Transcription of The QR Algorithm
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The QR Algorithm
Chapter 4. The QR Algorithm The QR Algorithm computes a Schur decomposition of a matrix. It is certainly one of the most important Algorithm in eigenvalue computations [9]. However, it is applied to dense (or: full) matrices only. The QR Algorithm consists of two separate stages. First, by means of a similarity transformation, the original matrix is transformed in a finite number of steps to Hessenberg form or in the Hermitian/symmetric case to real tridiagonal form. This first stage of the Algorithm prepares its second stage, the actual QR iterations that are applied to the Hessenberg or tridiagonal matrix. The overall complexity (number of floating points) of the Algorithm is O(n3 ), which we will see is not entirely trivial to obtain. The major limitation of the QR Algorithm is that already the first stage generates usually complete fill-in in general sparse matrices.
called LU factorization) is not stable without pivoting. Francis [5] noticed that the QR factorization would be the preferred choice and devised the QR algorithm with many of the bells and whistles used nowadays. Before presenting the complete picture, we start with a basic iteration, given in Algo-
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