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Power Iteration

Approximates the dominant eigenvector by repeatedly multiplying by a matrix.

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Quick Facts

Compact metadata for implementation and discovery.

Category: Numerical Methods
Difficulty: Intermediate
Family: Eigenvalue Method
Tier: Core
Status: Verified
Review: Source Checked
Sources: 1
Formulas: 1
Relationships: 3
Code example: Ready
Visualization: Ready

Overview

Power iteration converges toward the eigenvector associated with the largest-magnitude eigenvalue when the dominant direction is sufficiently separated.

Core Concepts

matrix-vector productnormalizationdominant eigenvalueiterative convergence

Quick facts

Complexity

Runtime and memory behavior at a glance.

Best

O(n^2)

Average

O(iterations * n^2)

Worst

O(iterations * n^2)

Space

O(n)

When to Use

Use when solving equations needs matrix-vector product.
Use when the input matches the assumptions behind eigenvalue Method.

When Not to Use

Avoid when the problem is ill-conditioned and no scaling or stability checks are available.
Avoid when a simpler algorithm gives the required result with less setup.

What It Does

Approximates the dominant eigenvector by repeatedly multiplying by a matrix
Uses matrix-vector product and normalization to guide the work.
Produces approximate numeric result.

Use Cases

Solving equations
Approximating integrals
Working with matrices and differential equations

Inputs

Numeric function, matrix, or sampled data
Initial estimate or discretization
Tolerance

Outputs

Approximate numeric result
Error estimate or residual

Key Formulas (1)

Power iteration

x_next = A x / ||A x||

x_next = A x / ||A x||

Repeatedly normalizes matrix-vector products to approach the dominant eigenvector.

Pseudocode

1.Prepare the inputs and state needed for Power Iteration.
2.Apply the matrix-vector product step while the stopping condition has not been met.
3.Update intermediate state and discard invalid candidates.
4.Return approximate numeric result.

Canonical Example

Input: Power Iteration input with a small representative case
Output: Approximate numeric result
Explanation: Power Iteration applies matrix-vector product to produce approximate numeric result.

Implementation Notes

Check convergence with residuals or estimated error, not iteration count alone.
Keep the matrix-vector product step separate enough to test independently.

Common Pitfalls

Ignoring conditioning and residuals can produce plausible-looking but wrong numeric answers.
Forgetting to check edge cases can make Power Iteration look correct on only ideal inputs.