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lecture15 - EECS 275 Matrix Computation Ming-Hsuan Yang...

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EECS 275 Matrix Computation Ming-Hsuan Yang Electrical Engineering and Computer Science University of California at Merced Merced, CA 95344 h ttp://faculty.ucmerced.edu/mhyang Lecture 15 1 / 21
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Overview Inverse iteration Rayleigh quotient iteration Condition Perturbation Stability 2 / 21
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Reading Chapter 27 and Chapter 12 of Numerical Linear Algebra by Llyod Trefethen and David Bau Chapter 7 of Matrix Computations by Gene Golub and Charles Van Loan 3 / 21
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Inverse iteration For any μ IR that is not an eigenvalue of A , the eigenvectors of ( A - μ I ) - 1 are the same as the eigenvectors of A , and the the corresponding eigenvalues are ( λ j - μ ) - 1 where λ j are the eigenvalues of A Suppose μ is close to an eigenvalue λ J of A , then ( λ J - μ ) - 1 may be much larger than ( λ j - μ ) - 1 for all j 6 = J If we apply power iteration to ( A - μ I ) - 1 , the process will converge rapidly to q J Algorithm: Initialize v (0) randomly with k v (0) k = 1 for k = 1 , 2 , . . . do Solve w = ( A - μ I ) - 1 v ( k - 1) // apply ( A - μ I ) - 1 v ( k ) = w k w k // normalize λ ( k ) = ( v ( k ) ) > A v ( k ) // Rayleigh quotient end for 4 / 21
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Inverse iteration (cont’d) what if μ is or nearly is an eigenvalue of A , so that A - μ I is singular? can be handled numerically exhibit linear convergence unlike power iteration, we can choose the eigenvector that will be found by supplying an estimate μ of the corresponding eigenvalue if μ is much closer to one eigenvalue of A than to the others, then the largest eigenvalue o ( A - μ I ) - 1 will be much larger than the rest Theorem Suppose λ J is the closest eigenvalue of μ and λ K is the second closest, that is | μ - λ J | < | μ - λ K | < | μ - λ j | for each j 6 = J . Furthermore, suppose q > J v (0) 6 = 0 , then after k iterations k v ( k ) - ( ± q J ) k = O μ - λ J μ - λ K k , | λ ( k ) - λ J | = O μ - λ J μ - λ K 2 k 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 5 / 21
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Inverse iteration and Rayleigh quotient iteration Inverse iteration: I one of the most valuable tools of numerical linear algebra I a standard method of calculating one or more eigenvectors of a matrix if the eigenvalues are already known Obtaining an eigenvalue estimate from an eigenvector estimate (the Rayleigh quotient) Obtaining an eigenvector estimate from an eigenvalue estimate (inverse iteration) Rayleigh quotient algorithm: Initialize v (0) randomly with k v (0) k = 1 λ (0) = ( v (0) ) > A v (0) = corresponding Rayleigh quotient for k = 1 , 2 , . . . do Solve w = ( A - λ ( k - 1) I ) - 1 v ( k - 1) // apply ( A - μ ( k - 1) I ) - 1 v ( k ) = w k w k // normalize λ ( k ) = ( v ( k ) ) > A v ( k ) // Rayleigh quotient end for 6 / 21
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Rayleigh quotient iteration Theorem Rayleigh quotient iteration converges to an eigenvalue/eigenvector pair for all except a set of measure zero of starting vectors v (0) . When it converges, the convergence is ultimately cubic in the sense that if λ J is an eigenvalue of A and v (0) is sufficiently close to the eigenvector q J , then
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