Interval propagation

Key takeaways

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

Reference excerpt

In numerical mathematics, interval propagation or interval constraint propagation is the problem of contracting interval domains associated to variables of R without removing any value that is consistent with a set of constraints (i.e., equations or inequalities). It can be used to propagate uncertainties in the situation where errors are represented by intervals. Interval propagation considers an estimation problem as a constraint satisfaction problem.

Atomic contractors A contractor associated to an equation involving the variables x1,...,xn is an operator which contracts the intervals [x1],..., [xn] (that are supposed to enclose the xi's) without removing any value for the variables that is consistent with the equation. A contractor is said to be atomic if it is not built as a composition of other contractors. The main theory that is used to build atomic contractors are based on interval analysis. Example. Consider for instance the equation

x 1 + x 2 = x 3 , {\displaystyle x_{1}+x_{2}=x_{3},}

which involves the three variables x1,x2 and x3. The associated contractor is given by the following statements

[ x 3 ] := [ x 3 ] ∩ ( [ x 1 ] + [ x 2 ] ) {\displaystyle [x_{3}]:=[x_{3}]\cap ([x_{1}]+[x_{2}])}

[ x 1 ] := [ x 1 ] ∩ ( [ x 3 ] − [ x 2 ] ) {\displaystyle [x_{1}]:=[x_{1}]\cap ([x_{3}]-[x_{2}])}

[ x 2 ] := [ x 2 ] ∩ ( [ x 3 ] − [ x 1 ] ) {\displaystyle [x_{2}]:=[x_{2}]\cap ([x_{3}]-[x_{1}])}

For instance, if

x 1 ∈ [ − ∞ , 5 ] , {\displaystyle x_{1}\in [-\infty ,5],}

x 2 ∈ [ − ∞ , 4 ] , {\displaystyle x_{2}\in [-\infty ,4],}

x 3 ∈ [ 6 , ∞ ] {\displaystyle x_{3}\in [6,\infty ]}

the contractor performs the following calculus

x 3 = x 1 + x 2 ⇒ x 3 ∈ [ 6 , ∞ ] ∩ ( [ − ∞ , 5 ] + [ − ∞ , 4 ] ) = [ 6 , ∞ ] ∩ [ − ∞ , 9 ] = [ 6 , 9 ] . {\displaystyle x_{3}=x_{1}+x_{2}\Rightarrow x_{3}\in [6,\infty ]\cap ([-\infty ,5]+[-\infty ,4])=[6,\infty ]\cap [-\infty ,9]=[6,9].}

x 1 = x 3 − x 2 ⇒ x 1 ∈ [ − ∞ , 5 ] ∩ ( [ 6 , ∞ ] − [ − ∞ , 4 ] ) = [ − ∞ , 5 ] ∩ [ 2 , ∞ ] = [ 2 , 5 ] . {\displaystyle x_{1}=x_{3}-x_{2}\Rightarrow x_{1}\in [-\infty ,5]\cap ([6,\infty ]-[-\infty ,4])=[-\infty ,5]\cap [2,\infty ]=[2,5].}

x 2 = x 3 − x 1 ⇒ x 2 ∈ [ − ∞ , 4 ] ∩ ( [ 6 , ∞ ] − [ − ∞ , 5 ] ) = [ − ∞ , 4 ] ∩ [ 1 , ∞ ] = [ 1 , 4 ] . {\displaystyle x_{2}=x_{3}-x_{1}\Rightarrow x_{2}\in [-\infty ,4]\cap ([6,\infty ]-[-\infty ,5])=[-\infty ,4]\cap [1,\infty ]=[1,4].}

For other constraints, a specific algorithm for implementing the atomic contractor should be written. An illustration is the atomic contractor associated to the equation

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Interval propagation

Start with the simplest possible case. Write down what Interval propagation claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Interval propagation 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 Interval propagation 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 Interval propagation

In research: Interval propagation appears in mathematics 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 Interval propagation 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: Interval propagation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebra of random variables, Numerical analysis, Statistical approximations, so understanding it makes those chapters shorter.
In everyday life: Look for Interval propagation 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.

How to study Interval propagation in 20 minutes

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

Frequently asked questions

What is Interval propagation in simple terms?

In numerical mathematics, interval propagation or interval constraint propagation is the problem of contracting interval domains associated to variables of R without removing any value that is consistent with a set of constraints (i.e., equations or inequalities). It can be used to propagate uncert…

Why does Interval propagation matter?

Because it connects several mathematics 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 Interval propagation?

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 Interval propagation.