QRIteration - QR Iterations Consider the iteration Q i R i...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.

Ask a homework question - tutors are online