ShaneLogic/chebfun-fundamentals
.claude/skills/chebfun-fundamentals/SKILL.md
Understand what chebfuns are, their philosophy, and how to construct them. Use when creating numerical representations of functions on intervals.
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SKILL.md
- name:
- chebfun-fundamentals
- description:
- Understand what chebfuns are, their philosophy, and how to construct them. Use when creating numerical representations of functions on intervals.
Chebfun Fundamentals
What is a Chebfun?
A chebfun is a numerical representation of a function of one variable defined on a finite interval [a, b]. It achieves for functions what floating-point arithmetic achieves for numbers—rapid computation where each operation is exact apart from a very small relative rounding error.
Core Philosophy: "Feel symbolic but run at the speed of numerics"
The syntax for chebfuns is almost exactly the same as MATLAB syntax for vectors, with familiar commands overloaded in natural ways.
Construction Syntax
Basic Constructor
f = chebfun('cos(20*x)') % String expression on default domain
f = chebfun(@(x) besselj(0,x)) % Anonymous function handle
Specifying Domain
f = chebfun('cos(20*x)', [0, 100]) % Custom domain [a, b]
f = chebfun(@(x) sin(x), [0, 2*pi]) % Function handle with domain
Default Domain
- If no domain is specified, defaults to
[-1, 1](mimics Chebyshev polynomials) - Explicitly pass domain interval for functions on other intervals
Internal Representation
Chebfuns are represented by polynomial interpolation in Chebyshev points (or equivalently, expansions in Chebyshev polynomials).
Chebyshev Points Formula (for interval [-1, 1]):
x_j = cos(j*pi/N) for j = 0, ..., N
- Points are clustered near endpoints (denser at -1 and 1)
- For other intervals, points are obtained by linear scaling
- Number of points (length) is determined adaptively
- Simple functions: 20-30 points
- Complicated functions: 1000 or 1,000,000+ points
- Evaluation uses barycentric formula (stable even for millions of points)
Accuracy Goal
Adaptive procedures aim to represent each function to roughly machine precision (about 15 digits of relative accuracy).
Use When
- Creating numerical representations of mathematical functions
- Need high-precision function approximation on an interval
- Working with functions that may be smooth or piecewise smooth
- Preparing functions for operations like integration, differentiation, root-finding
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