Recursive Difference Engine
Evaluates zero-th through third-order divided differences via two-point quotient recurrences.
Mathematics / Newton Divided Differences
FREE TO EXPLORELABORATORY 22 · MATHEMATICS
Newton Divided Differences & Triangular Tableau Interpolation
Implement the recursive Newton divided difference tableau, compute higher-order difference quotients, and verify the connection between the leading divided difference f[x_0, ..., x_n] and the n-th derivative of the underlying function. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
EMPIRICAL DIFFERENCE QUOTIENT VALUE—
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
Like all polynomial interpolation on equidistant nodes, high-degree Newton polynomials eventually diverge near boundaries due to Runge's phenomenon. The simulation evaluates intermediate and leading divided differences for the cubic test function across six successive observation trials. The linear regression fit of the response y = c_3 · k verifies the theoretical leading coefficient c_3 = 0.1667 (1/6) with R² = 1.000000.
Computational tools & numerical modules.
Triangular Tableau Buffer
Maintains the lower triangular tableau matrix of intermediate divided differences without redundant operations.
Newton Nested Multiplier
Evaluates the Newton polynomial form P_n(x) = c_0 + c_1(x-x_0) + c_2(x-x_0)(x-x_1) + ... in O(n) operations.
Mean Value Theorem Comparator
Compares the highest divided difference f[x_0, ..., x_n] against the theoretical derivative value f^{(n)}(ξ)/n!.
Dynamic Node Ingestion Port
Appends new observation nodes to the existing interpolant without recalculating previously computed table rows.
Linearized Quotient Tracker
Monitors the cumulative quotient progression across steps to fit the invariant leading coefficient c_3.
FOLLOW THE METHOD
A reason behind every step.
Implement the recursive Newton divided difference tableau, compute higher-order difference quotients, and verify the connection between the leading divided difference f[x_0, ..., x_n] and the n-th derivative of the underlying function. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 step.
01
Initialize Nodal Grid & Function Values
Set four distinct nodes x_0, x_1, x_2, x_3 and compute the base function values f[x_i] = f(x_i) in the first column of the tableau.
02
Execute First-Order Divided Differences
Compute first-order differences f[x_i, x_{i+1}] = (f[x_{i+1}] - f[x_i]) / (x_{i+1} - x_i) for all adjacent pairs.
03
Compute Higher-Order Difference Quotients
Iteratively construct second and third-order columns using the recurrence f[x_i, ..., x_{i+k}] = (f[x_{i+1}, ..., x_{i+k}] - f[x_i, ..., x_{i+k-1}]) / (x_{i+k} - x_i).
04
Extract Leading Newton Coefficients
Read the top diagonal entries c_k = f[x_0, ..., x_k], which constitute the coefficients of the Newton basis polynomials.
05
Verify Invariance of the Leading Quotient
Confirm that for the test cubic polynomial, the third divided difference c_3 = 1/6 ≈ 0.1667 remains constant regardless of node placement.
06
Record Telemetry & Linear Regression
Log the cumulative quotient response across six observation settings and perform linear regression to verify the theoretical slope c_3 = 0.1667.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Newton Divided Differences mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
y = c_3 · k + b
| # | Observation Step (k) (step) | Empirical Difference Quotient Value (quotient) | Observation Step Index k | Cumulative Quotient Metric y = c_3 · k | Remove |
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