Polynomial: Difference between revisions
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A computer program consisting of some number of <code>for</code> loops (and no recursion or <code>goto</code>s or other loops), where each <code>for</code> loop runs exactly <math>n</math> times, will take a number of steps to run which is given by some polynomial in <math>n</math>. In fact, any polynomial at all can be realized in this way. | A computer program consisting of some number of <code>for</code> loops (and no recursion or <code>goto</code>s or other loops), where each <code>for</code> loop runs exactly <math>n</math> times, will take a number of steps to run which is given by some polynomial in <math>n</math>. In fact, any polynomial at all can be realized in this way. | ||
Some generating functions in combinatorics or representation theory (those which truncate after a finite number of powers). | |||
A truncated Taylor series approximation to anything. Tangent line to a curve at a point, tangent conic to a curve at a point, tangent cubic to a curve at a point, etc. | |||
== Rational functions [TODO move to its own page] == | == Rational functions [TODO move to its own page] == | ||
There are many examples of useful, practical rational functions that arise as ratios between square distances in trigonometry. See Norman Wildberger's book. | There are many examples of useful, practical rational functions that arise as ratios between square distances in trigonometry. See Norman Wildberger's book. | ||
Revision as of 04:31, 23 April 2024
A polynomial is any function which obtains its result solely by some combination of: multiplying its inputs together, adding them together, or rescaling them.
Examples
The volume of a cube is a polynomial in its side length.
Every linear transformation is uniquely associated to a polynomial called its characteristic polynomial.
A computer program consisting of some number of for loops (and no recursion or gotos or other loops), where each for loop runs exactly times, will take a number of steps to run which is given by some polynomial in . In fact, any polynomial at all can be realized in this way.
Some generating functions in combinatorics or representation theory (those which truncate after a finite number of powers).
A truncated Taylor series approximation to anything. Tangent line to a curve at a point, tangent conic to a curve at a point, tangent cubic to a curve at a point, etc.
Rational functions [TODO move to its own page]
There are many examples of useful, practical rational functions that arise as ratios between square distances in trigonometry. See Norman Wildberger's book.