Chapter 12 Solving secular equations

Transcription

1 Chapter 12 Solving secular equations Gérard MEURANT January-February, 2012

2 1 Examples of Secular Equations 2 Secular equation solvers 3 Numerical experiments

3 Examples of secular equations What are secular equations? The term secular comes from the latin saecularis which is related to saeculum, which means century So secular refers to something that is done or happens every century. It is also used to refer to something that is several centuries old It appeared in mathematics to denote equations related to the motion of planets and celestial mechanics For instance, it appears in the title of a 1829 paper of A. L. Cauchy ( ) Sur l équation à l aide de laquelle on détermine les inégalités séculaires des mouvements des planètes (Oeuvres Complètes (IIème Série), v 9 (1891), pp ). There is also a paper by J. J. Sylvester ( ) whose title is On the equation to the secular inequalities in the planetary theory (Phil. Mag., v 5 n 16 (1883), pp )

4 Eigenvalues of a Tridiagonal Matrix We look for an eigenvalue λ and an eigenvector x = ( y ζ ) T of J k+1 where y is a vector of dimension k and ζ is a real number. Then J k y + η k ζe k = λy η k y k + α k+1 ζ = λζ where y k is the last component of y, α j, j = 1,..., k + 1 are the diagonal entries of J k+1 and η j, j = 1,..., k are the subdiagonal entries By eliminating the vector y from these two equations we have (α k+1 η 2 k (ek ) T (J k λi ) 1 e k )ζ = λζ

5 α k+1 η 2 k k j=1 (ξ j ) 2 θ j λ = λ where ξ j = z j k is the kth (i.e., last) component of the jth eigenvector of J k the θ j s are the eigenvalues of J k, that is, the Ritz values Too obtain the eigenvalues of J k+1 from those of J k we have to solve k f (λ) = λ α k+1 + ηk 2 ξj 2 θ j λ = 0 The secular function f has poles at the eigenvalues (Ritz values) of J k for λ = θ j = θ (k) j, j = 1..., k j=1

6 Example of secular function with k = 4

7 Modification by a Rank-One Matrix Assume that we know the eigenvalues of a matrix A and we would like to compute the eigenvalues of a rank-one modification of A We have Ax = λx where we know the eigenvalues λ and we want to compute µ such that (A + cc T )y = µy where c is a given vector (not orthogonal to an eigenvector of A) Clearly µ is not an eigenvalue of A Therefore A µi is nonsingular and we obtain an equation for µ Multiplying by c T to the left, y = (A µi ) 1 cc T y c T y = c T (A µi ) 1 cc T y

8 The secular equation is 1 + c T (A µi ) 1 c = 0 Using the spectral decomposition of A = QΛQ T with Q orthogonal and Λ diagonal and z = Q T c 1 + n j=1 where λ j are the eigenvalues of A (z j ) 2 λ j µ = 0

9 Constrained Eigenvalue Problem We wish to find a vector x of norm one which is the solution of max x x T Ax satisfying the constraint c T x = 0 where c is a given vector We introduce a functional ϕ with two Lagrange multipliers λ and µ corresponding to the two constraints ϕ(x, λ, µ) = x T Ax λ(x T x 1) + 2µx T c Computing the gradient of ϕ with respect to x, which must be zero at the solution, Ax λx + µc = 0 from which we have x = µ(a λi ) 1 c If λ is not an eigenvalue of A and using c T x = 0 we have c T (A λi ) 1 c = 0

10 Using the spectral decomposition of A = QΛQ T and d = Q T c f (λ) = n j=1 d 2 j λ j λ = 0 There are n 1 solutions to the secular equation When we have the values of λ that are solutions, we use the constraint x T x = 1 to remark that x T x = µ 2 c T (A λi ) 2 c = 1 Therefore, and µ 2 = 1 c T (A λi ) 2 c x = µ(a λi ) 1 c

11 Example of secular function

12 Eigenvalue Problem with a Quadratic Constraint We consider the problem min x T Ax 2c T x x with the constraint x T x = α 2 Introducing a Lagrange multiplier and using the stationary values of the Lagrange functional (A λi )x = c, x T x = α 2 Let A = QΛQ T The Lagrange equations can be written as ΛQ T x λq T x = Q T c, x T QQ T x = α 2 Introducing y = Q T x and d = Q T c Λy λy = d, y T y = α 2

13 Assume that all the eigenvalues λ i of A are simple If λ is equal to one of the eigenvalues, say λ j, we must have d j = 0 For all i j we have y i = d i λ i λ Then, there is a solution or not, whether we have ( ) 2 di = α 2 λ i λ j i j or not If λ is not an eigenvalue of A, the inverse of A λi exists and we obtain the secular equation c T (A λi ) 2 c = α 2 which is written as f (λ) = n i=1 ( ) 2 di α 2 = 0 λ i λ

14 Example of secular function

15 Secular equation solvers Consider solving the equation 1 + ρ n j=1 (c j ) 2 d j λ = 0 with ρ > 0 When we look for the solution in the interval ]d i, d i+1 [, we make a change of variable λ = d i + ρt Denoting δ j = (d j d i )/ρ, the secular equation is f (t) = 1 + Note that i j=1 c 2 j δ j t + ψ(t) = n j=i+1 i 1 j=1 c 2 j δ j t cj 2 δ j t c2 i t = 1 + ψ(t) + φ(t) = 0 since δ i = 0 The function ψ has poles δ 1,..., δ i 1, 0 with δ j < 0, j = 1,..., i 1

16 The solution is sought in the interval ]0, δ i+1 [ with δ i+1 > 0 Let us denote δ = δ i+1 In ]0, δ[ we have ψ(t) < 0 and φ(t) > 0 Assume we know all the poles of the secular function f and we are able to compute values of ψ and φ and their derivatives

17 BNS methods Bunch, Nielsen and Sorensen interpolated ψ to first order by a rational function p/(q t) and φ by r + s/(δ t) This is called osculatory interpolation The parameters p, q, r, s are determined by matching the exact values of the function and the first derivative of ψ or φ at some given point t (to the right of the exact solution) where f has a negative value q = t + ψ( t)/ψ ( t) p = ψ( t) 2 /ψ ( t) r = φ( t) (δ t)φ ( t) s = (δ t) 2 φ ( t)

18 For computing them we have and ψ (t) = φ (t) = i j=1 n j=i+1 c 2 j (δ j t) 2 c 2 j (δ j t) 2 Then the new iterate is obtained by solving the quadratic equation 1 + p q t + r + s δ t = 0 This equation has two roots, of which only one is of interest This method is called BNS1

19 Functions ψ (solid) and φ (dots) as function of t

20 To obtain an efficient method it remains to find a good initial guess Melman s strategy is to look for a zero of an interpolant of the function tf (t) When seeking the ith root, this function is written as ( ) tf (t) = t 1 c2 i t + c2 i+1 δ t + h(t) The function h is written in two parts h = h 1 + h 2 h 1 (t) = i 1 j=1 c 2 j δ j t, h 2(t) = n j=i+2 c 2 j δ j t.

21 The functions h 1 and h 2 do not have poles in the interval of concern They are interpolated by rational functions of the form p/(q t) The parameters are found by interpolating the function values at points 0 and δ This gives a function h = h 1 + h 2 The starting point is obtained by computing the zero (in the interval ]0, δ[) of the function ( ) t 1 c2 i t + c2 i+1 δ t + h(t) This can be done with a standard zero finder

22 BNS2 Interpolating ψ by r + s/t and φ by p/( q t), one obtains another method called BNS2 Then, r = ψ( t) (δ t)ψ ( t), s = (δ t) 2 ψ ( t) q = t + φ( t)/φ ( t) p = φ( t) 2 /φ ( t)

23 MW R. C. Li proposed to use r + s/(δ t) to interpolate both functions ψ and φ In fact, we interpolate ψ, which has a pole at 0 by r + s/t and φ, which has a pole at δ by r + s/(δ t) This method is not monotonic, but has quadratic convergence

24 Gragg s method Gragg interpolates f at t to second order with the function This gives a + b/t + c/(δ t) c = (δ t) 3 f + t(δ t) 3 f δ 2δ b = t 3 [ f (δ t) 2f ] 2δ a = f f t 2 (δ t) + (2 t δ)f Then we solve the quadratic equation The convergence is cubic at 2 t(aδ b + c) bδ = 0

25 Numerical experiments 1 + ρ n j=1 (c j ) 2 d j λ = 0 The dimension is n = 4 and d = [1, 1 + β, 3, 4] Defining v T = [γ, ω, 1, 1], we have c T = v T / v β = 1, γ = 10 2, ω = 1 Method No. it. Root 1 f (Root) BNS BNS MW GR

26 β = 1, γ = 1, ω = 10 2 Method No. it. Root 1 f (Root) BNS BNS MW GR

27 β = 10 2, γ = 1, ω = 1 Method No. it. Root 1 f (Root) BNS BNS MW GR

28 J.R. Bunch, C.P. Nielsen and D.C. Sorensen, Rank-one modification of the symmetric eigenproblem, Numer. Math., v 31 (1978), pp Ren-Cang Li, Solving secular equations stably and efficiently, Report UCB CSD , University of California, Berkeley (1994) A. Melman, A unifying convergence analysis of second-order methods for secular equation, Math. Comput., v 66 n 217 (1997), pp A. Melman, A numerical comparison of methods for solving secular equations, J. Comput. Appl. Math., v 86 (1997), pp