QR Algorithm - The Practical QR Algorithm

The Practical QR Algorithm

Formally, let A be a real matrix of which we want to compute the eigenvalues, and let A₀:=A. At the k-th step (starting with k = 0), we compute the QR decomposition A_k=Q_kR_k where Q_k is an orthogonal matrix and R_k is an upper triangular matrix. We then form A_k+1 = R_kQ_k. Note that

so all the A_k are similar and hence they have the same eigenvalues. The algorithm is numerically stable because it proceeds by orthogonal similarity transforms.

Under certain conditions, the matrices A_k converge to a triangular matrix, the Schur form of A. The eigenvalues of a triangular matrix are listed on the diagonal, and the eigenvalue problem is solved. In testing for convergence it is impractical to require exact zeros, but the Gershgorin circle theorem provides a bound on the error.

In this crude form the iterations are relatively expensive. This can be mitigated by first bringing the matrix A to upper Hessenberg form (which costs arithmetic operations using a technique based on Householder reduction), with a finite sequence of orthogonal similarity transforms, somewhat like a two-sided QR decomposition. (For QR decomposition, the Householder reflectors are multiplied only on the left, but for the Hessenberg case they are multiplied on both left and right.) Determining the QR decomposition of an upper Hessenberg matrix costs arithmetic operations. Moreover, because the Hessenberg form is already nearly upper-triangular (it has just one nonzero entry below each diagonal), using it as a starting point reduces the number of steps required for convergence of the QR algorithm.

If the original matrix is symmetric, then the upper Hessenberg matrix is also symmetric and thus tridiagonal, and so are all the A_k. This procedure costs arithmetic operations using a technique based on Householder reduction. Determining the QR decomposition of a symmetric tridiagonal matrix costs operations.

The rate of convergence depends on the separation between eigenvalues, so a practical algorithm will use shifts, either explicit or implicit, to increase separation and accelerate convergence. A typical symmetric QR algorithm isolates each eigenvalue (then reduces the size of the matrix) with only one or two iterations, making it efficient as well as robust.