Inverse Iteration

Definition

Inverse iteration is an iterative method used to find an approximation of an eigenvector corresponding to a particular eigenvalue of a matrix. This technique leverages the inverse of the shifted matrix to enhance the convergence towards the desired eigenvector, making it especially useful when combined with fixed-point iteration. By analyzing the condition number, it also sheds light on the sensitivity of the eigenvalue problem to perturbations in the matrix.

5 Must Know Facts For Your Next Test

1. Inverse iteration often requires selecting a suitable shift, which can greatly influence convergence speed and accuracy.
2. The process involves iteratively solving a linear system, typically using methods like Gaussian elimination or LU decomposition.
3. If the shift is chosen close to an actual eigenvalue, convergence can be significantly faster than other methods.
4. The condition number of the shifted matrix can indicate how well-behaved the inverse iteration will be, with lower values typically leading to more stable solutions.
5. Inverse iteration can be extended to simultaneously find several eigenvalues and their corresponding eigenvectors by employing deflation techniques.

Review Questions

• How does inverse iteration utilize fixed-point iteration principles to approximate eigenvalues and eigenvectors?
  • Inverse iteration builds on fixed-point iteration by reformulating the problem of finding an eigenvector as a fixed-point problem. In this method, we take a guess at an eigenvector and use it to solve for a new vector through the inverse of a shifted matrix. By repeatedly applying this process, we aim to converge on a true eigenvector that remains unchanged under the action of the matrix corresponding to its eigenvalue.
• What role does the condition number play in assessing the effectiveness of inverse iteration?
  • The condition number is crucial in inverse iteration as it measures the sensitivity of the shifted matrix to perturbations. A low condition number suggests that small changes in the input will result in small changes in the output, indicating stable behavior during iterations. Conversely, a high condition number implies potential instability and sensitivity, which can hinder convergence towards an accurate eigenvector during the inverse iteration process.
• Evaluate how the choice of shift impacts both convergence speed and stability in inverse iteration and discuss strategies for optimal shift selection.
  • The choice of shift in inverse iteration is critical because it directly affects both convergence speed and stability. A well-chosen shift that is close to an actual eigenvalue leads to rapid convergence, while a poor choice can result in slow convergence or divergence. Strategies for optimal shift selection include using prior knowledge about the spectrum of the matrix or employing techniques such as Rayleigh quotient or perturbation analysis to estimate suitable shifts that enhance overall performance in finding accurate eigenvectors.

Related terms

Eigenvalue: A scalar associated with a linear transformation represented by a matrix, indicating how much an eigenvector is stretched or compressed during that transformation.

Fixed-point iteration: A method for finding fixed points of a function, where a sequence of iterations converges to a point that remains unchanged when the function is applied.

Condition number: A measure of how sensitive a function or problem is to changes or perturbations in input, particularly important for understanding numerical stability in computations.