Calculator Inputs
Use real 2 × 2 or 3 × 3 matrices. The page layout is single-column, while the form fields adapt to 3, 2, or 1 columns.
Example Data Table
| Method | Matrix | Settings | Expected Approximation |
|---|---|---|---|
| Power | [[4, 1], [2, 3]] | Initial vector [1, 1], tolerance 1e-8 | Dominant eigenvalue ≈ 5 |
| Inverse | [[4, 1], [2, 3]] | Initial vector [1, 1], tolerance 1e-8 | Smallest-magnitude eigenvalue ≈ 2 |
| Shifted Inverse | [[2, 1, 0], [1, 2, 1], [0, 1, 2]] | Shift σ = 2, initial vector [1, 1, 1] | Eigenvalue near shift ≈ 2 |
| QR | [[2, 1, 0], [1, 2, 1], [0, 1, 2]] | 60 iterations, tolerance 1e-8 | Eigenvalues ≈ 3.414, 2, 0.586 |
Formula Used
Power Method: xk+1 = A xk / ||A xk|| and λk = (xkT A xk) / (xkT xk).
Inverse Power Method: solve A yk = xk, normalize yk, then estimate the eigenvalue using the Rayleigh quotient.
Shifted Inverse Iteration: solve (A - σI) yk = xk. The method converges toward the eigenvalue closest to the shift σ.
QR Iteration: factor Ak = Qk Rk, then update Ak+1 = Rk Qk. Diagonal entries approach eigenvalues when convergence occurs.
Residual Check: r = A v - λ v. A smaller residual norm indicates a more accurate approximation.
How to Use This Calculator
- Select a matrix size of 2 × 2 or 3 × 3.
- Choose the approximation method that matches your goal.
- Enter matrix values, then supply an initial vector.
- Set the iteration limit and tolerance. Enter a shift if using shifted inverse iteration.
- Press Approximate Eigenvalues to display results above the form.
- Review the summary, convergence diagnostics, and iteration history.
- Download the result as CSV or PDF when needed.
Frequently Asked Questions
1. What does the power method estimate?
It estimates the dominant eigenvalue, usually the one with largest magnitude. It also returns a corresponding eigenvector and convergence diagnostics.
2. When should I use inverse iteration?
Use inverse iteration when you want the smallest-magnitude eigenvalue and the matrix is nonsingular. It often converges faster than a plain power approach for that target.
3. Why is a shift useful?
A shift steers the iteration toward an eigenvalue near your chosen number. This is helpful when the dominant eigenvalue is not the one you need.
4. What does the residual norm mean?
The residual norm measures how closely the computed vector satisfies Av = λv. Smaller residuals usually mean a stronger approximation.
5. Can this calculator find all eigenvalues?
Yes, the QR option estimates all real eigenvalues for supported matrices when the iteration converges well. Other methods target one eigenvalue at a time.
6. Why did the method fail or stall?
Failure may happen with singular systems, poor shifts, unstable starting vectors, or slow convergence. Adjust the shift, tolerance, or matrix values and try again.
7. Are complex eigenvalues supported?
This version is designed for real-valued approximations on real matrices. QR may struggle when complex conjugate eigenvalues dominate the behavior.
8. Which settings usually improve accuracy?
Use a sensible starting vector, tighter tolerance, and enough iterations. For targeted roots, choose a shift close to the expected eigenvalue.