Piecewise linear continuation

Key takeaways

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

Reference excerpt

Simplicial continuation, or piecewise linear continuation (Allgower and Georg), is a one-parameter continuation method which is well suited to small to medium embedding spaces. The algorithm has been generalized to compute higher-dimensional manifolds by (Allgower and Gnutzman) and (Allgower and Schmidt). The algorithm for drawing contours is a simplicial continuation algorithm, and since it is easy to visualize, it serves as a good introduction to the algorithm.

Contour plotting The contour plotting problem is to find the zeros (contours) of f ( x , y ) = 0 {\displaystyle f(x,y)=0\,} ( f ( ⋅ ) {\displaystyle f(\cdot )\,} a smooth scalar valued function) in the square 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 {\displaystyle 0\leq x\leq 1,0\leq y\leq 1\,} ,

The square is divided into small triangles, usually by introducing points at the corners of a regular square mesh i h x ≤ x ≤ ( i + 1 ) h x {\displaystyle ih_{x}\leq x\leq (i+1)h_{x}\,} , j h y ≤ y ≤ ( j + 1 ) h y {\displaystyle jh_{y}\leq y\leq (j+1)h_{y}\,} , making a table of the values of f ( x i , y j ) {\displaystyle f(x_{i},y_{j})\,} at each corner ( i , j ) {\displaystyle (i,j)\,} , and then dividing each square into two triangles. The value of f ( x i , y j ) {\displaystyle f(x_{i},y_{j})\,} at the corners of the triangle defines a unique Piecewise Linear interpolant l f ( x , y ) {\displaystyle lf(x,y)\,} to f ( ⋅ ) {\displaystyle f(\cdot )\,} over each triangle. One way of writing this interpolant on the triangle with corners

( x 0 , y 0 ) , ( x 1 , y 1 ) , ( x 2 , y 2 ) {\displaystyle (x_{0},y_{0}),~(x_{1},y_{1}),~(x_{2},y_{2})\,} is as the set of equations

( x , y ) = ( x 0 , y 0 ) + ( x 1 − x 0 , y 1 − y 0 ) s + ( x 2 − x 0 , y 2 − y 0 ) t {\displaystyle (x,y)=(x_{0},y_{0})+(x_{1}-x_{0},y_{1}-y_{0})s+(x_{2}-x_{0},y_{2}-y_{0})t\,}

0 ≤ s {\displaystyle 0\leq s\,}

0 ≤ t {\displaystyle 0\leq t\,}

s + t ≤ 1 {\displaystyle s+t\leq 1\,}

l f ( x , y ) = f ( x 0 , y 0 ) + ( f ( x 1 , y 1 ) − f ( x 0 , y 0 ) ) s + ( f ( x 2 , y 2 ) − f ( x 0 , y 0 ) ) t {\displaystyle lf(x,y)=f(x_{0},y_{0})+(f(x_{1},y_{1})-f(x_{0},y_{0}))s+(f(x_{2},y_{2})-f(x_{0},y_{0}))t\,}

… excerpt ends here. Continue reading the full article.

Illustrations

Piecewise linear continuation illustration

Worked examples

Example 1 — a first encounter with Piecewise linear continuation

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

In research: Piecewise linear continuation appears in 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 Piecewise linear continuation 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: Piecewise linear continuation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Numerical analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Piecewise linear continuation 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 Piecewise linear continuation in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Piecewise linear continuation 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 Piecewise linear continuation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Piecewise linear continuation in simple terms?

Simplicial continuation, or piecewise linear continuation (Allgower and Georg), is a one-parameter continuation method which is well suited to small to medium embedding spaces. The algorithm has been generalized to compute higher-dimensional manifolds by (Allgower and Gnutzman) and (Allgower and Sc…

Why does Piecewise linear continuation matter?

Because it connects several 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 Piecewise linear continuation?

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 Piecewise linear continuation.