Calculator Inputs
Use a symmetric quadratic objective and a starting point. The calculator applies steepest descent with exact line search.
Example Data Table
| Parameter | Example Value | Meaning |
|---|---|---|
| a11 | 6 | Quadratic weight on x² |
| a12 | 2 | Cross-term weight between x and y |
| a22 | 5 | Quadratic weight on y² |
| b1 | 8 | Linear coefficient for x |
| b2 | 3 | Linear coefficient for y |
| c | 1 | Constant shift in the objective |
| Initial x | 0 | Starting x-coordinate |
| Initial y | 0 | Starting y-coordinate |
| Tolerance | 0.000001 | Stopping threshold for gradient norm |
| Maximum iterations | 25 | Upper limit for update steps |
Formula Used
Objective: f(x, y) = 0.5(a11x² + 2a12xy + a22y²) - b1x - b2y + c
Gradient: ∇f(x, y) = [a11x + a12y - b1, a12x + a22y - b2]
Descent direction: dk = -∇f(xk, yk)
Exact step size: αk = (gkTgk) / (gkTAgk)
Update rule: xk+1 = xk - αkgk
This page is designed for two-variable quadratic minimization. When the Hessian matrix A is symmetric positive definite, the method is stable and the exact line-search step is especially effective.
How to Use This Calculator
- Enter the symmetric quadratic coefficients a11, a12, and a22.
- Provide the linear coefficients b1 and b2, then the constant c.
- Choose an initial point (x, y) where the iteration should begin.
- Set a tolerance for the gradient norm and a maximum iteration count.
- Select the number of decimal places for displayed results.
- Click the compute button to generate the final point and full iteration history.
- Use the CSV button for spreadsheet work and the PDF button for printable reports.
FAQs
1. What does this calculator solve?
It minimizes a two-variable quadratic function using the steepest descent method with exact line search. You get the final point, gradient size, objective value, and every intermediate iteration.
2. Why does the calculator ask for a11, a12, and a22?
These values define the quadratic curvature matrix. They control how sharply the surface bends and how the variables interact through the cross term.
3. What is the best starting point?
Any numeric starting point can be tested. For positive definite problems, the method usually converges from many starts, though some points may require more iterations.
4. Why is positive definiteness important?
A positive definite Hessian makes the quadratic strictly convex. That gives one unique minimizer and supports reliable descent with a positive exact line-search denominator.
5. What does the tolerance control?
Tolerance sets the stopping threshold for the gradient norm. Smaller values ask for a more accurate answer but can increase the number of iterations.
6. Why might the method stop early?
It can stop because the initial point already satisfies the tolerance, because convergence was achieved, or because the denominator for exact line search became nonpositive.
7. What is the condition number shown in the result?
The condition number estimates how stretched the quadratic surface is. Larger values often indicate slower convergence and more zigzag behavior during descent.
8. Can I use the export files for reports?
Yes. The CSV file is useful for spreadsheet analysis, while the PDF export captures the result summary and iteration table for sharing or printing.