Perturbation Theory , B =

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1 Appendix D Perturbation Theory D. Simple Examples Let A = 3, B = 4 6. (D.) The eigenvalues of A are, and 3, where λ = has multiplicity. To find the eigenvalue of the perturbed matrix A + ǫb corresponding to the unperturbed eigenvalue λ =, we use non-degenerate perturbation theory, so λ = λ + ǫλ + O ( ǫ ) (D.) = + ǫ x T Bx x T x + O ( ǫ ) (D.3) = + ǫ [ ] 4 6 [ ] + O ( ǫ ) (D.4) = + 4ǫ + O ( ǫ ), (D.5) noting that A = A T, so x, the eigenvector of AT with eigenvalue, is the same as x, the eigenvector of A with eigenvalue. We can do likewise for λ = 3 to find λ = 3 + 6ǫ + O ( ǫ ). (D.6) 3

2 For λ =, we must use degenerate perturbation theory, which says that Bc = λ Mc (D.7) where so equation D.7 becomes [ B ij v T i Bv j M ij v T i v j, (D.8) ] c = λ [ hence λ is i or i, so the perturbed eigenvalues for λ = are ] c, (D.9) λ = + ǫ { i i + O ( ǫ ) (D.) In summary, the eigenvalues of A + ǫb are ± iǫ λ = + 4ǫ + O ( ǫ ) (D.) 3 + 6ǫ These are compared to numerically calculated eigenvalues in figure D.. If instead, B = 4, (D.) 6 then perturbation theory says that for ǫ small, the eigenvalues of A + ǫb are ± ǫ λ = + 4ǫ + O ( ǫ ) (D.3) 3 + 6ǫ These are compared to numerically calculated eigenvalues in figure D.. D. Wilkinson s Matrix We now choose to look at an example where perturbation theory is very poor at approximating the perturbed eigenvalues. Following Wilkinson [3], we let 9 8 A = (D.4) 4

3 Real Component Real Component of First Eigenvalue Imag. Component Imag. Component of st Eigenvalue...5. Real Component of Second Eigenvalue. Real Component Imag. Component of nd Eigenvalue. Imag. Component..5. Real Component Real Component of Third Eigenvalue Imag. Component of 3rd Eigenvalue. Imag. Component..5. Real Component of Fourth Eigenvalue 3. Real Component Imag. Component of 4th Eigenvalue. Imag. Component..5. calculated by MATLAB predicted by perturbation theory Figure D.: The real (first column) and imaginary (second column) components of the four eigenvalues of A + ǫb, plotted versus ǫ, with A and B defined by equation D.. 5

4 Eigenvalue Eigenvalue Eigenvalue Eigenvalue. First Eigenvalue Second Eigenvalue Third Eigenvalue Fourth Eigenvalue.5. calculated by MATLAB predicted by perturbation theory Figure D.: The four eigenvalues (all real) of A+ǫB, plotted versus ǫ, where B is defined by equation D.. 6

5 This has eigenvalues,,..., and the eigenvector corresponding to eigenvalue λ is ( ) λ ( λ)!. ( )! (D.5) }. λ zeroes The eigenvector of A T corresponding to eigenvalue λ is D.. Let Eigenvalues. ()!.. () λ (λ )! } λ zeroes B =..... (D.6) (D.7) Then perturbation theory says that the eigenvalues of A+ǫB are given, to order ǫ, by λ n = λ n + ǫλ n = n + ǫ x T Bx x T x 9 ( ) n = n + ǫ ( n)!(n )! (D.8) (D.9) (D.) or, if we generalise A and B to be m by m matrices, λ n = n + ǫ mm ( ) m n (m n)!(n )! (D.) 7

6 Real Component of Eigenvalue Real Component of Tenth Eigenvalue 5 5 predicted range of validity of perturbation theory calculated by MATLAB predicted by perturbation theory Figure D.3: The th eigenvalue in the m = case, as a function of ǫ, as computed numerically by MATLAB. The eigenvalue predicted by perturbation theory is also shown. The predicted range gives the number η for which we must have ǫ η in order for perturbation theory to be valid, as in equation D.. where this is valid for m m ǫ. (D.) (m n)!(n )! For example, when m = 5, perturbation theory predicts λ 8 = 8 + ǫ (D.3) so we expect this to be valid for ǫ 8. The eigenvalues in the m = case, as a function of ǫ, were computed numerically in MATLAB and are shown in figures D.4 D.9. Also, the th eigenvalues in the m = case, as a function of ǫ, is compared to that predicted by perturbation theory. This is shown in figure D.3. Error in Computation Since the Wilkinson matrix has such poorly behaved eigenvalues, we expect that numerical computations of its perturbed eigenvalues will be difficult. To illustrate this, we compute the eigenvalues of the perturbed Wilkinson matrix for different values of m and ǫ using two different programs, MATLAB and Mathematica. The calculations were performed to 5 digits. The result is shown 8

7 Real Component of Eigenvalues Real Component of Eigenvalues Imaginary Component of Eigenvalues Eigenvalues 5 5 Real Component of Eigenvalues Imaginary Component of Eigenvalues Imaginary Component of Eigenvalues predicted range of validity of perturbation theory calculated by MATLAB Figure D.4: The eigenvalues in the m = case, as a function of ǫ, as computed numerically by MATLAB. The predicted range gives the number η, as a function of λ, for which we must have ǫ η in order for perturbation theory to be valid, as in equation D.. 9

8 Eigenvalues Imaginary Component 5 Real Component 3 Figure D.5: A 3D representation of the eigenvalues in the m = case, as a function of ǫ, as computed by MATLAB.

9 Size of Matrix Difference Between Mathematica and MATLAB Eigenvalues Region Where Eigenvalues Become All Real e+ e e 4 e 6 e 8 e e e 4 Figure D.6: The difference between the computed eigenvalues of A + ǫb according to MATLAB and Mathematica, as a function of m and ǫ. The dashed lines indicate the region in which the eigenvalues, according to Mathematica, become entirely real.

10 Size of Matrix e+ e e 4 e 6 e 8 e 5 e e Difference Between Mathematica and Perturbation Theory Eigenvalues Region Where Eigenvalues Become All Real Figure D.7: The difference between the computed eigenvalues of A + ǫb according to Mathematica and the predicted eigenvalues from perturbation theory, as a function of m and ǫ. The dashed lines indicate the region in which the eigenvalues, according to Mathematica, become entirely real. This also corresponds to the approximate boundary of the region of validity of perturbation theory.

11 Size of Matrix e+ e e 4 e 6 e 8 e 5 e e Difference Between MATLAB and Perturbation Theory Eigenvalues Region Where Eigenvalues Become All Real Figure D.8: The difference between the computed eigenvalues of A + ǫb according to MATLAB and the predicted eigenvalues from perturbation theory, as a function of m and ǫ. The dashed lines indicate the region in which the eigenvalues, according to Mathematica, become entirely real. This also corresponds to the approximate boundary of the region of validity of perturbation theory. 3

12 in figure D.6. More precisely, the plot is of the maximum of the absolute value of the complex difference between the eigenvalues as computed by MATLAB and Mathematica. To determine which is most accurate, we also plot each as compared with perturbation theory (figures D.7 and D.8). Observe that, within the region of validity of perturbation theory, Mathematica agrees very well with perturbation theory. We assume then that the error depicted in figure D.7 is due to errors from perturbation theory, not Mathematica. D.. Sherman-Woodbury-Morrison Formula If we wish to solve the related problem Ax + ǫbx = b, with A and B as above, then perturbation theory predicts x = ( A A ǫba ) b. (D.4) Since for our choice of B, we can write B = uv T, then we can apply the Sherman- Woodbury-Morrison formula which says that the exact solution is x = (A A ǫba ) + ǫv T A b, (D.5) u so we expect perturbation theory to be valid for ǫv T A u. Now 9!! ( ) 8!! ( ) 7! 9!! ( )! 8! 9! ( ) 7! 9! A 7! = 8!... so v T A u = ( )9! for m =. We can generalise this for any m to obtain!! (D.6) (D.7) We then expect perturbation theory to be valid for v T A u = ( m)m. (D.8) m! ǫ ( v T A u ) (D.9) ( ) ( m) m ǫ. m! (D.3) For example, when m = 5, this means that perturbation theory is valid for ǫ 9. 4

13 Size of Perturbation Imaginary Component 5 pseudospectrum eigenvalues of perturbed matrix Real Component 3 Figure D.9: A 3D version of the pseudospectrum for the unperturbed matrix (the surface), in the m = case, in comparison with the eigenvalues of the perturbed matrix (the lines), as computed by MATLAB. The lines in this figure are identical to those in figure D.5. D..3 Pseudospectrum Following Embree and Trefethen [7], we define the pseudospectrum of A to be Λ ǫ (A) = {z C : z Λ(A + E) for some E with E ǫ} (D.3) where Λ(A + E) is the set of eigenvalues of the matrix (A + E) and is a matrix norm induced by a vector norm. We choose to be the -norm. Since ǫb = ǫ, then the eigenvalues of the perturbed matrix A + ǫb are elements of Λ ǫ (A). The pseudospectrum of A in the m = case is shown in figures D.9 and D.. Note that the pseudospectrum of A agrees very well with the eigenvalues of A + ǫb as computed by MATLAB, although the eigenvalues of A + ǫb are all strictly within the boundary predicted by the pseudospectrum. This is especially visible in figure D.. This suggests that there exist some matrices C with C = such that the eigenvalues of A + ǫc are slightly further away from the unperturbed eigenvalues than the eigenvalues of A + ǫb. 5

14 5 4 Imaginary Component Real Component eigenvalues of unperturbed matrix eigenvalues of perturbed matrix contour plot of pseudospectrum 3 Figure D.: The pseudospectrum for the unperturbed matrix, in the m = case. 6

15 D.3 Domain Perturbations Consider now the first order corrections to the eigenvalues of a differential equation where the domain has been perturbed by some perturbation of order ǫ. We look in specific at the problem φ + λφ = in Ω ǫ (D.3a) φ = on δω ǫ (D.3b) where and δω ǫ : R = + ǫf (θ) (D.33) f (θ) = n= a n e inθ, a n = a n. (D.34) We will now find the first order corrections to the eigenvalues by two different methods, and show that they produce the same result. D.3. General Method For the first method, we start by writing φ = φ + ǫφ + ǫ φ +... λ = λ + ǫλ + ǫ λ +... (D.35) (D.36) and equating terms of equal order from equation D.3a, we obtain that O () : φ + λ φ = (D.37) O (ǫ) : φ + λ φ = λ φ (D.38) L(φ) = λ φ. (D.39) Performing a Taylor expansion of the boundary conditions, = φ( + ǫf (θ),θ) (D.4) = φ(,θ) + ǫf (θ) φ,r (,θ) + ǫ f (θ) φ,rr (,θ) and equating terms of equal order, (D.4) O () : = φ (,θ) O (ǫ) : = φ (,θ) + f (θ) φ,r (,θ) (D.4) (D.43) 7

16 First Eigenvalue The unperturbed eigenfunction of the first eigenvalue is φ = J (j r) (D.44) where j mk is the kth positive zero of the Bessel function J m. Now we know that (φ, L(φ )) (φ, L(φ )) = (φ φ,r φ φ,r ) ds (D.45) where (f,g) D D fg dx. Using equations D.37, D.39, D.4 and D.43, equation D.45 becomes λ (φ,φ ) = f (θ)(φ,r ) ds so λ = = = j D D f (θ)(φ,r) ds D (φ ) dx ( an e inθ) (J (j ) j ) dθ θ=π θ= n= θ=π dθ r= θ= r= r (J (j r)) dθ = (J (j )) j θ=π ( a + ( θ= n= an e inθ + a n e inθ)) dθ π (J (j )) ( θ=π a + (Re (a n ) cos (nθ) + θ= n= Im (a n ) sin (nθ)) ) dθ (D.46) (D.47) (D.48) (D.49) (D.5) (D.5) = j a. (D.5) The first eigenvalue is then λ = j ǫj a + O ( ǫ ). (D.53) Second and Third Eigenvalues The second and third eigenvalues of the unperturbed system are the same, so we have φ = c v + c v (D.54) where c and c are arbitrary constants and v = J (j r) cos (θ), v = J (j r) sin (θ). (D.55) 8

17 As before, we can use (v, L(φ )) (φ, L(v )) = D (v φ,r φ v,r ) ds, (D.56) which, upon applying equations D.37, D.39, D.4 and D.43, becomes λ (v,φ ) = f (θ) φ,r v,r ds (D.57) D λ c (v,v ) λ c (v,v ) = c f v,r ds + c f v,r v,r ds. (D.58) D We can do likewise, with v in equation D.56 replaced with v to obtain λ c (v,v ) λ c (v,v ) = c f v,r v,r ds + c f v,r ds. (D.59) D We can express these two equations together as [ ][ ] f v,r f v,r v,r c f v,r v,r f v,r c D D = λ [ (v,v ) (v,v ) (v,v ) (v,v ) ] [ c (D.6) Using the definition of v and v from equation D.55 and the result that r= r= r (J (j r)) dr = J (j ) J (j ) this simplifies to [ ] [ f v,r f v,r v,r c f v,r v,r f v,r c ] c ]. = (J (j )), (D.6) = π (J (j )) λ [ c c ]. (D.6) If we re-write this as [ A B B C ] [ c c ] [ c = Q c ] (D.63) then the eigenvalues are given by Q = A + C ± A + C AC + 4B, (D.64) 9

18 where A = = D θ=π θ= f (θ) (v,r (,θ)) ds (D.65) ( a n e )(J inθ (j ) j cos (θ)) dθ (D.66) = (J (j )) j = (J (j )) j n= θ=π θ= θ=π θ= ( ( a + a + ( an e inθ + a n e inθ)) cos (θ) dθ (D.67) n= (Re (a n ) cos (nθ) n= +Im (a n ) sin(nθ)) ) cos (θ) dθ (D.68) = (J (j )) jπ (Re(a ) + a ) (D.69) A + ξ. (D.7) We likewise obtain that C = f (θ) (v,r (,θ)) ds (D.7) D ( θ=π = a n e )(J inθ (j ) j sin(θ)) dθ (D.7) θ= n= = (J (j )) jπ ( Re (a ) + a ) (D.73) = A + ξ (D.74) and B = = D θ=π θ= f (θ)v,r (,θ) v,r (,θ) ds (D.75) ( a n e )(J inθ (j ) j ) sin(θ)cos (θ) dθ (D.76) n= = (J (j )) j π Im(a ), (D.77) so that Q = A + C ± A + C AC + 4B = ξ ± A + B. (D.78) (D.79)

19 From the definition of Q, we obtain that λ π (J (j )) = Q (D.8) π λ (J (j )) = (J (j )) ja π ± (J (j )) jπ a (D.8) λ = j (± a a ) (D.8) So the second eigenvalue is and the third is D.3. λ = j + ǫj ( a a ) + O ( ǫ ) λ = j + ǫj ( a a ) + O ( ǫ ). Method of Assumed Solution (D.83) (D.84) For this method, we follow Wolf and Keller [3]. We start by assuming that the solution is of the form φ(r,θ,ǫ) = A n (ǫ) J n (kr) e inθ (D.85) where n= A n (ǫ) = δ n m α n + ǫβ n + ǫ γ n +... R (θ,ǫ) = + ǫ n a n e inθ + ǫ n b n e inθ + O ( ǫ 3) (D.86) (D.87) λ k = k + ǫk + ǫ k +... Applying boundary conditions, φ(r (θ,ǫ),θ,ǫ) = A n (ǫ) J n (kr (θ,ǫ))e inθ =. n (D.88) (D.89) Expanding this in a Taylor series, and writing kr = k + (k k ) + k (R ), we obtain A n {J n (k ) + J n (k ) (k k + k (R )) + n } J n (k ) (k k + k (R )) +... e inθ =. (D.9) Applying equations D.86 D.88 and equating terms of equal order, we obtain that O () : ( α m e imθ + α m e imθ) J m (k ) = O (ǫ) : ( ( α m J me imθ + α m J me imθ) k + k a l e )+ ilθ l β n J n e inθ =. n (D.9) (D.9)

20 From equation D.9, we obtain that k = j mp, so m =, p = corresponds to the first eigenvalue and m =, p = will give the second and third eigenvalues. First Eigenvalue Under the condition that m =, equation D.9 becomes ( ) = α J k + k a l e ilθ + β n J n e inθ. n Equating terms that have θ =, we obtain l (D.93) = α J (j ) (k + k a ) + β J (j ) (D.94) = α J (j ) (k + k a ) (D.95) = k + k a, (D.96) so and k k = a (D.97) λ = k which agrees with equation D.53. Second and Third Eigenvalues (D.98) = k + ǫk k + O ( ǫ ) (D.99) = j ǫj a + O ( ǫ ), (D.) Again following Wolf and Keller [3], we equate the coefficients of e imθ in equation D.9 to obtain and = α m J m (k ) k a m + α m J m (k ) (k + k a ) + β m J m (k ) (D.) = α m J m (k ) k a m + α m J m (k ) (k + k a ) (D.) = α m k a m + α m (k + k a ) (D.3) k k = α ma m α m a. (D.4) Alternately, equating the coefficients of e imθ in equation D.9, we obtain Equating equations D.4 and D.5, we obtain that α m a m α m eiω = α m a m α m k k = α ma m α m a. (D.5) e iω (D.6)

21 where so that ( ) αm a m ω = arg α m ( ) αm a m = arg α m e iω = e iω e iω = ±. (D.7) (D.8) (D.9) (D.) It then follows that k = ± α m a m k α m a (D.) = ± a m a, (D.) so the second and third eigenvalues are λ = j + ǫj (± a a ) + O ( ǫ ) (D.3) which agrees with equations D.83 and D.84. D.3.3 Result Similarly, Wolf and Keller [3] compute the second order correction to the first eigenvalue. Then under the condition that the area of the domain does not change under the perturbation (which causes a =, among other things), they obtain that Aλ II = πj π/ j j [ Aλ I πj ] / + j J (j +O [ Aλ I πj ) /J (j ) ]. (D.4) where A is the area of the domain, and λ I and λ II are the first and second eigenvalues, respectively. Normalising A to unity, we obtain the plot shown in figure D.. 3

22 8 Second Eigenvalue (λ II ) First Eigenvalue (λ I ) Figure D.: The first and second eigenvalues for perturbations to a circle. When there are no perturbations, λ I is at a minimum. 4