HomeCalculatorsMathNumerical CalculusSpline Interpolation Calculator
Math calculator

Spline Interpolation Calculator

Natural cubic spline through tabulated (x, y). Same engine as Numerical Interpolation. Not Akima, PCHIP, or regression. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published methods
Assurance
Core
Declared partition coverage
PASS · 9/9 declared partitions (ordinate-lagrange, linear, lagrange, spline, normalized-spline, node-identity, scaled-data, numerical-range, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
y at x from tabulated points by a natural cubic spline, with an engine-linked figure of the nodes and S(x).
Scope
The Result Card includes an engine-linked figure of the tabulated nodes and S(x) (query marker is published y; full sorted points are returned). Same natural cubic spline as REST/MCP. Not Akima / PCHIP / regression.
Verification
Engine tested · Source checked · v1.1.6 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods· View Manifest · CVP overview · Specification
Versions
Calculation 1.1.6 · CVP protocol 1.0.0-proposed · Evidence 2026-10-28.metric-scope-v3
Verification revision
2026-10-28.metric-scope-v3 · 1/1 property · digest 540f8d62cb85
Legacy regression
13/13 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 2/2 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
286/286 golden · 8/8 CVP boundary · 25/25 invalid · 1/1 property · 2/2 metamorphic · 10/10 cross-interface · 1/1 cross-calculator · 6/6 CVP contract · Manifest
Sources
Sources
Evidence
4 legacy golden · 2 legacy boundary · legacy regression suite · 286/286 oracle-backed golden · 25/25 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.1.6 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Natural endsS″(x₀) = S″(xₙ) = 0
Piecewise cubicSᵢ(x) = aᵢ + bᵢ(x−xᵢ) + cᵢ(x−xᵢ)² + dᵢ(x−xᵢ)³
iPoints as x,y; x,y (semicolon-separated). Unique x, sorted internally. Table ≤ 100 points. x outside the table needs extrapolate=true. Not clamped and not not-a-knot.

How to use

1

Enter tabulated points

At least two pairs with unique x. Linear and Lagrange live on the canonical interpolation page.

2

Evaluate at x

The spline passes through every node. Outside [x_min, x_max] requires extrapolate. The figure shows the nodes, S(x), and the query.

Example calculations

Common configurations with formula and result.

ϟ

Two-point = linear

(0,0) and (10,20) at x=5

S reduces to lerp
10
ϟ

Three-point natural spline

(0,0), (1,1), (2,0) at x=0.5

M₁ = −3
0.6875

Spline Interpolation calculator specification

Version 1.1.6 · Engine tested

Calculation status

Review policy · Evidence

Definition
A natural cubic spline is a piecewise cubic with continuous S and S′ and S″=0 at the ends. This page is method=spline on math.numerical.interpolate. Two points reduce to linear. Not a second interpolator. Not Akima, PCHIP, or least-squares regression.
What it calculates
y at x from tabulated points by a natural cubic spline, with an engine-linked figure of the nodes and S(x).
Inputs
  • x
  • points?
  • x0?
  • y0?
  • x1?
  • y1?
  • extrapolate?
Outputs
  • y
  • n
  • in_range
  • extrapolated
Formula
natural cubic spline (S″=0 at ends)
Assumptions
  • The Result Card includes an engine-linked figure of the tabulated nodes and S(x) (query marker is published y; full sorted points are returned). Same natural cubic spline as REST/MCP. Not Akima / PCHIP / regression.
  • Unique x; sorted internally
  • Natural ends, not clamped
  • Not Akima / PCHIP / regression
Units
  • dimensionless
Boundary conditions
  • Nonfinite table width or interpolated result → NUMERICAL_RANGE_ERROR
  • x outside table without extrapolate → VALUE_OUT_OF_RANGE
  • duplicate x → INVALID_INPUT
  • unknown method (including akima) → INVALID_MODE
Example
spline (0,0)(1,1)(2,0) x=0.5 → 0.6875
Validation cases

2 published on this page · 13/13 tests · Production surface contract 6/6 · View evidence

  • points=0,0;1,1;2,0 x=0.5 → y=0.6875
  • method=akima points=0,0;1,1 x=0.5 → INVALID_MODE
Sources
Calculation version
1.1.6

Background

Interpretation and common distinctions.

What does this spline interpolation calculator compute?

A natural cubic spline is a piecewise cubic with continuous S and S′ and S″=0 at the ends.

The published formula or method is natural cubic spline (S″=0 at ends). The following references support the stated mathematics or method:

Which inputs and limits should you check?

Inputs use finite decimal values (numbers or decimal text); hexadecimal, nonzero decimal underflow, incomplete/extra cells and conflicting aliases are rejected. Points accept pair text, arrays of exactly two cells, or objects with exactly x and y; scalar input requires all x0,y0,x1,y1. Equal aliases may coexist. Linear/spline allow 2–100 points; Lagrange allows 2–9. Defaults apply only to absent method/extrapolate; explicit empty fields remain invalid. Interpolation requires distinct node coordinates. Check the selected method and whether extrapolation is enabled; a curve through data is not statistical regression.

The worked example uses (0,0) and (10,20) at x=5 with S reduces to lerp. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.numerical.interpolate v1.1.6 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Primary interpolation evaluates the supplied binary64 inputs using exact rational intermediates, then publishes 12 significant digits. Intermediate numerator/denominator sizes are bounded at 16384 bits; nonrepresentable results, nonzero result underflow and unsupported spline coordinate normalization return NUMERICAL_RANGE_ERROR. Exact arithmetic cannot remove data uncertainty, polynomial oscillation or extrapolation risk. Curve sampling uses binary64 formulas and may be unavailable at extreme scales.

Calculations run locally in this page. Shared input URLs may appear in server or third-party logs; avoid putting sensitive data into a shared link.

Other calculators in this family: Brent Method Calculator, Definite Integral Calculator, Euler Method Calculator, Gauss Quadrature Calculator, Golden Section Search Calculator, Nonlinear Equation Calculator, Numerical Derivative Calculator, Numerical Interpolation Calculator . Explore all Numerical Calculus.

→ Brent Method Calculator Brent’s method (golden section plus inverse parabolic interpolation) for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not root-finding Brent. Runs locally. › → Definite Integral Calculator Composite trapezoid, Simpson 1/3, or Gauss–Legendre of a polynomial, sin, or exp on [a, b]. Not indefinite symbolic integrate. Runs locally. › → Euler Method Calculator Forward Euler for a scalar first-order IVP y′=f(t,y). Same engine as Ordinary Differential Equation. Not RK4, not rk45, not a system. Runs locally. › → Gauss Quadrature Calculator n-point Gauss–Legendre quadrature of a polynomial, sin, or exp on [a, b]. Same engine as Definite Integral. Not indefinite CAS. Runs locally. › → Golden Section Search Calculator Golden-section search for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not Brent, not multivariate. Runs locally. › → Nonlinear Equation Calculator Solve a polynomial or sin x for one approximate real zero using Newton, bisection, or secant. Inspect the residual and stopping reason. Calculate locally. › → Numerical Derivative Calculator Finite-difference derivative of a polynomial, sin, or exp, plus a numeric two-sided limit. Central, forward, backward, or limit. Discovery /two-sided-limit. Not symbolic d/dx. Runs locally. › → Numerical Interpolation Calculator Piecewise-linear, Lagrange, or natural cubic spline of tabulated (x, y). Not regression, not CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second interpolation engine?

No. It is method=spline on math.numerical.interpolate, the same engine as /calc/math/numerical-interpolation.

Is this Akima, PCHIP, or regression?

No. This seed is a natural cubic spline only. Clamped, not-a-knot, Akima, PCHIP, and least-squares fit stay out of scope.