Zero localization: from 17th-century algebra to challenges of today

Transcription

1 Zero localization: from 7th-century algebra to challenges of today UC Berkeley & TU Berlin Bremen summer school July 203

2 Descartes rule of signs Theorem [Descartes]. The number of positive zeros of a real univariate polynomial does not exceed the number of sign changes in its sequence of coefficients. Moreover, it has the same parity. René Descartes Theorem [Sturm]. The number of zeros of a real univariate polynomial p on the interval (a, b] is given by V (a) V (b), with V () the number of sign changes in its Sturm sequence p, p, p 2,....

3 Descartes rule of signs Theorem [Descartes]. The number of positive zeros of a real univariate polynomial does not exceed the number of sign changes in its sequence of coefficients. Moreover, it has the same parity. René Descartes Theorem [Sturm]. The number of zeros of a real univariate polynomial p on the interval (a, b] is given by V (a) V (b), with V () the number of sign changes in its Sturm sequence p, p, p 2,....

4 Euclidean algorithm and continued fractions Starting from f 0 := p, f := q (b 0 /a 0 )p, form the Euclidean algorithm sequence f j = q j f j + f j+, j =,..., k, f k+ = 0. Then f k is the greatest common divisor of p and q. This gives a continued fraction representation R(z) = f (z) f 0 (z) =. q (z) + q 2 (z) + q 3 (z) q k (z)

5 Euclidean algorithm and continued fractions Starting from f 0 := p, f := q (b 0 /a 0 )p, form the Euclidean algorithm sequence f j = q j f j + f j+, j =,..., k, f k+ = 0. Then f k is the greatest common divisor of p and q. This gives a continued fraction representation R(z) = f (z) f 0 (z) =. q (z) + q 2 (z) + q 3 (z) q k (z)

6 Generalized Jacobi matrices J (z) := q k (z) q k (z) q k 2 (z) q 2 (z) q (z) Remark. h j (z) := f j (z)/f k (z) is the leading principal minor of J (z) of order k j. In particular, h 0 (z) = det J (z). Remark 2. Eigenvalues of the generalized eigenvalue problem J (z)u = 0 are closely related to properties of R(z).

7 Generalized Jacobi matrices J (z) := q k (z) q k (z) q k 2 (z) q 2 (z) q (z) Remark. h j (z) := f j (z)/f k (z) is the leading principal minor of J (z) of order k j. In particular, h 0 (z) = det J (z). Remark 2. Eigenvalues of the generalized eigenvalue problem J (z)u = 0 are closely related to properties of R(z).

8 Generalized Jacobi matrices J (z) := q k (z) q k (z) q k 2 (z) q 2 (z) q (z) Remark. h j (z) := f j (z)/f k (z) is the leading principal minor of J (z) of order k j. In particular, h 0 (z) = det J (z). Remark 2. Eigenvalues of the generalized eigenvalue problem J (z)u = 0 are closely related to properties of R(z).

9 Jacobi continued fractions In the regular case, q j (z) = α j z + β j, α j, β j C, α j 0. The polynomials f j satisfy the three-term recurrence relation f j (z) = (α j z + β j )f j (z) + f j+ (z), j =,..., r. R(z) = f (z) f 0 (z) = α z + β + α 2 z + β 2 + α 3 z + β α r z + β r.

10 Jacobi continued fractions In the regular case, q j (z) = α j z + β j, α j, β j C, α j 0. The polynomials f j satisfy the three-term recurrence relation f j (z) = (α j z + β j )f j (z) + f j+ (z), j =,..., r. R(z) = f (z) f 0 (z) = α z + β + α 2 z + β 2 + α 3 z + β α r z + β r.

11 Stieltjes continued fractions In the doubly regular case, q 2j (z) = c 2j, k j =,..., 2 q 2j (z) = c 2j z, j =,..., r. R(z) = f (z) f 0 (z) = c z + c 2 + c 3 z T { c2r if R(0) <, T := c 2r z if R(0) =., where

12 Stieltjes continued fractions In the doubly regular case, q 2j (z) = c 2j, k j =,..., 2 q 2j (z) = c 2j z, j =,..., r. R(z) = f (z) f 0 (z) = c z + c 2 + c 3 z T { c2r if R(0) <, T := c 2r z if R(0) =., where

13 Sturm algorithm Sturm s algorithm is a variation of the Euclidean algorithm f j (z) = q j (z)f j (z) f j+ (z), j = 0,,..., k, where f k+ (z) = 0. The polynomial f k is the greatest common divisor of p and q. The Sturm algorithm is regular if the polynomials q j are linear. Theorem [Sturm]. Ind + ( f f0 ) = n 2V (h 0,..., h n ) where h k is the leading coefficient of f k.

14 Sturm algorithm Sturm s algorithm is a variation of the Euclidean algorithm f j (z) = q j (z)f j (z) f j+ (z), j = 0,,..., k, where f k+ (z) = 0. The polynomial f k is the greatest common divisor of p and q. The Sturm algorithm is regular if the polynomials q j are linear. Theorem [Sturm]. Ind + ( f f0 ) = n 2V (h 0,..., h n ) where h k is the leading coefficient of f k.

15 Cauchy indices Definition. Ind ω (F) := { +, if F(ω 0) < 0 < F(ω + 0),, if F(ω 0) > 0 > F(ω + 0), is the index of the function F at its real pole ω of odd order. Theorem [Gantmacher]. If a rational function R with exactly r poles is represented by a series R(z) = s + s 0 z + s z 2 +, then Ind + = r 2V (D 0(R), D (R), D 2 (R),..., D r (R)).

16 Cauchy indices Definition. Ind ω (F) := { +, if F(ω 0) < 0 < F(ω + 0),, if F(ω 0) > 0 > F(ω + 0), is the index of the function F at its real pole ω of odd order. Theorem [Gantmacher]. If a rational function R with exactly r poles is represented by a series R(z) = s + s 0 z + s z 2 +, then Ind + = r 2V (D 0(R), D (R), D 2 (R),..., D r (R)).

17 Hankel and Hurwitz matrices Let R(z) be a rational function expanded in its Laurent series at R(z) = s + s 0 z + s z 2 + s 2 z 3 +. Introduce the infinite Hankel matrix S :=[s i+j ] i,j=0 and consider the leading principal minors of S: D j (S) := det s 0 s s 2... s j s s 2 s 3... s j s j s j s j+... s 2j 2, j =, 2, 3,.... These are Hankel minors or Hankel determinants.

18 Hankel and Hurwitz matrices Let R(z) be a rational function expanded in its Laurent series at R(z) = s + s 0 z + s z 2 + s 2 z 3 +. Introduce the infinite Hankel matrix S :=[s i+j ] i,j=0 and consider the leading principal minors of S: D j (S) := det s 0 s s 2... s j s s 2 s 3... s j s j s j s j+... s 2j 2, j =, 2, 3,.... These are Hankel minors or Hankel determinants.

19 Hurwitz determinants Let R(z) = q(z) p(z), p(z) = a 0z n + + a n, a 0 0, For each j =, 2,..., denote 2j (p, q) := det q(z) = b 0 z n + + b n, a 0 a a 2... a j a j... a 2j b 0 b b 2... b j b j... b 2j 0 a 0 a... a j 2 a j... a 2j 2 0 b 0 b... b j 2 b j... b 2j a 0 a... a j b 0 b... b j These are the Hurwitz minors or Hurwitz determinants..

20 Hankel Hurwitz Theorem [Hurwitz]. Let R(z) = q(z)/p(z) with notation as above. Then 2j (p, q) = a 2j 0 D j(r), j =, 2,.... Corollary. Let T (z) = /R(z) with notation as above. Then D j (S) = s 2j D j(t ), j =, 2,....

21 Hankel Hurwitz Theorem [Hurwitz]. Let R(z) = q(z)/p(z) with notation as above. Then 2j (p, q) = a 2j 0 D j(r), j =, 2,.... Corollary. Let T (z) = /R(z) with notation as above. Then D j (S) = s 2j D j(t ), j =, 2,....

22 Even and odd parts of polynomials Let p be a real polynomial p(z) = a 0 z n +a z n +...+a n =: p 0 (z 2 ) + zp (z 2 ), a 0 > 0, a i IR, Let n = deg p and m = for n = 2m n 2. p 0 (u) = a 0 u m + a 2 u m a n 2 u + a n, p (u) = a u m + a 3 u m a n 3 u + a n, for n = 2m + p 0 (u) = a u m + a 3 u m a n 2 u + a n, p (u) = a 0 u m + a 2 u m a n 3 u + a n.

23 Even and odd parts of polynomials Let p be a real polynomial p(z) = a 0 z n +a z n +...+a n =: p 0 (z 2 ) + zp (z 2 ), a 0 > 0, a i IR, Let n = deg p and m = for n = 2m n 2. p 0 (u) = a 0 u m + a 2 u m a n 2 u + a n, p (u) = a u m + a 3 u m a n 3 u + a n, for n = 2m + p 0 (u) = a u m + a 3 u m a n 2 u + a n, p (u) = a 0 u m + a 2 u m a n 3 u + a n.

24 Even and odd parts of polynomials Let p be a real polynomial p(z) = a 0 z n +a z n +...+a n =: p 0 (z 2 ) + zp (z 2 ), a 0 > 0, a i IR, Let n = deg p and m = for n = 2m n 2. p 0 (u) = a 0 u m + a 2 u m a n 2 u + a n, p (u) = a u m + a 3 u m a n 3 u + a n, for n = 2m + p 0 (u) = a u m + a 3 u m a n 2 u + a n, p (u) = a 0 u m + a 2 u m a n 3 u + a n.

25 Even and odd parts of polynomials Let p be a real polynomial p(z) = a 0 z n +a z n +...+a n =: p 0 (z 2 ) + zp (z 2 ), a 0 > 0, a i IR, Let n = deg p and m = for n = 2m n 2. p 0 (u) = a 0 u m + a 2 u m a n 2 u + a n, p (u) = a u m + a 3 u m a n 3 u + a n, for n = 2m + p 0 (u) = a u m + a 3 u m a n 2 u + a n, p (u) = a 0 u m + a 2 u m a n 3 u + a n.

26 Associated function and the main theorem of stability Introduce the function associated with the polynomial p(z) = p 0 (z 2 ) + zp (z 2 ) Φ(u) = p (u) p 0 (u) Definition. A polynomial is Hurwitz stable if all its zeros lie in the open left half-plane. Main Theorem of Stability A polynomial z p(z) = p 0 (z 2 ) + zp (z 2 ) is Hurwitz stable if and only if m α j n Φ(u) = β +, β 0, α j, ω j > 0, m =. u + ω j 2 j=

27 Associated function and the main theorem of stability Introduce the function associated with the polynomial p(z) = p 0 (z 2 ) + zp (z 2 ) Φ(u) = p (u) p 0 (u) Definition. A polynomial is Hurwitz stable if all its zeros lie in the open left half-plane. Main Theorem of Stability A polynomial z p(z) = p 0 (z 2 ) + zp (z 2 ) is Hurwitz stable if and only if m α j n Φ(u) = β +, β 0, α j, ω j > 0, m =. u + ω j 2 j=

28 Associated function and the main theorem of stability Introduce the function associated with the polynomial p(z) = p 0 (z 2 ) + zp (z 2 ) Φ(u) = p (u) p 0 (u) Definition. A polynomial is Hurwitz stable if all its zeros lie in the open left half-plane. Main Theorem of Stability A polynomial z p(z) = p 0 (z 2 ) + zp (z 2 ) is Hurwitz stable if and only if m α j n Φ(u) = β +, β 0, α j, ω j > 0, m =. u + ω j 2 j=

29 Hurwitz and Lienard Chipart Theorems Hurwitz Theorem A polynomial p of degree n is Hurwitz stable if and only if (p) > 0, 2 (p) > 0,..., n (p) > 0. Lienard and Chipart Theorem A polynomial p of degree n is Hurwitz stable if and only if and n (p) > 0, n 3 (p) > 0, n 5 (p) > 0,... a n > 0, a n 2 > 0, a n 4 > 0,...

30 Hurwitz and Lienard Chipart Theorems Hurwitz Theorem A polynomial p of degree n is Hurwitz stable if and only if (p) > 0, 2 (p) > 0,..., n (p) > 0. Lienard and Chipart Theorem A polynomial p of degree n is Hurwitz stable if and only if and n (p) > 0, n 3 (p) > 0, n 5 (p) > 0,... a n > 0, a n 2 > 0, a n 4 > 0,...

31 Stable polynomials and Stieltjes continued fractions Using properties of functions mapping the UHP to LoHP and the main theorem of stability, one can obtain Stieltjes criterion of stability A polynomial p of degree n is Hurwitz stable if and only if its associated function Φ has the following Stieltjes continued fraction expansion Φ(u) = p (u) p 0 (u) = c 0 + c u + c 2 + c 3 u + where c 0 0 and c i > 0, i =,..., 2m, and m =... + c 2m n 2.,

32 Further criteria of stability Theorem. A polynomial p of degree n is Hurwitz stable if and only if the infinite Hankel matrix S = s i+j i,j=0 is a sign-regular matrix of n rank m, where m =. 2 Theorem. A polynomial f = p(z 2 ) + zq(z 2 ) is stable if and only if its infinite Hurwitz matrix H(p, q) is totally nonnegative.

33 Further criteria of stability Theorem. A polynomial p of degree n is Hurwitz stable if and only if the infinite Hankel matrix S = s i+j i,j=0 is a sign-regular matrix of n rank m, where m =. 2 Theorem. A polynomial f = p(z 2 ) + zq(z 2 ) is stable if and only if its infinite Hurwitz matrix H(p, q) is totally nonnegative.

34 Factorization of infinite Hurwitz matrices Theorem If g(z) = g 0 z l + g z l g l, then H(p g, q g) = H(p, q)t (g), T (g) := Here we set g i = 0 for all i > l. g 0 g g 2 g 3 g g 0 g g 2 g g 0 g g g 0 g g where.

35 Another factorization Theorem If the Euclidean algorithm for the pair p, q is doubly regular, then H(p, q) factors as J(c) := H(p, q) = J(c ) J(c k )H(0, )T (g), c c c H(0, ) =

36 Stable polynomials and Stieltjes continued fractions Using properties of functions mapping the UHP to LoHP and the main theorem of stability, one can obtain Stieltjes criterion of stability A polynomial p of degree n is Hurwitz stable if and only if its associated function Φ has the following Stieltjes continued fraction expansion Φ(u) = p (u) p 0 (u) = c 0 + c u + c 2 + c 3 u + where c 0 0 and c i > 0, i =,..., 2m, and m =... + c 2m n 2.,

37 Stable polynomials and Jacobi continued fractions Jacobi criterion of stability A polynomial p of degree n is Hurwitz stable if and only if its associated function Φ has the following Jacobi continued fraction expansion Φ(u) = p (u) p 0 (u) = αu+β+, α u+β α 2 u+β α n u+β n where α 0, α j > 0 and β, β j IR.

38 Other topics Stielties, Jacobi, other continued fractions. Padé approximation Orthogonal polynomials. Moment problems Nevanlinna functions Pólya frequency sequences and functions Laguerre-Pólya class and its generalizations Total nonnegativity Matrix factorizations Hurwitz rational and meromorphic functions

39 Other topics Stielties, Jacobi, other continued fractions. Padé approximation Orthogonal polynomials. Moment problems Nevanlinna functions Pólya frequency sequences and functions Laguerre-Pólya class and its generalizations Total nonnegativity Matrix factorizations Hurwitz rational and meromorphic functions

40 Other topics Stielties, Jacobi, other continued fractions. Padé approximation Orthogonal polynomials. Moment problems Nevanlinna functions Pólya frequency sequences and functions Laguerre-Pólya class and its generalizations Total nonnegativity Matrix factorizations Hurwitz rational and meromorphic functions

41 Other topics Stielties, Jacobi, other continued fractions. Padé approximation Orthogonal polynomials. Moment problems Nevanlinna functions Pólya frequency sequences and functions Laguerre-Pólya class and its generalizations Total nonnegativity Matrix factorizations Hurwitz rational and meromorphic functions

42 Other topics Stielties, Jacobi, other continued fractions. Padé approximation Orthogonal polynomials. Moment problems Nevanlinna functions Pólya frequency sequences and functions Laguerre-Pólya class and its generalizations Total nonnegativity Matrix factorizations Hurwitz rational and meromorphic functions

43 Other topics Stielties, Jacobi, other continued fractions. Padé approximation Orthogonal polynomials. Moment problems Nevanlinna functions Pólya frequency sequences and functions Laguerre-Pólya class and its generalizations Total nonnegativity Matrix factorizations Hurwitz rational and meromorphic functions

44 Other topics Stielties, Jacobi, other continued fractions. Padé approximation Orthogonal polynomials. Moment problems Nevanlinna functions Pólya frequency sequences and functions Laguerre-Pólya class and its generalizations Total nonnegativity Matrix factorizations Hurwitz rational and meromorphic functions

45 Other topics Stielties, Jacobi, other continued fractions. Padé approximation Orthogonal polynomials. Moment problems Nevanlinna functions Pólya frequency sequences and functions Laguerre-Pólya class and its generalizations Total nonnegativity Matrix factorizations Hurwitz rational and meromorphic functions

46 Zero localization... is everywhere Zeros of entire and meromorphic functions from number theory (e.g., Riemann ζ-function and other L-functions) Zeros of partition functions for Ising, Potts and other models of statistical mechanics (Lee-Yang program) Zeros arising as eigenvalues in matrix/operator eigenvalue problems (e.g., random matrix theory) Recent generalizations of stability and hyperbolicity to the multivariate case and applications to Pólya-Schur-Lax type problems (Borcea, Brändén, B. Shapiro, etc.)

47 Zero localization... is everywhere Zeros of entire and meromorphic functions from number theory (e.g., Riemann ζ-function and other L-functions) Zeros of partition functions for Ising, Potts and other models of statistical mechanics (Lee-Yang program) Zeros arising as eigenvalues in matrix/operator eigenvalue problems (e.g., random matrix theory) Recent generalizations of stability and hyperbolicity to the multivariate case and applications to Pólya-Schur-Lax type problems (Borcea, Brändén, B. Shapiro, etc.)

48 Zero localization... is everywhere Zeros of entire and meromorphic functions from number theory (e.g., Riemann ζ-function and other L-functions) Zeros of partition functions for Ising, Potts and other models of statistical mechanics (Lee-Yang program) Zeros arising as eigenvalues in matrix/operator eigenvalue problems (e.g., random matrix theory) Recent generalizations of stability and hyperbolicity to the multivariate case and applications to Pólya-Schur-Lax type problems (Borcea, Brändén, B. Shapiro, etc.)

49 Zero localization... is everywhere Zeros of entire and meromorphic functions from number theory (e.g., Riemann ζ-function and other L-functions) Zeros of partition functions for Ising, Potts and other models of statistical mechanics (Lee-Yang program) Zeros arising as eigenvalues in matrix/operator eigenvalue problems (e.g., random matrix theory) Recent generalizations of stability and hyperbolicity to the multivariate case and applications to Pólya-Schur-Lax type problems (Borcea, Brändén, B. Shapiro, etc.)

50 References Structured matrices, continued fractions, and root localization of polynomials, O. H. & M. Tyaglov, SIAM Review, 54 (202), no.3, arxiv: Lectures on the Routh-Hurwitz problem, Yu. S. Barkovsky, translated by O. H. & M. Tyaglov, arxiv: Generalized Hurwitz matrices, multiple interlacing and forbidden sectors of the complex plane, O. H., S. Khrushchev, O. Kushel, M. Tyaglov, coming soon Thank you!

51 References Structured matrices, continued fractions, and root localization of polynomials, O. H. & M. Tyaglov, SIAM Review, 54 (202), no.3, arxiv: Lectures on the Routh-Hurwitz problem, Yu. S. Barkovsky, translated by O. H. & M. Tyaglov, arxiv: Generalized Hurwitz matrices, multiple interlacing and forbidden sectors of the complex plane, O. H., S. Khrushchev, O. Kushel, M. Tyaglov, coming soon Thank you!

52 References Structured matrices, continued fractions, and root localization of polynomials, O. H. & M. Tyaglov, SIAM Review, 54 (202), no.3, arxiv: Lectures on the Routh-Hurwitz problem, Yu. S. Barkovsky, translated by O. H. & M. Tyaglov, arxiv: Generalized Hurwitz matrices, multiple interlacing and forbidden sectors of the complex plane, O. H., S. Khrushchev, O. Kushel, M. Tyaglov, coming soon Thank you!

53 References Structured matrices, continued fractions, and root localization of polynomials, O. H. & M. Tyaglov, SIAM Review, 54 (202), no.3, arxiv: Lectures on the Routh-Hurwitz problem, Yu. S. Barkovsky, translated by O. H. & M. Tyaglov, arxiv: Generalized Hurwitz matrices, multiple interlacing and forbidden sectors of the complex plane, O. H., S. Khrushchev, O. Kushel, M. Tyaglov, coming soon Thank you!

54 References Structured matrices, continued fractions, and root localization of polynomials, O. H. & M. Tyaglov, SIAM Review, 54 (202), no.3, arxiv: Lectures on the Routh-Hurwitz problem, Yu. S. Barkovsky, translated by O. H. & M. Tyaglov, arxiv: Generalized Hurwitz matrices, multiple interlacing and forbidden sectors of the complex plane, O. H., S. Khrushchev, O. Kushel, M. Tyaglov, coming soon Thank you!

55 References Structured matrices, continued fractions, and root localization of polynomials, O. H. & M. Tyaglov, SIAM Review, 54 (202), no.3, arxiv: Lectures on the Routh-Hurwitz problem, Yu. S. Barkovsky, translated by O. H. & M. Tyaglov, arxiv: Generalized Hurwitz matrices, multiple interlacing and forbidden sectors of the complex plane, O. H., S. Khrushchev, O. Kushel, M. Tyaglov, coming soon Thank you!