Numerical Methods · Module 1 of 10
Modeling, Error, and Taylor Series
Every numerical answer is approximate. This module sets up the two languages that run through the course: error, the gap between the estimate and the truth, and the Taylor series, the source of most of it.
Readiness check
This module opens the course. Tick only what you can do closed-notes.
- Compute a percentage difference between two numbers.
- Recall the Taylor series of a function about a point.
- Evaluate a factorial and a power.
- Recall that a derivative is a local slope.
- Distinguish an exact value from an estimate.
The core idea
A numerical result carries an error: the difference between the estimate and the true value. Most of it is truncation, from stopping an infinite process early, and the Taylor series both creates it and lets you bound it.
Et = true − approximationεt = |Et| / true × 100%f(x+h) = f(x) + f′(x)h + f″(x)h2/2 + …An engineering model turns a physical problem into equations, and numerical methods turn those equations into numbers. Because the methods are approximate, every result has an error. The true error Et is the exact value minus the estimate; the true relative error εt expresses it as a fraction of the true value, the more useful measure. When the true value is unknown, an approximate relative error εa compares successive estimates instead, and an iteration stops once it falls below a chosen tolerance. Two sources dominate: roundoff error, from representing numbers in finite precision, and truncation error, from cutting an infinite process short. The Taylor series is central to both: it expresses a function as an infinite sum of derivative terms, and truncating it after a few terms is exactly what most numerical methods do, so its remainder is the truncation error.
The skills, taught in order
Five skills build the language of error and the Taylor series that underlies it.
1.1 Mathematical models and approximation
A model represents a physical system as equations. Most realistic models cannot be solved exactly, so a numerical method produces an approximate solution. Accepting and quantifying that approximation, rather than pretending it is exact, is the foundation of the subject.
1.2 True and relative error
The true error is Et = true value − approximation. The true relative error εt = |Et|/true × 100% scales it against the quantity itself. When the true value is unknown, the approximate relative error εa compares the current and previous estimates, and a computation iterates until εa falls below a stopping tolerance.
| Quantity | Definition | Use |
|---|---|---|
| True error Et | true − approx | when the true value is known |
| True relative error εt | |Et|/true × 100% | scaled accuracy |
| Approximate error εa | |(new − old)/new| × 100% | stopping an iteration |
The three error measures. The relative forms are dimensionless, so they compare across problems of any scale.
1.3 Roundoff and truncation error
Roundoff error comes from storing numbers in a finite number of digits; it grows as operations accumulate. Truncation error comes from approximating an exact mathematical process, such as cutting an infinite series short. Total numerical error is the sum of the two, and reducing one can increase the other.
1.4 The Taylor series
The Taylor series expresses a function near a point as f(x + h) = f(x) + f′(x)h + f″(x)h2/2! + …, an infinite sum of derivative terms. Keeping the first few terms gives a polynomial approximation; the first dropped term sets the order of the truncation error, usually written as O(hn).
1.5 Error propagation
When an uncertain input x ± Δx feeds a function f, the output uncertainty is estimated from the first Taylor term: Δf ≈ |f′(x)| Δx. This propagates a known input error to the result, the basis of estimating how measurement uncertainty affects a computed quantity.
Engineering connection: every method in this course is a truncated Taylor series in disguise, so its accuracy is read straight from the order of the first dropped term.
Worked example 1: a Taylor series estimate and its error
Estimate e0.5 with the Taylor series about zero, using three terms and then four. The true value is 1.648721. Find the true relative error of each and the approximate relative error between them.
- ProblemEstimate e0.5 to three and four terms and find the errors, as in Figure 1.
- Given / findx = 0.5, true value 1.648721. Find the three-term and four-term estimates, their εt, and εa.
- AssumptionsTaylor series about zero, where every derivative of ex is 1.
- Modelex = 1 + x + x2/2! + x3/3! + …; εt = |true − approx|/true × 100%.
- Equations3 terms: 1 + x + x2/24 terms: + x3/6εa = |(new − old)/new| × 100%
- SolveThree terms: 1 + 0.5 + 0.125 = 1.625, εt = |1.648721 − 1.625|/1.648721 × 100 = 1.44%. Four terms: 1.625 + 0.020833 = 1.645833, εt = 0.175%. εa = |(1.645833 − 1.625)/1.645833| × 100 = 1.27%.
- CheckThe approximate error (1.27 percent) is close to the true error of the coarser estimate, as expected: εa tracks εt and signals when to stop without knowing the true value.
- ConclusionFour terms reach about 0.2 percent error. Adding terms shrinks the truncation error predictably, and εa tells you when it is small enough.
Worked example 2: propagating an input error
A quantity is computed as f(x) = x3, with x = 2.5 known only to ±0.01. Estimate the resulting uncertainty in f and the relative error.
- ProblemFind the uncertainty in f = x3 and the relative error for the input in Figure 2.
- Given / findf(x) = x3, x = 2.5, Δx = 0.01. Find f, Δf, and the relative error.
- AssumptionsThe error is small, so the first-order Taylor estimate of propagation holds.
- ModelΔf ≈ |f′(x)| Δx with f′(x) = 3x2; relative error = Δf/f.
- Equationsf = x3Δf ≈ 3x2 Δxrelative = Δf/f
- Solvef = 2.53 = 15.625. Δf ≈ 3(2.5)2(0.01) = 3(6.25)(0.01) = 0.1875. Relative error = 0.1875/15.625 = 1.2%.
- CheckThe exact bounds, 2.493 = 15.438 and 2.513 = 15.813, span about ±0.187 around 15.625, matching the first-order estimate closely.
- ConclusionA 0.4 percent input error (0.01 in 2.5) becomes a 1.2 percent output error, tripled by the cube. Error propagation shows how a function magnifies input uncertainty.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| Reporting no error | A bare number with no qualifier | "How accurate is this estimate?" | Always state a relative or absolute error. |
| Absolute vs relative confused | Large absolute error called small | "Error relative to what?" | Scale by the true value for relative error. |
| Ignoring truncation order | Accuracy claimed beyond the method | "What is the first dropped term?" | The order O(hn) sets the accuracy. |
| Forgetting the slope in propagation | Output error set equal to input error | "How steep is f at this point?" | Δf ≈ |f′(x)| Δx, scaled by the derivative. |
Practice ladder
A true value is 10 and an estimate is 9.2. Find the true relative error.
Show answer
εt = |10 − 9.2|/10 × 100 = 8%.
Estimate e0.5 with only two terms (1 + x). Find the true relative error against 1.648721.
Show answer
Estimate 1.5; εt = |1.648721 − 1.5|/1.648721 × 100 = 9.02%. Far coarser than the three-term estimate.
A volume is V = s3 with s = 4.0 ± 0.05 m. Estimate the uncertainty in V and its relative error.
Show answer
V = 64 m³. ΔV ≈ 3s2Δs = 3(16)(0.05) = 2.4 m³. Relative error = 2.4/64 = 3.75 percent, about three times the 1.25 percent input error.
A measured quantity feeds a strongly nonlinear formula. Explain how error propagation tells you whether to refine the measurement or the model first.
What good work looks like
Compute Δf ≈ |f′|Δx: if the slope is large, the input error dominates and the measurement must improve; if small, the input is fine and the model or method error matters more. The derivative points to the limiting source.
Working with AI, and proving it yourself
Use AI as an examiner, not a solver
Portfolio task
Approximate a function with a Taylor series, tabulate the error as you add terms, and propagate a realistic input uncertainty through a formula you use.
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. Define true relative error.
εt = |true − approx|/true × 100%.
2. What does the approximate error compare?
Successive estimates, used to stop an iteration when the true value is unknown.
3. Name the two main error sources.
Roundoff (finite precision) and truncation (cutting a process short).
4. Write the Taylor series.
f(x + h) = f(x) + f′(x)h + f″(x)h2/2! + …
5. How is input error propagated?
Δf ≈ |f′(x)| Δx, scaled by the local slope.
Textbook mapping
This module follows Chapra and Canale, Numerical Methods for Engineers, 7th edition. Use these references to read further.
| Topic in this module | Where to read more |
|---|---|
| Mathematical models and error | Chapra & Canale, Chapters 1 and 3 |
| Roundoff and truncation error | Chapra & Canale, Chapter 4 |
| Taylor series and error propagation | Chapra & Canale, Chapter 4 |
Chapter numbers refer to the 7th edition. The error definitions are standard, so any recent edition will align closely.