Calculator Inputs
Example Data Table
| Quadratic surface example | ||||||
|---|---|---|---|---|---|---|
| a | b | c | d | e | f | Expected |
| 1 | -1 | 0 | 0 | 0 | 0 | Saddle at (0,0), value 0 |
| 2 | 2 | 0 | -4 | 8 | 1 | Minimum at (1,-2), value -7 |
| Payoff matrix example (A payoff) | ||||
|---|---|---|---|---|
| A \ B | B1 | B2 | B3 | Notes |
| A1 | 2 | 1 | 3 | Check row minima and column maxima |
| A2 | 0 | 2 | 4 | |
| A3 | 1 | 0 | 2 | |
Formula Used
1) Quadratic surface: We use
f(x,y)=ax²+by²+cxy+dx+ey+f
Critical points satisfy the gradient equations: ∂f/∂x = 2ax + cy + d = 0 and ∂f/∂y = 2by + cx + e = 0.
This is a linear system: [2a c; c 2b][x;y] = [-d;-e]. The Hessian is H = [[2a, c],[c, 2b]] with determinant Δ = 4ab − c².
- If Δ < 0 → the critical point is a saddle point.
- If Δ > 0 and 2a > 0 → local minimum.
- If Δ > 0 and 2a < 0 → local maximum.
- If Δ ≈ 0 → classification is inconclusive.
2) Payoff matrix game: For matrix A (payoff to Player A):
- Row minimum for row i: rᵢ = minⱼ Aᵢⱼ.
- Maximin value: v₁ = maxᵢ rᵢ.
- Column maximum for column j: cⱼ = maxᵢ Aᵢⱼ.
- Minimax value: v₂ = minⱼ cⱼ.
- If v₁ = v₂, there is a pure-strategy saddle point.
How to Use This Calculator
- Select a mode: quadratic surface or payoff matrix.
- Enter values. For the matrix, build rows and columns first.
- Press Submit. Results appear above the form.
- Review the classification, steps, and checks shown.
- Use the CSV or PDF buttons to export your result summary.
Saddle points in optimization workflows
In multivariable optimization, a saddle point is a critical point where the surface rises in one direction and falls in another. In quadratic models, the determinant of the Hessian, Δ=4ab−c², is the fastest diagnostic: negative Δ indicates mixed curvature. This calculator reports Δ, second derivatives, and the critical coordinates produced by solving a 2×2 linear system.
Interpreting curvature with measurable signals
Curvature is not a visual guess; it is quantified. When Δ>0, the sign of fxx=2a separates minima from maxima. When Δ≈0, the model can be flat along some direction, so small numeric changes can shift classification. Using an adjustable tolerance ε helps you flag borderline cases that deserve sensitivity testing or higher precision arithmetic.
Critical coordinate solving as a linear system
The gradient equations, 2ax+cy+d=0 and cx+2by+e=0, form a compact matrix equation. If Δ≠0, the inverse exists and the solution is unique. The tool returns (x*,y*) and f(x*,y*), then formats results with controlled rounding so reports remain consistent across teams and datasets.
Game matrices and pure-strategy equilibrium checks
For payoff matrices, a saddle point represents a pure-strategy equilibrium. The calculator computes row minima, then takes their maximum (maximin). It also computes column maxima, then takes their minimum (minimax). Equality implies a stable value where Player A can guarantee at least v and Player B can hold A to at most v.
Plot-driven validation for faster reviews
Graphs accelerate verification. The surface plot centers around the computed (x*,y*) to show opposing slopes that characterize a saddle. For matrix games, the heatmap makes dominance patterns and candidate saddle cells obvious. Visual checks are especially helpful when communicating results to non-specialists or reviewing many scenarios quickly.
Exportable outputs for audit and reporting
Decision models often require traceability. CSV export captures inputs, diagnostics, and the final classification for spreadsheet audits. PDF export packages the same fields into a clean summary suitable for approvals. Together with the step display, these exports support reproducible analysis and reduce mistakes when the same model is revisited later. Include version notes, parameter ranges, and the chosen ε in every export. When models feed downstream automation, these metadata fields prevent silent regressions and make it clear why two runs differ. across environments and settings.
FAQs
1) What does a negative Δ mean?
A negative Hessian determinant indicates mixed curvature: the surface bends upward in one direction and downward in another. If a unique critical point exists, that point is classified as a saddle point.
2) Why is the result sometimes inconclusive?
When Δ is near zero, the quadratic can be nearly flat along a direction. Small coefficient changes or rounding may change the classification. Increase precision, adjust ε, or analyze the function along candidate directions.
3) What if the matrix game has no saddle point?
If maximin and minimax differ, a pure-strategy saddle point does not exist. That typically means the equilibrium requires mixed strategies. The reported bounds still show what A can guarantee and what B can enforce.
4) How is the critical point computed?
The calculator sets the gradient to zero and solves a 2×2 linear system. If the determinant is nonzero, the solution is unique. If the system is singular, the tool warns that no unique critical point was found.
5) What do the plots help me verify?
The surface plot shows whether slopes change sign around the critical point, which is typical for a saddle. The heatmap highlights candidate saddle cells and payoff structure, making maximin and minimax patterns easier to review.
6) What is included in CSV and PDF exports?
Exports include the selected mode, inputs, key diagnostics like Δ or maximin/minimax, and the final classification. Use them for audits, sharing, and reproducing results without re-entering values.