Numerical Analysis Questions Medium
The QR algorithm is an iterative method used to compute eigenvalues and eigenvectors of a square matrix. It is based on the QR decomposition of a matrix, where a matrix A is decomposed into the product of an orthogonal matrix Q and an upper triangular matrix R.
The QR algorithm starts by taking an initial matrix A and decomposing it into Q and R. Then, it repeatedly applies the QR decomposition to the resulting R matrix, until R becomes upper triangular and converges to a diagonal matrix. The diagonal elements of the final R matrix are the eigenvalues of the original matrix A.
To compute the eigenvectors, the algorithm uses the fact that the eigenvectors of a matrix are the same as the eigenvectors of its upper triangular form. Therefore, during each iteration, the algorithm accumulates the orthogonal transformations from the QR decompositions and applies them to an initial set of eigenvector estimates. This process is known as the implicit QR iteration.
The QR algorithm can be summarized in the following steps:
1. Start with an initial matrix A.
2. Compute the QR decomposition of A: A = QR.
3. Compute the product RQ to obtain a new matrix A'.
4. Repeat steps 2 and 3 until A' becomes upper triangular or until convergence is achieved.
5. Extract the diagonal elements of the final upper triangular matrix as the eigenvalues of A.
6. Use the accumulated orthogonal transformations to compute the eigenvectors corresponding to the eigenvalues.
The QR algorithm is known for its robustness and efficiency in computing eigenvalues and eigenvectors, even for matrices that are ill-conditioned or have complex eigenvalues. It is widely used in various fields, including physics, engineering, and computer science, for solving eigenvalue problems and analyzing the behavior of linear systems.