Overview
Polynomial interpolation builds a function that exactly matches known samples, but high-degree fits can oscillate strongly.
CoreSource checked
Polynomial Interpolation
Finds a polynomial that passes through a set of sample points.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Numerical Methods
- Difficulty
- Intermediate
- Family
- Interpolation
- Tier
- Core
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 5
- Code example
- Ready
- Visualization
- Ready
Core Concepts
sample pointinterpolating polynomialbasis functionapproximation error
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(n^2)
Average
O(n^2)
Worst
O(n^3)
Space
O(n)
When to Use
- Use when solving equations needs sample point.
- Use when the input matches the assumptions behind interpolation.
When Not to Use
- Avoid when the problem is ill-conditioned and no scaling or stability checks are available.
- Avoid when a simpler algorithm gives the required result with less setup.
What It Does
- Finds a polynomial that passes through a set of sample points
- Uses sample point and interpolating polynomial to guide the work.
- Produces approximate numeric result.
Use Cases
- Solving equations
- Approximating integrals
- Working with matrices and differential equations
Inputs
- Numeric function, matrix, or sampled data
- Initial estimate or discretization
- Tolerance
Outputs
- Approximate numeric result
- Error estimate or residual
Key Formulas (1)
Interpolating polynomial
p(xi) = yi for every sample i
p(x_i) = y_i for every sample iDefines the polynomial by matching all provided samples.
Pseudocode
- 1.Prepare the inputs and state needed for Polynomial Interpolation.
- 2.Apply the sample point step while the stopping condition has not been met.
- 3.Update intermediate state and discard invalid candidates.
- 4.Return approximate numeric result.
Canonical Example
- Input
- Polynomial Interpolation input with a small representative case
- Output
- Approximate numeric result
- Explanation
- Polynomial Interpolation applies sample point to produce approximate numeric result.
Implementation Notes
- Check convergence with residuals or estimated error, not iteration count alone.
- Keep the sample point step separate enough to test independently.
Common Pitfalls
- Ignoring conditioning and residuals can produce plausible-looking but wrong numeric answers.
- Forgetting to check edge cases can make Polynomial Interpolation look correct on only ideal inputs.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow the numeric approximation improving against the target function, matrix, or state.
Step 1 of 4: Prepare the inputs and state needed for Polynomial Interpolation.
numerical-methods-mechanismNotes
Polynomial Interpolation should be used with the assumptions and tradeoffs of interpolation in mind.
Sources (1)
What each reference supports for this entry.
Numerical Recipes
BookNumerical methods reference covering interpolation formulas and practical polynomial interpolation.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Relationships
View full relationship pageGeneralizesLagrange Interpolation
Lagrange interpolation is one polynomial interpolation form.
GeneralizesNewton Interpolation
Newton interpolation is one polynomial interpolation form.
Related Algorithms
Tags
Numerical MethodsInterpolationPolynomialsApproximation