Power & Inverse Power Method Calculator
Enter a real 3 by 3 matrix, a starting vector, and a stopping tolerance. The calculator estimates one eigenvalue and eigenvector, then shows the residual at every iteration.
Estimated eigenvalue4
- Normalized eigenvector
- [0.5773502692, 0.5773502692, 0.5773502692]
- Residual norm
- 0
- Relative residual
- 0
- Iterations used
- 1
- Convergence status
- converged
Iteration table
| Iteration | lambda estimate | x1 | x2 | x3 | Residual norm | Relative residual |
|---|---|---|---|---|---|---|
| 1 | 4 | 0.5773502692 | 0.5773502692 | 0.5773502692 | 0 | 0 |
Uses real arithmetic on a 3 by 3 matrix and reports one eigenpair, not the full spectrum.
Inverse methods solve a linear system at each step; they do not explicitly form the matrix inverse.
How to use this calculator
- Choose power iteration, inverse iteration, or shifted inverse iteration.
- Enter the nine matrix entries and a nonzero starting vector.
- For shifted inverse iteration, enter a target shift near the eigenvalue you want.
- Set the residual tolerance and maximum iterations, then read the estimate and iteration table.
How the iteration is calculated
This calculator works with a real 3 by 3 matrix A and a real starting vector x0. The starting vector is scaled so ||x0||2 = 1. At each step the method produces a new vector y, normalizes it to x = y / ||y||2, and estimates the eigenvalue with the Rayleigh quotient:
lambda = (x^T A x) / (x^T x)
The standard power method uses y = A x, so it usually moves toward the eigenvector for the eigenvalue with largest magnitude. The inverse power method solves A y = x, which is equivalent to multiplying by A^-1 without forming the inverse. It usually moves toward the eigenvalue whose magnitude is smallest. The shifted inverse method solves (A - sigma I)y = x, so it targets the eigenvalue nearest the shift sigma.
After each normalized step, the calculator computes the eigenpair residual:
r = ||A x - lambda x||2
It also reports the scaled relative residual rho = r / max(||A||inf ||x||2, epsilon). Iteration stops when rho is at or below the tolerance, when the maximum iteration count is reached, or when the selected linear system cannot be solved reliably.
What changes convergence
Eigenvalue separation matters most. Power iteration is fast when one eigenvalue has a clearly larger magnitude than the others. It slows down or can fail to settle when two leading eigenvalues have nearly equal magnitude, when the starting vector has little component in the desired eigenvector direction, or when the desired eigenvalue is complex. Inverse methods have the same starting-vector issue, and they also depend on solving a well-conditioned linear system at each step.
The shift controls shifted inverse iteration. A shift close to the desired eigenvalue can converge quickly, but a shift equal to an eigenvalue makes A - sigma I singular. A shift that is too close for floating-point arithmetic can also make the solve unreliable.
What this calculator leaves out
The tool estimates one accessible real eigenpair. It does not compute the full spectrum, complex eigenvalues, generalized eigenproblems, defective-matrix diagnostics, or high-precision arithmetic. A small residual is the main accuracy check here; reaching the iteration limit means only that this run did not meet the requested tolerance.
Worked example
For the diagonal matrix with entries 5, 2, and 1, the standard power method starts from [1, 1, 1] and moves toward the first coordinate. With the default tolerance style, the estimated eigenvalue approaches 5, the normalized eigenvector approaches [1, 0, 0], and the residual falls toward zero.
For the same kind of hand-check, inverse iteration on the diagonal matrix with entries 5, 2, and 0.5 moves toward the third coordinate because 0.5 is the smallest-magnitude eigenvalue. Shifted inverse iteration on diag(1, 3, 6) with sigma = 2.8 moves toward the middle coordinate and estimates the eigenvalue 3.
Common questions
What is the difference between the power method and inverse power method?
The power method repeatedly multiplies by A and usually finds the eigenvalue with largest magnitude. The inverse power method solves A y = x at each step and usually finds the eigenvalue with smallest magnitude. The calculator uses the Rayleigh quotient for the displayed eigenvalue in both cases.
How do I choose an initial vector?
Use any nonzero vector that is not deliberately orthogonal to the eigenvector you want. A simple vector such as [1, 1, 1] is often enough for exploratory work. If convergence stalls, try a different starting vector.
How do I choose a shift for shifted inverse iteration?
Choose a shift near the eigenvalue you want, but not exactly on it. A closer shift can make convergence faster, while a shift equal or extremely close to an eigenvalue can make A - sigma I singular or badly conditioned.
Why is the largest eigenvalue not always the most positive eigenvalue?
The power method is governed by magnitude, not signed value. An eigenvalue of -10 has larger magnitude than an eigenvalue of 7, so the standard power method tends toward the -10 eigenpair if the starting vector reaches it.
Does inverse iteration calculate the matrix inverse?
No. Each inverse step solves a linear system, which is numerically preferable to forming A inverse explicitly. The result is mathematically equivalent to applying the inverse when the system is nonsingular.
How can I check whether the result is accurate?
Look at the relative residual. A small relative residual means the reported vector nearly satisfies A x = lambda x at the scale of the matrix. The iteration table is useful for checking whether the residual is still falling or has stalled.