# The sign means that at each step k one or the other

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Unformatted text preview: ubic in the sense that if λJ is an eigenvalue of A and v(0) is suﬃciently close to the eigenvector qJ , then v(k +1) − (±qJ ) = O ( v(k ) − (±qJ ) 3 ) and |λ(k +1) − λJ | = O (|λ(k ) − λJ |3 ) as k → ∞. The ± sign means that at each step k , one or the other choice of sign is to be taken, and then the indicated bound holds. 7 / 21 Example Consider the symmetric matrix 211 A = 1 3 1 114 √ and let v(0) = (1, 1, 1) / 3 the initial eigenvector estimate When Rayleigh quotient iteration is applied to A, the following values λ(k ) are computed by the ﬁrst 3 iterations: λ(0) = 5, λ(1) = 5.2131 . . . , λ(2) = 5.214319743184 . . . The actual value of the eigenvalue corresponding to the eigenvector closest to v(0) is λ = 5.214319743377 After three iterations, Rayleigh quotient iteration has produced a result accurate to 10 digits With three more iterations, the solution increases this ﬁgure to about 270 digits, if our machine precision were high enough 8 / 21 Operation counts Flops (ﬂoating point operations): addition, subtraction, multiplication, division, or square root counts as one ﬂop Suppose A ∈ IRm×m is a full matrix, each step of power iteration involves a matrix-vector multiplication, requiring O (m2 ) ﬂops Each step of inverse iteration involves the solution of a linear system, which might seem to require O (m3 ) ﬂops, but this can be reduced to O (m2 ) if the matrix is processed in advance by LU or QR factorization In Rayleigh quotient iteration, the matrix to be inverted changes at each step, and beating O (m3 ) ﬂops per step These ﬁgures improve greatly if A is tridiagonal, and all three iterations requires just O (m) ﬂops per step For non-symmetric matrices, we must deal with Hessenberg instead of tridiagona...
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