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Hessenberg - Hessenberg Similarity One of the most...

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Hessenberg Similarity One of the most important condensed forms in numerical linear algebra is the Hessenberg matrix. An upper Hessenberg matrix is almost upper triangular, having zeros below the first subdiagonal. Every square matrix is orthogonally similar to a Hessenberg matrix. There are methods for reducing A R n × n to H = V t AV based on Gram-Schmidt (the Arnoldi method ) and on Householder reflectors (a slight modification to the Householder QR method). These methods are about twice as expensive as their QR analogs (the Householder Hessenberg reduction requires about 10 3 n 3 flops), but the resulting matrix is similar to A . There are several reasons Hessenberg matrices are important, but I would suggest that the following is the most fundamental: Suppose that you wanted to solve ( A - sI ) x = b for m different values of s . Gaussian elimination (or QR ) would require a new factorization for each s , resulting in a cost of O( mn 3 ) flops. But ( A - sI ) x = b V t ( A - sI ) V V t x = V t b ( H - sI ) y = z,
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