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  • Appendix D

    Perturbation Theory

    D.1 Simple Examples

    Let

    A =

    1 0 0 0 0 1 0 0 0 0 2 0 0 0 0 3

    , B =

    0 1 10 10 −1 0 10 10 10 10 4 10 10 10 10 6

    . (D.1)

    The eigenvalues of A are 1, 2 and 3, where λ = 1 has multiplicity 2. To find the eigenvalue of the perturbed matrix A + ǫB corresponding to the unperturbed eigenvalue λ0 = 2, we use non-degenerate perturbation theory, so

    λ = λ0 + ǫλ1 + O (

    ǫ2 )

    (D.2)

    = 2 + ǫ x∗T0 Bx0 x∗T0 x0

    + O (

    ǫ2 )

    (D.3)

    = 2 + ǫ

    [

    0 0 1 0 ]

    0 1 10 10 −1 0 10 10 10 10 4 10 10 10 10 6

    0 0 1 0

    [

    0 0 1 0 ]

    0 0 1 0

    + O (

    ǫ2 )

    (D.4)

    = 2 + 4ǫ + O (

    ǫ2 )

    , (D.5)

    noting that A = AT , so x∗0, the eigenvector of A T with eigenvalue 2, is the same

    as x0, the eigenvector of A with eigenvalue 2. We can do likewise for λ0 = 3 to find

    λ = 3 + 6ǫ + O (

    ǫ2 )

    . (D.6)

    203

  • For λ0 = 1, we must use degenerate perturbation theory, which says that

    Bc = λ1Mc (D.7)

    where

    Bij ≡ v∗Ti Bvj Mij ≡ v∗Ti vj , (D.8)

    so equation D.7 becomes [

    0 1 −1 0

    ]

    c = λ1

    [

    1 0 0 1

    ]

    c, (D.9)

    hence λ1 is i or −i, so the perturbed eigenvalues for λ0 = 1 are

    λ = 1 + ǫ

    {

    i −i + O

    (

    ǫ2 )

    (D.10)

    In summary, the eigenvalues of A + ǫB are

    λ =

    1 ± iǫ 2 + 4ǫ 3 + 6ǫ

    + O (

    ǫ2 )

    (D.11)

    These are compared to numerically calculated eigenvalues in figure D.1. If instead,

    B =

    1 0 10 10 0 −1 10 10 10 10 4 10 10 10 10 6

    , (D.12)

    then perturbation theory says that for ǫ small, the eigenvalues of A + ǫB are

    λ =

    1 ± 1ǫ 2 + 4ǫ 3 + 6ǫ

    + O (

    ǫ2 )

    (D.13)

    These are compared to numerically calculated eigenvalues in figure D.2.

    D.2 Wilkinson’s Matrix

    We now choose to look at an example where perturbation theory is very poor at approximating the perturbed eigenvalues. Following Wilkinson [31], we let

    A =

    20 20 19 20 0

    18 20 . . .

    . . .

    0 2 20 1

    . (D.14)

    204

  • 0 0.005 0.01 0.98

    1

    1.02 Real Component of First Eigenvalue

    R ea

    l C om

    po ne

    nt

    Size of Perturbation (ε) 0 0.005 0.01

    −0.01

    0

    0.01 Imag. Component of 1st Eigenvalue

    Im ag

    . C om

    po ne

    nt

    Size of Perturbation (ε)

    0 0.005 0.01 0.98

    1

    1.02 Real Component of Second Eigenvalue

    R ea

    l C om

    po ne

    nt

    Size of Perturbation (ε) 0 0.005 0.01

    −0.01

    0

    0.01 Imag. Component of 2nd Eigenvalue

    Im ag

    . C om

    po ne

    nt

    Size of Perturbation (ε)

    0 0.005 0.01 1.98

    2

    2.02

    2.04

    Real Component of Third Eigenvalue

    R ea

    l C om

    po ne

    nt

    Size of Perturbation (ε) 0 0.005 0.01

    −0.01

    0

    0.01 Imag. Component of 3rd Eigenvalue

    Im ag

    . C om

    po ne

    nt

    Size of Perturbation (ε)

    0 0.005 0.01

    3

    3.05

    3.1 Real Component of Fourth Eigenvalue

    R ea

    l C om

    po ne

    nt

    Size of Perturbation (ε) 0 0.005 0.01

    −0.01

    0

    0.01 Imag. Component of 4th Eigenvalue

    Im ag

    . C om

    po ne

    nt

    Size of Perturbation (ε)

    calculated by MATLAB

    predicted by perturbation theory

    Figure D.1: 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.1.

    205

  • 0 0.005 0.01 0.98

    1

    1.02 First Eigenvalue

    E ig

    en va

    lu e

    Size of Perturbation (ε)

    0 0.005 0.01 0.98

    1

    1.02 Second Eigenvalue

    E ig

    en va

    lu e

    Size of Perturbation (ε)

    0 0.005 0.01 1.98

    2

    2.02

    2.04

    Third Eigenvalue

    E ig

    en va

    lu e

    Size of Perturbation (ε)

    0 0.005 0.01

    3

    3.05

    3.1 Fourth Eigenvalue

    E ig

    en va

    lu e

    Size of Perturbation (ε)

    calculated by MATLAB

    predicted by perturbation theory

    Figure D.2: The four eigenvalues (all real) of A+ ǫB, plotted versus ǫ, where B is defined by equation D.12.

    206

  • This has eigenvalues 1, 2, . . . , 20 and the eigenvector corresponding to eigenvalue λ is

    (−20)20−λ

    (20−λ)!

    ... (−20)2

    2! −20 1 0 ... 0

    }

    λ − 1 zeroes

    (D.15)

    The eigenvector of AT corresponding to eigenvalue λ is

    0 ... 0 1 20

    (20)2

    2! ...

    (20)λ−1

    (λ−1)!

    }

    20 − λ zeroes

    (D.16)

    D.2.1 Eigenvalues

    Let

    B =

    0 0 ... . .

    .

    0 0 1 0 · · · 0

    . (D.17)

    Then perturbation theory says that the eigenvalues of A+ǫB are given, to order ǫ, by

    λn = λ0n + ǫλ1n (D.18)

    = n + ǫ x∗T0 Bx0 x∗T0 x0

    (D.19)

    = n + ǫ 2019 (−1)n

    (20 − n)! (n − 1)! (D.20)

    or, if we generalise A and B to be m by m matrices,

    λn = n + ǫ mm−1 (−1)m−n (m − n)! (n − 1)! (D.21)

    207

  • 10 −15

    10 −10

    10 −5

    10

    10.1

    10.2

    10.3

    10.4

    10.5

    10.6

    Size of Perturbation (ε)

    R ea

    l C om

    po ne

    nt o

    f E ig

    en va

    lu e

    Real Component of Tenth Eigenvalue

    predicted range of validity of perturbation theory calculated by MATLAB predicted by perturbation theory

    Figure D.3: The 10th eigenvalue in the m = 20 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.22.

    where this is valid for

    ǫ mm−1

    (m − n)! (n − 1)! ≪ 1. (D.22)

    For example, when m = 50, perturbation theory predicts

    λ8 = 8 + ǫ · 2.5085 × 1028 (D.23)

    so we expect this to be valid for ǫ ≪ 10−28. The eigenvalues in the m = 20 case, as a function of ǫ, were computed

    numerically in MATLAB and are shown in figures D.4–D.9. Also, the 10th eigenvalues in the m = 20 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 15 digits. The result is shown

    208

  • 10 −15

    10 −10

    10 −5

    −5

    0

    5

    10

    15

    20

    25 Real Component of Eigenvalues

    R ea

    l C om

    po ne

    nt o

    f E ig

    en va

    lu es

    Size of Perturbation (ε) 0 10 20

    −10

    −5

    0

    5

    10 Eigenvalues

    Im ag

    in ar

    y C

    om po

    ne nt

    o f E

    ig en

    va lu

    es

    Real Componen