Transformed Companion Matrices as a Theoretical Tool in the Numerical Analysis of Differential Equations

Transcription

1 Transformed Companion Matrices as a Theoretical Tool in the Numerical Analysis of Differential Equations Winfried Auzinger Vienna University of Technology (work in progress, with O. Koch and G. Schranz-Kirlinger) CBC Workshop, Széchenyi István University, Györ October 22, 2004 Györ, Oct p.1/18

2 Contents Companion matrix Algorithmic relevance Companion matrix as a theoretical tool Linear Multistep Methods Example: Stability of BDF 2 (model problem, stiff) Bidiagonal and Bidiagonal-Frobenius form, confluent divided differences Stability of Linear Multistep Methods (stiff case) Further applications, Remarks Györ, Oct p.2/18

3 Companion matrix Companion (or Frobenius) matrix (nonderogatory): 0 1 C = C n n 0 1 γ 0 γ 1... γ n 2 γ n 1... associated with characteristic polynomial charpoly(c) = p(ζ) = n in monomial (or Taylor ) representation i=0 γ i ζ i (γ n := 1) Györ, Oct p.3/18

4 Algorithmic relevance Solving the eigenvalue problem for C (lower Hessenberg) gives the zeros of p Monomial representation of p may be numerically unfavorable Use other representations (Lagrange, Newton,... ) transformed companion matrix Literature: Numerical Polynomial Algebra, by Hans J. Stetter (SIAM Press, 2004)... multivariate polynomial systems We do not consider such algorithmic aspects here Rather: Companion matrices as a theoretical tool Györ, Oct p.4/18

5 Companion matrix as a theoretical tool Example 1: n - step Linear Multistep Method characterized by a polynomial p of degree n ( p not identical with stability function ) Companion matrix C represents equivalent one-step method in a higher dimensional space Example 2: Linear ODE of order n Companion matrix C represents equivalent 1st order system In such cases, the characteristic polynomial p is a symbol for the method or the problem Stability estimates reduce to norm estimates for ϕ(c), e.g. ϕ(c) = C ν or ϕ(c) = exp(tc) ( = constant coefficient case) Györ, Oct p.5/18

6 Linear Multistep Methods ODE: y (t) = f(t, y(t)) n - step linear multistep method (stepsize h) n α k y ν+k = h k=0 n β k f(t ν+k, y ν+k ) k=0 ( y ν+k y(t ν+k ) ) Backward Differentiation Formulas (BDF): β k = 0, k < n, and β n = 1 simplest cases ( n = 1, n = 2, A-stable ): y ν + y ν+1 = hf(t ν+1, y ν+1 ) (Backward Euler) 1 2 y ν 2 y ν y ν+2 = hf(t ν+2, y ν+2 ) (BDF 2) Györ, Oct p.6/18

7 Example: Stability of BDF 2 (i) To begin with: model problem y = λ y, Re λ < 0, stiff Note: BDF 2 is A - stable y ν 0 for ν But: Estimates for finite ν not directly available? y ν?... This is not an open problem... Write BDF 2 as a one-step method in C 2 using companion matrix Apply the Kreiss Matrix Theorem, or Estimate using G-stability But: Both approaches have a very restricted scope Györ, Oct p.7/18

8 Example: Stability of BDF 2 (ii) Let µ := hλ, Y ν := (y ν, y ν+1 ) T BDF 2 Y ν+1 = C(µ) Y ν, with companion matrix 0 1 C(µ) = 1 4, EV 1 (Re µ 0) 3 2µ 3 2µ Wanted: Estimate C(µ) 1, uniformly for Re µ 0 For = 2 this does not hold (consider µ = 0) G-stability: Estimate O.K. for Y = G Y, Y 2, G =... But: G-stable n - step LMM for n > 2 not generalizable Kreiss Matrix Theorem: Restricted to const. coeff. Györ, Oct p.8/18

9 Example: Stability of BDF 2 (iii)... look for general, more flexible approach to stability Jordan canonical form for C = C(µ): C is diagonalizable for µ 1/2, C = X Ξ X X = Vandermonde matrix ξ 1 ξ, ξ k = EV of C 2 X becomes singular (confluent) for µ 1/2 Estimate C X Ξ X 1 useless near µ = 1/2 Note ( charpoly(c) = p(ζ) = (ζ ξ 1 )(ζ ξ 2 ) ) : Jordan form of C is discontinuous w.r.t. parameter µ corresponds to Lagrange representation of p for ξ 1 ξ 2 undefined for ξ 1 = ξ 2 ( µ = 1/2 ) Györ, Oct p.9/18

10 Example: Stability of BDF 2 (iv) Our approach: Consider LU-decomposition of X, X = L U, with 1 0 L = ξ 1 1 Then, for all µ : C = L J L 1, with ξ 1 1 J = 0 ξ 2... Bidiagonal canonical form of C continuous w.r.t. parameter µ cond(l) uniformly bounded for Re µ 0 Györ, Oct p.10/18

11 Example: Stability of BDF 2 (v) We can prove: After appropriate diagonal scaling, J J = D J D 1, J 1 Re µ 0 uniform stability estimate C ν K, ν > 0, Re µ 0 but:... does not work for = 2 Generalization: Transform C to Bidiagonal-Frobenius form, C = L H L 1, with appropriate η 1, η 2 and η 1 1 H = p[η 1 ] p[η 1, η 2 ] + η, L = η 1 1 Györ, Oct p.11/18

12 Example: Stability of BDF 2 (vi) We can prove: With η 1 = η 2 = 1 2 trace(c), and after appropriate diagonal scaling, H H = D H D 1, H 2 1 Re µ 0 ( and: scaled version L of L uniformly well-conditioned ) Proof: Apply the Cohn-Schur-criterion to H T H Interpretation in terms of p = charpoly(c) : H is associated with a (scaled) Newton-Taylor representation of p = Newton representation allowed to be confluent (always well-defined) Györ, Oct p.12/18

13 Bidiagonal-Frobenius form (i) For each n n companion matrix C : C = L H L 1, η 1 1 H = η η n 1 1 p [1] p [1 2]... p [1 n 1] p [1 n] +η n η 1,..., η n arbitrary (not necessarily distinct) p [j k]... [confluent] divided differences of p w.r.t. the η j L from LU-decomposition of Vandermonde matrix X = X(η 1,..., η n ) Györ, Oct p.13/18

14 Confluent divided differences η 1,..., η n arbitrary (not necessarily distinct) [Confluent] divided differences (ϕ smooth enough, j k ): ϕ[η j,..., η k ] := ϕ[η j+1,..., η k ] ϕ[η j,..., η k 1 ], η j η k η k η j lim ε 0 ϕ[η j+1,..., η k +ε] ϕ[η j,..., η k 1 ], η j = η k ε Includes all possible combinations of confluent ( derivative-like ) and nonconfluent cases Györ, Oct p.14/18

15 Bidiagonal-Frobenius form (ii) BF-form always well-defined, continuous w.r.t. varying C L (lower triangular): L = x[η 1 ] x[η 1, η 2 ] x[η 1,..., η n ] where x[η] = (1, η,..., η n 1 ) T, x[η j ] = j - th column of X Interpretation in terms of p = charpoly(c) : H associated with Newton-Taylor representation of p = Newton representation allowed to be confluent Györ, Oct p.15/18

16 Bidiagonal-Frobenius form (iii) Special case: η j ξ j = EV of C Bidiagonal form : ξ 1 1 H = J = ξ ξ n 1 1 ξ n Remark: Matrix functions ϕ(j) are upper tridiagonal, with (ϕ(j)) j,k = ϕ[ξ j,..., ξ k ] Trivial case: η j 0 H = C BF-form can be generalized to companion matrix w.r.t. arbitrary basis polynomials Györ, Oct p.16/18

17 Stability of Linear Multistep Methods Approach much more versatile than G-stability or Kreiss Matrix Theorem Bidiagonal canonical form has been used for stability analysis of A(α)-stable BDF methods applied to stiff ODEs y = λ(t)y y = A(t)y + φ(t, y) (A. Eder, G. Schranz-Kirlinger) To be done: Stability/convergence analysis for highly nonlinear problems Györ, Oct p.17/18

18 Further applications, Remarks Sharp growth estimates for higher order ODEs [Numerical] analysis of singular ODE systems: Well-posedness of BVP depends on Jordan structure of boundary conditions... Use of bidiagonal form simplifies the analysis, e.g. for parameter-dependent problems General linear methods: [generalized] companion matrices play a role in the analysis may be worth considering (Example: Butcher, Wright ESIRK methods characterized by doubly companion matrices) General matrices: Frobenius canonical form can be transformed blockwise Györ, Oct p.18/18