Nick Higham. Director of Research School of Mathematics

Transcription

1 Exploiting Research Tropical Matters Algebra in Numerical February 25, Linear 2009 Algebra Nick Higham Françoise Tisseur Director of Research School of Mathematics The School University of Mathematics of Manchester Joint work with James Hook, Vanni Noferini, Meisam Sharify, Jennifer Pestana Householder Symposium XIX, June / 6

2 Tropical Semirings By tropical" we refer to a semiring in which the addition operation is min or max. In this talk, consider max-plus semiring (R max,, ), where R max = R { }, a b = max(a, b), a b = a + b, a, b R max, and additive and multiplicative identities and 0: a = a, a 0 = a. Tropical algebra is the tropical analogue of linear algebra, working with matrices with entries in R max. If A, B R n n (A B) ij = a ij b ij, (A B) ij = n a ik b kj. k=1 max, Françoise Tisseur Tropical algebra 2 / 20

3 When/How Can Tropical Algebra Help NLA? When? Tropical algebra can help NLA when there are large variations in the magnitude of the data. How? By providing order of magnitude approximation to roots, modulus of ei vals and singular values. Offer good starting points for iterative algorithms. Can help to reduce condition numbers/backward errors. error in solution < condition number backward error. Françoise Tisseur Tropical algebra 3 / 20

4 Tropicalization" of Linear Algebra Problems We use valuations (provide a measure of size or multiplicity of elements of the field). E.g., x C V(x) = log x R max (log 0 = ). When a b or a b with a, b C, V(a + b) = log a + b V(ab)= log ab max(log a, log b ) = log a + log b = V(a) V(b), = V(a) V(b). Tropicalized linear algebra problems can be easier/cheaper to solve and, does not suffer much from numerical instabilities. Françoise Tisseur Tropical algebra 4 / 20

5 Scalar Polynomials: Classical/Max-Plus Tropicalize" p(x) = d i=0 a ix i, a i C, i.e., construct tp(x) = d i=0 log a i x i = max 0 i d (log a i + ix). Let α 1 < < α p be roots of tp with α j of multiplicity m j. Theorem (Sharify 11) If max(α j α j 1, α j+1 α j ) log for 1 j p then p(x) has exactly m j roots in the annulus A(x) = {x C : 1 3 exp(α j) x 3 exp(α j ) }. Max-plus roots of tp(x) offer order of magnitude approx. to roots of p as long as the α j are well separated. Françoise Tisseur Tropical algebra 5 / 20

6 Extension to Matrix Polynomials Let P(λ) = d A i λ i C[λ] n n and tp(x) = i=0 d log A i x i with max-plus roots α 1 < < α q, α j of multiplicity m j. k 0 < < k q : corresponding indices in Newton polygon. Theorem (Noferini, Sharify, T. 14) If α l α l 1 2 log(1 + 2κ ( A kl ) ), l = j 1, j then P(λ) has exactly nm j ei vals inside the annulus A ( (1 + 2κ(A kj 1 )) 1 exp(α j ), (1 + 2κ(A kj ))exp(α j ) ). i=0 For A kj 1, A kj well conditioned and α j 1, α j, α j+1 sufficiently well separated, P has nm j ei vals of modulus close to exp(α j ). Françoise Tisseur Tropical algebra 6 / 20

7 Illustration with Spring Problem spring: quadratic matrix polynomial from NLEVP. κ 2 (A j ) 5, j = 0: α 2 λ j α Eigenvalue index j Françoise Tisseur Tropical algebra 7 / 20

8 Use of Max-Plus Roots in NLA Max-plus roots used to select starting points in the Ehrlich-Aberth method for polynomial eigenproblems. [Bini, Noferini, Sharify 13]. Useful for Betcke s diagonal scaling aimed at improving the conditioning of P s ei vals near ω. Define eigenvalue parameter scalings (λ = α j µ) for polynomial eigenvalue solvers based on linearizations. P(µ) := δ 1 P(α j µ), δ = A kj 1 exp(k j 1 α j ). Allow computation of ei pairs with small b err for λ near α j. Linearization process does not affect ei val condition number of ei vals near α j. Françoise Tisseur Tropical algebra 8 / 20

9 Orr-Sommerfeld Problem Quartic matrix polynomial from NLEVP collection λ j α Eigenvalue index j α 1 Here e vals of large magnitude are not captured by the max-plus roots. Françoise Tisseur Tropical algebra 9 / 20

10 Max-Plus Eigenvalues The max-plus ei vals of M R n n max are the roots of χ M (λ) = perm(m λ I). Here I is the identity matrix in R n n max. The n max-plus ei vals of M can be computed in O(zn) ops, where z = nnz(m) using a network flow algorithm [Gassner & Klinz 10] The max-plus ei vals of a max-plus matrix polynomial P(λ) = P 0 P 1 λ P d λ d, P j R n n max are the max-plus roots of χ P (λ) = perm ( P(λ) ). Françoise Tisseur Tropical algebra 10 / 20

11 Matrices Depending on a Parameter Let A(t) ij = b ij exp(m ij t), B C n n, M R n n max (exp( ) = 0). Use valuation V[f (t)] = lim t log f (t) t Theorem (Akian, Bapat, Gaubert 04) = V[A] = M. For all M R n n max and all generic B C n n, the ei vals λ 1 (t),..., λ n (t) of nonsing. A(t) satisfy V[λ i (t)] = lim t log λ i (t) t = µ i, where µ 1..., µ n are max-plus ei vals of M s. For A C n n use V[x] = log x = ( V[A] ) ij = log a ij. Françoise Tisseur Tropical algebra 11 / 20

12 Classical/Max-Plus Eigenvalues steam 3 from Florida sparse, cd player, orr sommerfeld from the NLEVP. steam3 cd player orr sommerfeld λ i, e µ i λ i, e µ i λ i, e µ i max plus classical i 10 2 max plus classical i 10 3 max plus classical i Françoise Tisseur Tropical algebra 12 / 20

13 Max-Plus Singular Values Let M R n n max, B C n n, A(t) = (a ij (t)), a ij (t) = b ij exp(m ij t). Theorem (De Schutter, De Moor 02) Let A(t) = U(t)Σ(t)V (t) be the analytic SVD of A(t) with Σ = diag(σ 1 (t),..., σ n (t)), σ j (t) = 0, j = n k + 1: n. Then for all G and all generic B log σ i (t) lim t t =: s i, i = 1: n k exists and is independent of the choice of B. Definition The max-plus singular values of M are s 1,..., s n, with s 1,..., s n k defined as above and s n k+1,..., s n =. Françoise Tisseur Tropical algebra 13 / 20

14 Max-Plus Singular Values (cont.) Theorem (Hook 14) The max-plus singular values of M R n n max are the max-plus ei vals of the max-plus pencil, M σ 0. Charaterization extends to rectangular case. Max-plus singular values and e vals of symmetric max-plus matrices are equal. If A C n n with ith singular value σ i we expect exp(s i ) σ i, where s i is the ith max-plus singular value of M = (log a ij ) R n n max. Françoise Tisseur Tropical algebra 14 / 20

15 Matrices from Florida Sparse Collection west lns_ d_dyn σ i, e s i 10 0 σ i, e s i σ i, e s i max plus classical i 10 5 max plus classical i 10 4 max plus classical i Françoise Tisseur Tropical algebra 15 / 20

16 Hungarian Pair Optimal assignment problem for M Rmax: n n compute n perm(m) = max m π(i),i. π P(n) Can be expressed as a linear programming problem (LPP) { n n n } max m ij d ij : d ij > 0, d ij = d ji = 1 i, i,j=1 with dual problem n min{ u i + v i : u, v R n : m ij u i v j 0}. i=1 An Hungarian pair is a solution (u, v) to the dual LPP. j=1 i=1 j=1 Françoise Tisseur Tropical algebra 16 / 20

17 Hungarian Scaling For A C n n let M = V[A] R n n max with V[x] = log x. Let (u, v) be a Hungarian pair for M. Define diagonal matrices D u, D v R n n by (D u ) ii = exp( u i ), (D v ) ii = exp( v i ). The Hungarian scaling of A is D u AD v =: H ( h ij 1). Implemented in HSL-MC64 (together with some reordering). Commonly used before solving highly indefinite and nonsymmetric linear systems. Can show κ 2 (H) n min D1,D 2 D n κ 2 (D 1 AD 2 ) for normal A (tends to also be the case for nonnormal A). Françoise Tisseur Tropical algebra 17 / 20

18 Numerical Experiments H = diag(e u )A diag(e v ) : Hungarian scaling. Row and column scalings with DGEEQU, B = D R AD C. min κ 2 (D 1 AD 2 ) ρ( A A 1 ) n min κ 2 (D 1 AD 2 ), D 1,D 2 D n D 1,D 2 D n [Rump 03]. Problem κ 2 (A) κ(b) κ 2 (H) ρ( A A 1 ) d dyn 7.4e+6 8.4e+2 1.6e+1 6.2e+0 rotor1 2.4e e+8 3.0e+1 1.3e+0 lns e e+6 6.0e+1 1.8e+1 west e e+6 2.1e+2 6.5e+1 nnc e e e+4 1.4e+4 Françoise Tisseur Tropical algebra 18 / 20

19 Hungarian Scaling and Max-Plus Sing. Val. Theorem (Hook 14) Let u, v R n and M R n n max. The matrix diag( u) M diag( v) has all of its max-plus singular values equal to zero iff (u, v) is a Hungarian pair for M. Now if A C n n has entries which varies a lot in magnitude and (u, v) is a Hungarian pair for M = V[A] R n n max then (heuristically) diag(e u )A diag(e v ) should have its singular values map to exp(0) = 1. Françoise Tisseur Tropical algebra 19 / 20

20 Summary Max-plus roots, eigenvalues and singular values can offer order of magnitude approximations to their classical" analogs for problems with large variations in the magnitude of the data. Plan to use tropical algebra to produce a polynomial eigensolver with better numerical stability property than eigensolvers such as polyeig. Investigate the effect of Hungarian scaling on eigenvalue condition numbers and backward errors. Françoise Tisseur Tropical algebra 20 / 20