Chebyshev Polynomial Calculator
Enter a non-negative degree and the point x to evaluate precisely.
| Degree | โ |
|---|---|
| x | โ |
| Tn(x) | โ |
Introduction to Chebyshev Polynomials
Chebyshev polynomials of the first kind, denoted , occupy a prominent place in approximation theory and numerical analysis. They arise naturally in the study of trigonometric identities and minimize the maximum error when approximating smooth functions on the interval . This minimax property means Chebyshev polynomials oscillate between -1 and 1 with the smallest possible amplitude growth for a given degree. Engineers and mathematicians leverage them when designing algorithms for interpolation, spectral methods, and digital signal processing.
Definition Using Cosine
The most direct definition uses the cosine function. For a non-negative integer , we set
This relationship immediately reveals that equals 1 and equals . Because cosine is periodic, higher-degree polynomials oscillate increasingly rapidly across the interval yet remain bounded in magnitude by one. Plotting reveals zero crossings between -1 and 1, a pattern that underpins their interpolation capabilities.
Recurrence Relation
Chebyshev polynomials satisfy a simple three-term recurrence that facilitates efficient computation:
Implementing this recurrence in code ensures numerical stability and avoids repeatedly evaluating expensive trigonometric functions. The calculator follows this iterative pattern, updating only the two most recent values to reach the requested degree.
Orthogonality and Weighting
A remarkable property is that these polynomials are orthogonal on with respect to the weight , producing the integral identity , where ฮด is the Kronecker delta.
This orthogonality makes Chebyshev polynomials ideal basis functions in spectral methods, providing numerically stable approximations for smooth functions. Engineers often pair them with Clenshawโs algorithm to evaluate series quickly while minimizing floating-point error.
Common Reference Values
Many numerical libraries store a handful of evaluations for quick interpolation checks. The table below illustrates how the polynomials behave at a typical design point of .
| Degree n | Tn(0.5) | Oscillation note |
|---|---|---|
| 0 | 1.0000 | Constant baseline |
| 1 | 0.5000 | Linear growth |
| 4 | -0.5000 | First negative swing |
| 8 | -0.1914 | Rapid oscillation onset |
| 12 | 0.8688 | Return toward crest |
When implementing the recurrence in software, guard against overflow by using iterative updates instead of recursion. Many developers employ a rolling buffer to store and before generating the next degree. Our calculator follows that pattern to maintain stability up to high orders.
Chebyshev series underpin fast cosine transforms, filter design, and minimax polynomial approximations. For deeper dives, explore the legendre-polynomial-calculator, hermite-polynomial-calculator, and polynomial-regression-calculator to compare orthogonal bases and regression strategies across the AgentCalc library.