Piecewise Knot Mesh Discretizer
Divides the domain into N sub-intervals with knot coordinates x_0 < x_1 < ... < x_N and spacing h_i = x_{i+1} - x_i.
Mathematics / Cubic Spline Interpolation
FREE TO EXPLORELABORATORY 24 · MATHEMATICS
Cubic Spline Interpolation: Natural and Clamped Boundary Conditions
Construct piecewise cubic spline interpolants satisfying C² continuity, assemble the strictly diagonally dominant tridiagonal continuity system, solve via the Thomas algorithm, and verify the optimal O(h⁴) uniform error convergence under mesh refinement. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
MAXIMUM SPLINE ERROR ||F - S||_∞—
COMPUTATION STATUS
Ready to execute Step = 1.
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Mathematical operations are computed deterministically using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations.
Numerical assumptions and limits
Natural cubic splines exhibit boundary layer degradation to O(h²) error near endpoints unless the underlying function naturally satisfies f''(a) = f''(b) = 0. The simulation solves clamped cubic splines for mesh spacings h = 0.20 / 2^{k-1}. The maximum error ||f - S||_∞ is tracked over 1500 interior points. The log-log regression ln(E) = α·ln(h) + ln(C) extracts the asymptotic order α = 4.000 with R² = 1.000000.
Computational tools & numerical modules.
C² Continuity Matrix Assembler
Formulates the tridiagonal system for second derivatives M_i = S''(x_i) based on slope matching at interior knots.
Thomas Algorithm Engine
Solves the symmetric, strictly diagonally dominant tridiagonal system in O(N) operations without pivoting.
Boundary Condition Switcher
Applies either natural free-edge constraints (M_0 = M_N = 0) or clamped derivative constraints.
Piecewise Hermite Cubic Evaluator
Interpolates any query point x within its respective knot interval using the local cubic polynomial representation.
Curvature Energy Integrator
Computes the total bending energy ∫ (S''(x))² dx to verify the Holladay minimum curvature variational property.
FOLLOW THE METHOD
A reason behind every step.
Construct piecewise cubic spline interpolants satisfying C² continuity, assemble the strictly diagonally dominant tridiagonal continuity system, solve via the Thomas algorithm, and verify the optimal O(h⁴) uniform error convergence under mesh refinement. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 step.
01
Define Domain Knots & Mesh Spacing
Set up N = 2^{k+1} knot intervals on [0, 1] with uniform spacing h = 1.0 / N across refinement levels k = 1 to 6.
02
Evaluate Nodal Coordinates & Function Values
Compute exact function ordinates y_i = f(x_i) at all knots x_i = i · h for i = 0, ..., N.
03
Formulate Tridiagonal Curvature System
Set up the continuity equations h_{i-1} M_{i-1} + 2(h_{i-1} + h_i) M_i + h_i M_{i+1} = 6[(y_{i+1} - y_i)/h_i - (y_i - y_{i-1})/h_{i-1}].
04
Impose Boundary Conditions & Solve via TDMA
Apply clamped boundary conditions and execute the forward elimination and backward substitution of the Thomas algorithm to obtain all moments M_i.
05
Evaluate Supremum Norm Spline Error
Sample S(x) across 1500 interior points and extract the maximum absolute deviation E_max = ||f - S||_∞.
06
Linearize Logarithmic Coordinates & Fit Slope
Plot ln(E_max) versus ln(h) and perform linear regression to quantitatively confirm the theoretical order α = 4.000.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Cubic Spline Interpolation mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
ln(E_max) = α · ln(h) + ln(C)
| # | Mesh Refinement Level (k) (step) | Maximum Spline Error ||f - S||_∞ (error norm) | Logarithmic Mesh Spacing ln(h) | Logarithmic Spline Error ln(E_max) | Remove |
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