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Unformatted text preview: QR Iterations Consider the iteration Q i R i ← A i A i +1 ← R i Q i Here we have first computed the QR factorization of A i , and then reversed their product to form A i +1 . From A i = Q i R i we have R i = Q t i A i , and substituting that into A i +1 = R i Q i gives A i +1 = Q t i A i Q i and thus the QR step is a similarity transformation! If the eigenvalues of A are all real, then this iteration almost always converges to an upper triangular matrix T . In this limit, the eigenvalues of T (and hence A , right?) are t 11 ,t 22 ,...t nn . T is called a Schur form for A and the eigenvectors of T are Schur vectors of A . Every matrix is unitarily similar to an upper triangular matrix, and T = Q * AQ is called a Schur decomposition of A . As it stands, this QR iteration requires O( n 3 ) flops per iteraton. We can reduce this by an order of magnitude by first reducing A to Hessenberg form H = Q t AQ . The following iteration preserves the Hessenberg form, and if we use a Householder (or...
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This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.
- Fall '10