Calculator
Enter a multivariable function, choose variable names, set the point, and calculate numerical gradient and Hessian values. Trigonometric functions use radians.
Example Data Table
| Function | Variables | Point | Gradient | Hessian | Quick Insight |
|---|---|---|---|---|---|
| x² + 3xy + y² | x, y | (1, 2) | [8, 7] | [[2, 3], [3, 2]] | Indefinite Hessian, so curvature is saddle-like. |
| x² + y² + z² | x, y, z | (1, -2, 3) | [2, -4, 6] | 2I₃ | Positive definite curvature everywhere. |
| sin(xy) + x² | x, y | (1, 1) | [2.5403, 0.5403] | Approximate numeric matrix | Mixed curvature depends on location. |
Formula Used
Gradient
The gradient vector collects all first partial derivatives:
∇f(x) = [∂f/∂x₁, ∂f/∂x₂, ..., ∂f/∂xₙ]
With central differences, each component is approximated by: (f(x + h eᵢ) - f(x - h eᵢ)) / (2h)
Hessian Matrix
The Hessian stores all second partial derivatives:
H(f) = [∂²f / (∂xᵢ ∂xⱼ)]
For diagonal terms with central differences: (f(x + h eᵢ) - 2f(x) + f(x - h eᵢ)) / h²
For mixed terms with central differences: (f(x+h eᵢ+h eⱼ) - f(x+h eᵢ-h eⱼ) - f(x-h eᵢ+h eⱼ) + f(x-h eᵢ-h eⱼ)) / (4h²)
Interpretation
The gradient points toward steepest ascent. Its norm measures local steepness.
The Hessian describes curvature. Positive curvature often suggests local convexity, while mixed signs can indicate saddle behavior.
In two variables, the second derivative test uses D = fxx·fyy - (fxy)².
How to Use This Calculator
- Type your multivariable function using standard expression syntax.
- List variables in order, such as x,y or x,y,z.
- Enter the coordinate value for each listed variable.
- Choose a numerical differentiation method and a suitable step size.
- Set the number of decimal places for displayed results.
- Click the calculate button to generate values and graphs.
- Review the function value, gradient vector, Hessian matrix, determinant, and local point classification.
- Use the CSV or PDF buttons to export the current result set.
FAQs
1. What does the gradient tell me?
The gradient shows the local direction of fastest increase. Its size indicates how quickly the function changes near the selected point.
2. What is the Hessian used for?
The Hessian measures local curvature through second derivatives. It helps study convexity, saddle behavior, and optimization conditions around a point.
3. Why are the results numerical instead of symbolic?
This page uses numerical finite differences. That makes it flexible for many custom expressions without requiring a symbolic algebra engine.
4. Which step size should I choose?
Small step sizes improve local accuracy, but extremely tiny values can magnify floating-point noise. A practical starting value is 0.0001.
5. Why is my Hessian slightly asymmetric?
Numerical approximation can introduce tiny symmetry errors. This usually comes from rounding, step size choice, or difficult function behavior near the point.
6. Can I analyze functions with three or more variables?
Yes. The calculator supports several variables, provided you list them clearly and supply a value for each coordinate.
7. What does a near-zero gradient mean?
A near-zero gradient suggests a stationary point. Further classification depends on the Hessian, especially for optimization and curvature analysis.
8. What does the two-variable classification mean?
For two variables, the determinant and second derivative test can indicate local minimum, local maximum, saddle point, or an inconclusive case.