The kernel structure of rectangular Hankel and Toeplitz matrices

Transcription

1 The ernel structure of rectangular Hanel and Toeplitz matrices Martin H. Gutnecht ETH Zurich, Seminar for Applied Mathematics ØØÔ»»ÛÛÛºÑ Ø º Ø Þº» Ñ 3rd International Conference on Structured Matrices and Tensors Hong Kong, January 2010 With some help from Karla Rost

2 Outline Heinig and Rost Structure of the ernel Padé approximants Cabay and Melescho Conclusions M.H. Gutnecht HK09 p. 2

3 Heinig and Rost (1984) Given: a sequence of ν complex numbers φ 1,...,φ ν, which may be moments of a matrix A: φ : ỹ T 0 A y 0. For = 1,...,ν let l() : ν + 1 (so + l() = ν + 1) and consider the l() Hanel matrix (moment matrix) H [l(),] : φ 1 φ 2... φ 1 φ φ 2 φ 3... φ φ φ l φ l+1... φ ν 1 φ ν. M.H. Gutnecht HK09 p. 3

4 In particular, H [ν,1] : φ 1 φ 2 φ 3. φ ν 1 φ ν, H [ν 1,2] : φ 1 φ 2 φ 2 φ 3. φ ν 2 φ ν 1. φ ν 1 φ ν [ ] H [2,ν 1] φ1 φ : 2... φ ν 2 φ ν 1, φ 2 φ 3... φ ν 1 φ ν H [1,ν] : [ φ 1 φ 2 φ 3... φ ν 1 φ ν ].,..., M.H. Gutnecht HK09 p. 4

5 If ν = 2m 1 is odd, we get a square matrix φ 1 φ 2... φ m 1 φ m H [m,m] φ 2 φ 3... φ m φ m+1 :.... φ m φ m+1... φ 2m 2 φ 2m 1, while if f ν = 2n is even, φ 1 φ 2... φ n φ n+1 H [n,n+1] φ 2 φ 3... φ n+1 φ n+2 ( :.... = H [n+1,n]) T. φ n φ n+1... φ 2n 1 φ 2n M.H. Gutnecht HK09 p. 5

6 Clearly, just due to the size of the matrices, ran H [l(),] min{l(), }, and so the maximum ran is at most m if l() = and ν = 2m 1 is odd, n if l() = ± 1 and ν = 2n is even. max {ϕ i } ran H [l(),] m n max {ϕ i } ran H [l(),] 1 ν 1 ν But, for some sets of moments {φ 1,...,φ ν } the maximum ran can be smaller. M.H. Gutnecht HK09 p. 6

7 For a particular set {φ 1,...,φ ν } the ran dependence is as indicated above if, respectively, ran H [m,m] = m ran H [n,n+1] = n and ν = 2m 1 is odd, and ν = 2n is even. ran H[l(),] m n ran H[l(),] 1 m ν 1 n ν Here, H [m,m] and H [n,n+1] have maximum possible ran. M.H. Gutnecht HK09 p. 7

8 Heinig & Rost (1984): case where ran not maximum: ran H[l(),] m 1 1 m ν ran H[l(),] n 1 1 n ν Clearly, these discrete functions must be symmetric about the vertical line = m or = n + 1 2, respectively, since H [,l()] = ( H [l(),]) T. What is special (due to the Hanel structure) is the plateau. M.H. Gutnecht HK09 p. 8

9 By the well-nown dimension formula the dimension of the ernel (null space) of H [l(),] is dim er H [l(),] = ran H [l(),] max{ l(), 0} = max{2 ν 1, 0} m 1 dim er H[l(),] ν 1 1 m ν n 1 dim er H[l(),] ν 1 1 n ν M.H. Gutnecht HK09 p. 9

10 By the well-nown dimension formula the dimension of the ernel (null space) of H [l(),] is dim er H [l(),] = ran H [l(),] max{ l(), 0} = max{2 ν 1, 0} m 1 dim er H[l(),] ν 1 1 m ν n 1 dim er H[l(),] ν 1 1 n ν M.H. Gutnecht HK09 p. 10

11 Partly, this behavior can be explained easily: 1. If l : ran H [l,] = l = ran H [l 1,+1] = l 1. Hence, by dimension formula: as + 1, dimension of ernel increases by 2 if H [l,] is not ran-deficient. 2. Due to the Hanel structure, if ernel is nonzero: H [l,] q = o H [l 1,+1] [ q 0 0 q ] = [ o o ] Hence, the dimension of the ernel increases by at least 1. Generalization: Prop. 5.2 of Heinig & Rost (1984). 3. Due to the Hanel structure (but not so straightforward): If H [l,] is ran-deficient, ran H [l,] = ran H [l 1,+1] = ran H [l+1, 1]. Hence, if H [l,] is ran-deficient, the ran stays the same as ± 1 ; i.e., the dimension of the ernel changes by ±1. M.H. Gutnecht HK09 p. 11

12 The structure of the ernel Heinig & Rost (1984) as well as Mastronardi, Van Barel, & Vandebril (2009) also describe the structure of the ernel. Their description is in terms of column vectors; ours will be in terms of polynomials. Their basic tool are the so-called U-chains and their linear hull. This is just polynomial multiplication in disguise! M.H. Gutnecht HK09 p. 12

13 Definition. Given q : [ ρ 1... ρ s ] T C s \{o} and its embedding q : [ ρ 1... ρ s ] T C r+s 1, a U-chain of length r in C r+s 1 generated by q is the sequence of columns of the (r + s 1) r matrix ρ ρ 2 ρ.. 1. [ q Sq S 2 q... S r 1 q ].. ρ =. ρ s... ρ1, 0 ρ s ρ ρ s where S is the (square) downshift matrix of appropriate order. (Here: r + s 1.) M.H. Gutnecht HK09 p. 13

14 Now we can cite Prop. 5.2 of Heinig & Rost (1984), the (easy) generalization of Item 2: The U-chain of length r generated by q belongs to the ernel of H [l,] if and only if q er H [l+r 1, r+1]. Example with = 4, r = 3, s = 2 (note: = r + s 1): φ 1 φ 2 φ 3 φ 4 ρ φ 2 φ 3 φ 4 φ 5... ρ 2 ρ ρ 2 ρ 1 = φ l φ l+1 φ l+2 φ ν 0 0 ρ 2 φ 1 φ 2 φ 2 φ 3 0 φ 3 φ 4 [ ] 0 ρ1.. =.. ρ 2 0. φ l+1 φ l φ l+2 φ ν M.H. Gutnecht HK09 p. 14

15 Here is a reformulation of the principal result from Heinig & Rost (1984) regarding the ernel of H [l,] : THEOREM If H [l,] is ran deficient, its ernel is the linear hull of a single U-chain. If l = 1 and H [l,] has full ran, its one-dimensional ernel is trivially the linear hull of a U-chain of length 1. If l < 1 and H [l,] has full ran, its ( l)-dimensional ernel is the linear hull of two U-chains of length 1 2 l and l + 1, respectively. 1 2 M.H. Gutnecht HK09 p. 15

16 Padé approximants and Padé forms Let P m be the set of polynomials of degree m. Let R m,n : {r = p/q ; p P m, q P n \{0}}. Definition (Padé, 1892). Given: formal power series (FPS) f(z) : φ z =0 (also called the generating function of the moments). For every pair (m, n) N 2 the (m, n) Padé approximant (PA) r = r m,n R m,n is defined by the property f(z) r(z) = O(z κ ) with κ as large as possible. (1) Here, O(z κ ) is a FPS starting at the term z κ or later. M.H. Gutnecht HK09 p. 16

17 The Padé approximant is a unique rational function; but it has various representations p/q. Normally, κ = m + n + 1 number of free parameters. If κ m + n + 1 : true PA. If κ < m + n + 1 : deficient PA. To find r m,n = p/q, we loo at the linearized problem. Definition. For every pair (m, n) N 2 an (m, n) Padé form (PF) (p, q) = (p m,n, q m,n ) P m (P n \{0}) is defined by the property f(z)q(z) p(z) = O(z m+n+1 ) (2) (2) always has nontrivial solutions, and they all yield the same rational function p/q = r m,n. M.H. Gutnecht HK09 p. 17

18 THEOREM (GRAGG (1972)) The condition (2) has in P m P n the general solution (p(z), q(z)) = (z σ p(z) w(z), z σ q(z) w(z)). (3) Here, p : p m,n P m and q : q m,n P n with q(0) = 1 are uniquely determined and relatively prime (except when p 0 and q 1), and r m,n = p/ q. Moreover, σ : σ m,n : max{0, m + n + 1 order of (f q p)}, is the deficiency, and w P δ σ is arbitrary, where is the defect of p/ q in R m,n. δ : δ m,n : min{m p, n q} M.H. Gutnecht HK09 p. 18

19 COROLLARY (CHARACTERIZATION THEOREM) r R m,n is the (m, n) PA of f iff ( f(z) r(z) = O z m+n+1 δ), (4) i.e., iff r matches m + n + 1 δ Taylor coefficients (moments) of f. COROLLARY (BLOCK STRUCTURE THEOREM) The set of (m, n) pairs for which a particular nonzero rational function is a PA of f covers a (possibly empty) square in the (m, n) plane. If r = f, the square is infinite. M.H. Gutnecht HK09 p. 19

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22 In addition to we let p(z) : f(z) : φ z =0 m π z, q(z) : =0 m ρ z. =0 Comparing coefficients in fq p = O(z m+n+1 ) we obtain an infinite linear system for the coefficients of p and q. Comparing only the coefficients of z m+1,...,z m+n yields a homogeneous linear n (n + 1) Toeplitz system for the coefficients of q. The set of its solutions is the ernel of the n (n + 1) Toeplitz matrix, which maps 1-to-1 into the ernel of an n (n + 1) Hanel matrix. M.H. Gutnecht HK09 p. 22

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24 In particular, if m = n, we have or, Since, by (3), φ 1 φ 2... φ n φ n+1 φ 2 φ 3... φ n+1 φ n φ n φ n+1... φ 2n 1 φ 2n H [n,n+1] (Jq) }{{} :q J = o,. q(z) = z σ q(z) w(z) ρ n ρ n 1. ρ 1 ρ 0 = with fixed σ and q(z), but arbitrary w P δ σ, we see that the ernel of H [n,n+1] is a single U-chain of length δ σ + 1 generated by the vector of coefficients of q in reverse order. M.H. Gutnecht HK09 p

25 Hence, on the basis of this observation we fully understand the case where the ernel of H [n,n+1] is a single U-chain: this case corresponds to a singular bloc in the Padé table, and the structure of the ernel results from Gragg s theorem. Since the Padé approximant is the same in the whole bloc, one can conclude that some neighboring matrices H [l,] have as ernel a U-chain generated by the the same vector q and determined by the same polynomial q. But where do the ernels made up of two U-chains come from? Here we apply another Padé approximation result. M.H. Gutnecht HK09 p. 25

26 Note that whenever the dimension of the ernel grows by two when we proceed from H [l,] to H [l 1,+1], then l and H [l,] has maximum ran: m 1 dim er H[l(),] ν 1 1 m ν n 1 dim er H[l(),] ν 1 1 n ν M.H. Gutnecht HK09 p. 26

27 Cabay and Melescho (1993) Cabay and Melescho (1993) came up with a Schur-lie algorithm for computing diagonal sequences of Padé approximants in Padé tables with singular blocs. Soon after it was generalized to many other situations, including the one of row sequences (see G. (1993), G./Hochbruc (1994, 1995, 1996), Bultheel/Van Barel (...)) superfast Toeplitz solvers. One introduces regular pairs of PAs for which the Padé forms (p n, q n ) and (p n 1, q n 1 ) are uniquely defined up to scaling, i.e., H [n,n] is nonsingular. Heinig and Rost (1984) have the related concept of a pair of fundamental solutions. M.H. Gutnecht HK09 p. 27

28 If (p n+, q n+ ) and (p n+ 1, q n+ 1 ) is another regular pair, i.e., H [n+,n+] is also nonsingular, then q n+ is of the form q n+ (z) = z 2 q n 1 (z)u (n) (z) + q n (z)v (n) (z). Here, (u (n), v (n) ) is a (, ) Padé form corresponding to another function, but in our situation it will be arbitrary, since we have underdetermined Padé approximants. Hence, this formula gives rise to two U-chains. M.H. Gutnecht HK09 p. 28

29 Conclusions Results on the structure of the ernel of square or rectangular Hanel or Toeplitz matrices are closely related to results from the constructive theory of Padé approximation. These results can be used to construct the U-chains that form a basis of these ernels. M.H. Gutnecht HK09 p. 29