Inverse Power Method Calculator

Find smallest eigenpairs fast with shift‑invert, normalization, and safeguards against divergence loops. Adjust tolerance, iterations, and shift to target specific spectral regions with precision. Paste a matrix and initial vector with automatic sizing support. Track convergence using Rayleigh quotients, residual norms, and iteration tables for transparency. Export clean CSV and PDF reports for sharing.

Matrix A (rows by newline; values by space or comma)
Initial Vector x₀ (length n)
Shift μ (shift‑invert)
Tolerance
Max Iterations
Stop When
Example Data Table

A 3×3 symmetric matrix with a simple initial vector.

Matrix A
a₁₁a₁₂a₁₃
410
131
012
Initial vector x₀
111
Tip Click “Fill Example” to populate inputs.
Formula Used

Given a square matrix A and shift μ, the shift‑invert iteration computes xₖ by solving:

(A − μI) yₖ = xₖ₋₁,   xₖ = yₖ / ‖yₖ‖₂

The Rayleigh quotient estimates λₖ:

λₖ = (xₖᵀ A xₖ) / (xₖᵀ xₖ)

Stopping criteria options:

  • Residual norm: ‖A xₖ − λₖ xₖ‖₂ < tol.
  • Eigenvalue change: |λₖ − λₖ₋₁| < tol.

With μ=0, the inverse iteration converges to the smallest‑magnitude eigenvalue. For a chosen μ close to a target eigenvalue, convergence is typically rapid to that eigenpair.

How to Use This Calculator
  1. Paste a square matrix A with rows separated by newlines.
  2. Enter an initial vector x₀ of matching length.
  3. Choose a shift μ. Use 0 for smallest eigenvalue.
  4. Set tolerance and maximum iterations appropriate for your problem.
  5. Pick a stopping rule based on residual or eigenvalue change.
  6. Click Compute to view the estimated eigenpair and iterations.
  7. Export results as CSV or PDF for reporting or sharing.
FAQs

The shift μ changes the system to (A − μI). Choosing μ near a desired eigenvalue accelerates convergence to its eigenpair via shift‑invert iteration.

The Rayleigh quotient provides a stable, inexpensive estimate of the eigenvalue corresponding to the current iterate xₖ and is standard in inverse iteration methods.

If (A − μI) is singular or ill‑conditioned, the linear solve may fail. Adjust μ slightly, scale A, or use higher precision or regularization techniques.

With μ=0, inverse iteration typically converges to the smallest magnitude eigenvalue. Shift‑invert can target other eigenvalues near the chosen shift μ.

Start with tol = 1e‑8 and max iterations around 200. Tougher problems or tighter accuracy may require stricter tolerance and more iterations.

Related Calculators

Inverse Function Finder CalculatorPolynomial Long Division Calculatorroots of cubic equation calculatorquadratic function from 3 points calculatorWeighted linear regression calculatorremainder and factor theorem calculatordivide using long division calculatorsynthetic division remainder calculatorLCM fraction Calculatorfactor polynomials by grouping calculator

Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.