Lecture 11 - 11 The QR Algorithm 11.1 QR Algorithm without...

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11 The QR Algorithm 11.1 QR Algorithm without Shifts In the previous chapter (in the Maple worksheet 473 Hessenberg.mws ) we investigated two different attempts to tackling the eigenvalue problem. In the first attempt (which we discarded) the matrix A was multiplied from the left and right by a unitary House- holder matrix Q . We now consider the following Algorithm (“Pure” QR) Initialize A (0) = A for k = 1 , 2 , . . . Compute the QR factors Q ( k ) R ( k ) = A ( k - 1) Reassemble the factors A ( k ) = R ( k ) Q ( k ) end Note that R ( k ) = Q ( k ) T A ( k - 1) and so A ( k ) = Q ( k ) T A ( k - 1) Q ( k ) , the similarity transform from “approach 1” mentioned above. We recall that this algorithm will not produce a triangular (or, in our special real symmetric case, a diagonal) matrix in one step. However, it will succeed in an iterative setting. In fact, we will see below that the “pure” QR algorithm is equivalent to the orthogonal simultaneous iteration algorithm presented earlier. Remark In the next section we will discuss a “practical” QR algorithm that will use shifts and converge cubically like the Rayleigh quotient iteration. We now show the equivalence of the “pure” QR algorithm and orthogonal simulta- neous iteration. Recall the orthogonal simultaneous iteration algorithm: Initialize ˆ Q (0) with an arbitrary m × n matrix with orthonormal columns for k = 1 , 2 , . . . Z = A ˆ Q ( k - 1) Compute the QR factorization ˆ Q ( k ) ˆ R ( k ) = Z end Now consider the special case of square (real symmetric) matrices, which is what we will be working with in eigenvalue problems. We start the orthogonal simultaneous iteration with Q (0) = I R m × m , (S1) 88
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where we no longer need the hat (since the matrices are square), i.e., we will be perform- ing full QR factorizations. The underline is used to distinguish between the Q -matrices appearing in this algorithm and those in the “pure” QR below. Next, we compute Z = AQ ( k - 1) (S2) Q ( k ) R ( k ) = Z (S3) so that A ( k ) = Q ( k ) T AQ ( k ) . (S4) For the “pure” QR algorithm, on the other hand, we have A (0) = A (Q1) Q ( k ) R ( k ) = A ( k - 1) (Q2) A ( k ) = R ( k ) Q ( k ) (Q3) For this algorithm we also define Q ( k ) = Q (1) Q (2) . . . Q ( k ) , (Q4) and for both algorithms R ( k ) = R ( k ) R ( k - 1) . . . R (1) . (R5) Here we make the convention that empty products yield the identity. Theorem 11.1 Orthogonal simultaneous iteration (S1) (S4) and the “pure” QR algo- rithm (Q1) (Q4) are equivalent, i.e., both algorithms produce identical sequences R ( k ) , Q ( k ) and A ( k ) . In fact, the k -th power of A is given by A k = Q ( k ) R ( k ) , (28) and the k -th iterate by A ( k ) = Q ( k ) T AQ ( k ) . (29) Proof The proof uses induction on k .
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