

Derivative bound method

Maximum derivative M

Bound samples

Maximum search degree

Displayed precision

Show every polynomial term

Infinite Series Remainder Estimator

Supports custom terms, geometric series, alternating bounds, and convergence checks.

General term a(n)

Use n, factorial(n), powers, and standard functions.

Starting index

Included terms

Series type

Exact infinite sum

Target tolerance

Known geometric ratio r

Custom remainder bound

Maximum search terms

Numerical Integration Truncation Error

Evaluate composite rules, adaptive refinement, error bounds, and convergence order.

Integrand f(x)

Lower limit a

Upper limit b

Method

Subintervals n

Exact integral

Maximum derivative bound

Target tolerance

Maximum refinements

Batch Calculations and Data Import

Paste CSV rows or import a local file. Results can be exported after processing.

CSV input

Drop a CSV file here.

Download Sample CSV

Calculation History

Saved locally in this browser. Rename, restore, compare, export, or delete entries.

Charts and Detailed Analysis

Plots update after supported calculations.

Formula Used

Taylor: |Rₙ(x)| ≤ M|x-a|ⁿ⁺¹/(n+1)!

Alternating series: |Rₙ| ≤ |aₙ₊₁|

Geometric remainder: Rₙ = arⁿ⁺¹/(1-r)

Forward difference error: O(h)

Central difference error: O(h²)

Richardson: A ≈ Aₕ + (Aₕ-A₂ₕ)/(2ᵖ-1)

How to Use This Calculator

Step 1

Choose the numerical method.

Select the tab matching your approximation problem.

Step 2

Enter inputs and precision.

Provide exact values when available for measured errors.

Step 3

Review bounds and charts.

Export calculations or save them in local history.

Understanding Truncation Error

Truncation error appears when an infinite or continuous process is shortened. Numerical methods replace exact operations with finite approximations. The remaining difference is the truncation error.

Series methods stop after a selected number of terms. Differential formulas replace derivatives using nearby function values. Integration rules replace curved areas with simpler geometric pieces.

Truncation Versus Rounding

Truncation removes information by limiting terms, steps, or digits. Rounding changes a stored value because computers have finite precision. Both errors may interact during long calculations.

Smaller steps often reduce theoretical truncation error. Extremely small steps can amplify floating-point cancellation. A practical calculation balances both error sources carefully.

Local and Global Error

Local truncation error measures one numerical step. Global error measures accumulated differences across all steps. Stable higher-order methods usually reduce global error faster.

Convergence order explains how error changes during refinement. An order-two method often reduces error fourfold after halving the step. Observed rates may differ near singularities.

Error Bounds and Reliability

An error bound gives a guaranteed maximum under stated assumptions. An actual error requires an exact reference value. An estimated error uses asymptotic behavior or omitted terms.

Bounds can be conservative because they use worst-case derivatives. Charts and refinement studies reveal practical convergence. Multiple methods provide stronger confidence in difficult problems.

Reducing Truncation Error

Increase retained terms for convergent series expansions. Reduce step sizes while monitoring roundoff effects. Choose higher-order formulas when smoothness assumptions remain valid.

Adaptive methods concentrate effort where the function changes rapidly. Richardson extrapolation combines refined solutions for improved accuracy. Always verify convergence before trusting results.