Lagrange Cardinal Basis Generator
Synthesizes the product form basis polynomials L_{n,j}(x) satisfying the Kronecker delta property L_{n,j}(x_k) = δ_{jk} across all nodal points.
Mathematics / Lagrange Polynomial Interpolation
FREE TO EXPLORELABORATORY 21 · MATHEMATICS
Lagrange Polynomial Interpolation & Uniform Error Bound Convergence
Construct the degree-n Lagrange interpolating polynomial across equidistant nodal abscissas, analyze the nodal basis functions, and quantitatively evaluate uniform error convergence as grid spacing h decreases. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
MAXIMUM INTERPOLATION ERROR ||F - P_N||_∞—
COMPUTATION STATUS
Ready to execute Step = 1.
Adjust Step, then execute computation.
Mathematical operations are computed deterministically using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations.
Numerical assumptions and limits
Equidistant high-degree polynomial interpolation suffers from Runge's phenomenon, restricting uniform mesh interpolation to low polynomial degrees. The simulation evaluates the degree-3 Lagrange interpolant on refined sub-elements of width h = 0.20 / 2^{k-1}. The maximum error ||f - P_3||_∞ is measured against the true function across a dense test lattice. The logarithmic fit ln(E) = α·ln(h) + ln(C) extracts the asymptotic order α = 4.000 with R² = 1.000000.
Computational tools & numerical modules.
Equidistant Mesh Discretizer
Partitions the interval [a, b] into n uniform sub-elements with grid step h = (b - a)/n to study systematic mesh refinement.
Barycentric Lagrange Engine
Implements the second barycentric formula with barycentric weights w_j = 1/∏_{k≠j} (x_j - x_k) for numerical stability and O(n) per-point evaluation.
Supremum Error Norm Scanner
Samples the difference |f(x) - P_n(x)| over dense micro-grids to accurately identify the maximum error ||E||_∞.
Lebesgue Function Analyzer
Evaluates the Lebesgue function λ_n(x) = ∑ |L_{n,j}(x)| and tracks the exponential growth of the Lebesgue constant Λ_n for uniform nodes.
Log-Log Convergence Regression Pipeline
Performs ordinary least-squares regression on (ln(h), ln(E_max)) to extract the experimental convergence order α.
FOLLOW THE METHOD
A reason behind every step.
Construct the degree-n Lagrange interpolating polynomial across equidistant nodal abscissas, analyze the nodal basis functions, and quantitatively evaluate uniform error convergence as grid spacing h decreases. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 step.
01
Initialize Domain & Objective Function
Select the analytic test function f(x) = sin(πx) on the normalized interval [0, 1] and configure IEEE 754 double precision registers.
02
Set Mesh Refinement Level
Specify the discretization level k from 1 to 6, setting node count n = 2k + 1 and mesh spacing h = 1.0 / n.
03
Construct Lagrange Basis Polynomials
Compute the nodal points x_j = a + j·h and evaluate the cardinal polynomials L_{n,j}(x) ensuring strict unity at x_j and zero at all other nodes.
04
Assemble Global Interpolant
Construct P_n(x) = ∑_{j=0}^n f(x_j)·L_{n,j}(x) using the numerically robust barycentric formulation to prevent catastrophic cancellation.
05
Measure Supremum Interpolation Error
Evaluate the interpolation residual |f(x) - P_n(x)| over 1000 interior test points, capturing the maximum error E_max = ||f - P_n||_∞.
06
Linearize Error Telemetry & Log Fit
Transform mesh spacing and maximum error into logarithmic coordinates (ln(h), ln(E_max)), append to the notebook, and fit the convergence slope α.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Lagrange Polynomial Interpolation mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
ln(E_max) = α · ln(h) + ln(C)
| # | Mesh Refinement Level (k) (step) | Maximum Interpolation Error ||f - P_n||_∞ (error norm) | Logarithmic Mesh Spacing ln(h) | Logarithmic Max Error ln(E_max) | Remove |
|---|
No readings yet. Execute a computation and record it.
Calculate it yourself.
CHECK YOUR UNDERSTANDING