3D Optimization Calculator

Explore three-variable surfaces with exact stationary analysis. Adjust coefficients, review determinants, and compare sampled values. Download clean outputs for study, teaching, validation, and sharing.

Calculator Inputs

Use the quadratic model f(x,y,z) = ax² + by² + cz² + dxy + eyz + fzx + gx + hy + iz + j.

x minimum

x maximum

y minimum

y maximum

manual z slice

plot samples

tolerance

Example Data Table

Case	Quadratic form summary	Stationary point	Expected type
Example 1	x² + y² + z² - 2x - 4y + 6z	(1, 2, -3)	Local minimum
Example 2	-x² - 2y² - 3z² + 4x - 2y + z	(2, -0.5, 0.166667)	Local maximum
Example 3	x² - y² + z² + xy - 3x + 2y + z	Unique point exists	Saddle point

Formula Used

This calculator uses the quadratic function:

f(x,y,z) = ax² + by² + cz² + dxy + eyz + fzx + gx + hy + iz + j

The stationary point is found by solving these first-derivative equations:

∂f/∂x = 2ax + dy + fz + g = 0
∂f/∂y = dx + 2by + ez + h = 0
∂f/∂z = fx + ey + 2cz + i = 0

The Hessian matrix is:

H = [ 2a   d   f ]
    [  d  2b   e ]
    [  f   e  2c ]

Classification uses the leading principal minors:

If D1 > 0, D2 > 0, D3 > 0, the stationary point is a local minimum.
If D1 < 0, D2 > 0, D3 < 0, the stationary point is a local maximum.
If the determinant pattern is mixed and D3 is nonzero, the point is a saddle point.
If a determinant is near zero, the test is degenerate or inconclusive.

How to Use This Calculator

Enter the coefficients a through j for your three-variable quadratic function.
Choose the x and y plotting ranges used for the surface slice.
Set plot samples to control graph detail.
Enter a manual z slice if you want a custom slice when no unique stationary point exists.
Set a tolerance for determinant and classification sensitivity.
Press Optimize Now.
Review the stationary point, function value, Hessian minors, and classification.
Download the result as CSV or PDF for reporting or revision.

FAQs

1) What does this calculator optimize?

It analyzes a quadratic function of three variables. The tool solves for the stationary point, checks the Hessian, classifies the point, and plots an XY surface slice.

2) Does it handle nonlinear constraints?

No. This version focuses on unconstrained quadratic optimization in three variables. For constrained problems, Lagrange multipliers or numerical optimization methods are usually required.

3) Why might no unique stationary point appear?

If the Hessian determinant is zero or nearly zero, the linear derivative system may not have one unique solution. That usually means degeneracy, flat directions, or dependent equations.

4) What does the graph represent?

The graph shows an XY slice of the full three-variable function. The z variable is fixed to the stationary z value, or to your manual slice when uniqueness fails.

5) How is minimum or maximum classification decided?

The calculator uses the leading principal minors of the Hessian matrix. Their signs indicate positive definiteness, negative definiteness, saddle behavior, or a degenerate case.

6) Why is the gradient check useful?

The gradient check confirms the computed stationary point satisfies the derivative equations numerically. Values near zero show the solution is internally consistent.

7) Can I use decimals and negative coefficients?

Yes. All coefficient fields accept decimal and negative values, so you can test many quadratic surfaces without changing the code structure.

8) What do the CSV and PDF files contain?

They include the entered coefficients, determinant values, classification, slice setting, and stationary-point summary. That makes the output easier to archive, share, or compare.