Data Points & Settings
Presets
Data Points (max 10)
x
y
Methods
Query x value
Lagrange Basis
$$L(x)=\sum_{i=0}^n y_i \prod_{j\neq i}\frac{x-x_j}{x_i-x_j}$$—
Linear value
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Lagrange value
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Spline value
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Condition no. (est.)
Interpolated values at query x = 0.5
Natural Cubic Spline
Piecewise cubic $S_i(x)$ on each interval $[x_i, x_{i+1}]$, continuity conditions:
$$S_i(x_{i+1})=S_{i+1}(x_{i+1}),\quad S_i'(x_{i+1})=S_{i+1}'(x_{i+1}),\quad S_i''(x_{i+1})=S_{i+1}''(x_{i+1})$$Natural boundary: $S''(x_0)=S''(x_n)=0$
Runge's phenomenon: equally spaced high-degree polynomial interpolation ($n \geq 7$) produces large oscillations near the interval endpoints.
CAE Relevance: Spline fitting of stress-strain curves for FEM material models / LS-DYNA *DEFINE_CURVE / field quantity interpolation between mesh nodes / NURBS and B-spline geometry (rational extensions of spline interpolation). PCHIP is recommended for monotone material data.