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Numerical error

Numerical error is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Numerical error rather than just read about it. In short: Numerical error is the error that arises in numerical computations due to limitations in representing real numbers and performing arithmetic operation on computers. These errors commonly appear in software engineering and mathematics.

Numerical error — main illustration
Numerical error — illustration

Key takeaways

  • Numerical error belongs to computer science; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Numerical error to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Numerical error from memory before moving on to harder problems.

Reference excerpt

Numerical error is the error that arises in numerical computations due to limitations in representing real numbers and performing arithmetic operation on computers. These errors commonly appear in software engineering and mathematics.

Types Numerical error in computations generally arises from several sources. The most common sources are round-off error and truncation error. It can be the combined effect of two kinds of error in a calculation. The first is referred to as round-off error and is caused by the finite precision of computations involving floating-point numbers. The second, usually called truncation error, is the difference between the exact mathematical solution and the approximate solution obtained when simplifications are made to the mathematical equations to make them more amenable to calculation.

Measure Floating-point numerical error is often measured in ULP (unit in the last place), which represents the distance between two adjacent floating-point numbers.

See also Loss of significance Numerical analysis Error analysis (mathematics) Round-off error Kahan summation algorithm Numerical sign problem

References

Accuracy and Stability of Numerical Algorithms, Nicholas J. Higham, ISBN 0-89871-355-2 "Computational Error And Complexity In Science And Engineering", V. Lakshmikantham, S.K. Sen, ISBN 0444518606

Illustrations

Numerical error: Time series of the Tent map for the parameter m=2.0 which shows numerical error: "the plot of time series (plot of x variable with respect to number of iterations) stops fluctuating and no values are observed after n=50". Parameter m= 2.0, initial point is random.
Time series of the Tent map for the parameter m=2.0 which shows numerical error: "the plot of time series (plot of x variable with respect to number of iterations) stops fluctuating and no values are observed after n=50". Parameter m= 2.0, initial point is random.

Worked examples

Example 1 — a first encounter with Numerical error

Start with the simplest possible case. Write down what Numerical error claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Numerical error before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Numerical error ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Numerical error

In research
Numerical error appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Numerical error in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Numerical error is common in secondary-school and first-year university syllabi. It links to neighbouring topics Applied mathematics stubs, Computer arithmetic, Numerical analysis, so understanding it makes those chapters shorter.
In everyday life
Look for Numerical error outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Numerical error in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Numerical error means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Numerical error out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Numerical error in simple terms?

Numerical error is the error that arises in numerical computations due to limitations in representing real numbers and performing arithmetic operation on computers. These errors commonly appear in software engineering and mathematics.

Why does Numerical error matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Numerical error?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Numerical error.

Tags

  • Applied mathematics stubs
  • Computer arithmetic
  • Numerical analysis
  • Software engineering stubs

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