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Intro Subdivision Numerical Comparison Goals Certified Complex Numerical Root Finding Alexander Kobel Max Planck Institute for Informatics Dep. 1: Algorithms and Complexity Seminar on Computational Geometry and Geometric Computing May 6,

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Intro Subdivision Numerical Comparison Goals

Certified Complex Numerical Root Finding

Alexander Kobel

Max Planck Institute for Informatics

  • Dep. 1: Algorithms and Complexity

Seminar on Computational Geometry and Geometric Computing May 6, 2010

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals

Errata & Additions

Bugs squashed in the version of the talk on May 6, 2010

The explanation of the Obreshkoff’s lens criterion was wrong w.r.t. the radii and centers of the circles. A slide explaining the correct theorem is added. A proof for the inclusion predicate property of the Sf -test has been added. The statement that the denominator in the Durand-Kerner iteration actually converges to f ′(z(k)

i

) is correct: f (Z) = an

  • i

(Z − ζi) ⇒ f ′(Z) = an

  • i
  • j=i

(Z − ζj) ⇒ f ′(ζi) = an

  • j=i

(ζi − ζj) since all other factors are 0. Thus z(k)

i

→ ζi implies f ′(z(k)

i

) → f ′(ζi) (for simple roots).

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals

1

Traditional Subdivision Solvers Generic Subdivision Algorithm Sturmian Sequence Solver Descartes Solver EVAL CEVAL

2

Numerical Solvers Eigenvalue Computation Newton-Raphson Method Durand-Kerner Method Aberth-Ehrlich Iteration Other Methods

3

Comparison Subdivision-based vs. Numerical Solvers

4

Goals of Further Work

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 4

Intro Subdivision Numerical Comparison Goals

Vocabulary and Notation

Throughout the talk: Input: Polynomial f = f (Z) = n

i=0 aiZ i ∈ R[Z]

Degree: n Roots: {ζ1, . . . , ζn} Coefficients:

R = Z: integer coefficients of arbitrary length R = Z[ 1

2]: “bigfloats”

R = R: “bitstreams” R = float, double, long double (R = Z[i], R = Z[ 1

2][i], R = C)

Bitsize: L

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals

Vocabulary and Notation

Predicate (test): “fuzzy” boolean function (returns true, false or uncertain) Construction (calculation): computes a number, point, area, . . . (exact or approximated) Certificate: guarantees (by a predicate) that a calculation yields a valid and correct result (up to given precision) Isolating interval (box, disc): interval ⊂ R (or box, disc ⊂ C) containing exactly one (possibly multiple) root of f in it’s interior and none on the boundary

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals

Basic Arithmetic Operations

Bit complexity of integer and polynomial arithmetic

Operation na¨ ıve best known CGAL +, − O(nL) ×, ÷ (integers) O(L2) O˜(L) O˜(L) ×, ÷ (polynomials) O(n2L2) O˜(nL) ≤ O˜(n2L) polynomial evaluation on n points f → f (b1), . . . , f (bn) O(n2L2) O˜(nL) O˜(n2L) Taylor shift f (Z) → f (Z + b) O(n3L2) O˜(n2L) O˜(n3L) polynomial gcd O(n2L2) O˜(nL) O˜(n2L) Numerical stability is a different issue!

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Generic Subdivision Algorithm

1 Calculate a root bound B of f . 2 Start subdivision algorithm on interval [−B, B]

(or box [−B − Bi, B + Bi] in the complex case).

3 If an exclusion predicate holds on the current region,

it cannot contain a root ⇒ throw it away.

4 If an inclusion predicate holds on the current region,

it contains a root ⇒ report it.

5 If neither succeeds: check border cases, subdivide and recurse.

Choices inclusion and exclusion predicates subdivision method (bisection vs. more involved methods)

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Root Bounds

Theorem (Cauchy root bound) All (complex) roots of f have an absolute value ≤ 1 + max

i<n

  • ai

an

  • .

Theorem (Fujiwara root bound) All (complex) roots of f have an absolute value ≤ 2 max

  • an−1

an

  • ,
  • an−2

an

  • 1

2

, . . . ,

  • a1

an

  • 1

n−1

,

  • a0

2an

  • 1

n

  • .

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Sturmian Sequence Solver

Definition (Sturm chain / Sturmian sequence) Apply Euclidean division with remainder: fn = fn−2 − fn−1qn−2 f0 = f f1 = f ′ f2 = −(f0 mod f1) = q0f1 − f0 f3 = −(f1 mod f2) = q1f2 − f1 . . . fs = −(fs−2 mod fs−1) = qs−2fs−1 − fs−2 until fs is constant.

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Sturmian Sequence Solver

Theorem (Sturm’s theorem) Let σ(x) denote the number of sign changes in the Sturmian remainder sequence f0(x), f1(x), . . . , fs(x). Then, an interval (a, b] contains exactly σ(b) − σ(a) real roots of f (counted without multiplicities). ⇒ both inclusion and exclusion predicate.

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Comparison of Subdivision Solvers

Sturm theoretical complexity O˜( · ) n4L2 predicate / tree size ++ local / adaptive ✪ ✦ ✦ ✦ numerical stability −− multiple roots ✦ ✪ ✪ ✪ bitstream coefficients ✪ ✦ ✦ ✦ sparse polynomials ✪ ✪ ✦ ✦ complex roots ✪ ✪ ✪ ✦ analytical functions ✪ ✪ ✦ ✦ easy to implement (well) (✪) ✦ ✦ ✦

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Descartes Solver

Theorem (Descartes’ Rule of Signs) Let p denote the # of distinct positive real roots of f , and v be the # of sign changes in it’s (= 0) coefficient sequence. Then, p ≤ v and p ≡ v (mod 2). In particular, v = 1 ⇒ p = 1 and v = 0 ⇒ p = 0. How to apply to arbitrary intervals?

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Descartes Solver

M¨

  • bius transformations

(a, b) → (0, b − a) → (0, 1) → (1, ∞) → (0, ∞)

Z → Z − a Z → Z/b−a Z → 1/Z Z → Z − 1

f (Z) f (Z + a) f ((b − a)Z + a) f b−a

Z

+ a

  • f
  • aZ+b

Z+1

  • Theorem

Let g(Z) := f

  • aZ+b

Z+1

  • (Z + 1)n ∈ R[Z].

Then, roots of f in (a, b) correspond to roots of g in (0, ∞).

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Descartes Solver (in Bernstein Basis)

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 15

Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Descartes Solver (in Bernstein Basis)

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 16

Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Descartes Solver (in Bernstein Basis)

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Descartes Solver (in Bernstein Basis)

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 18

Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Descartes Solver (in Bernstein Basis)

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 19

Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Descartes Solver (in Bernstein Basis)

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Obreshkoff’s Lens

My explanation in the talk was wrong regarding which roots are “counted” in v. The correct statements are as follows: Theorem (in somewhat informal notation) Let (a, b) ⊂ R be some finite interval, g ∈ R[Z] the “localization” of f in (a, b) (M¨

  • bius transformations), and v the number of sign changes of g.

If the closed disc with diameter (a, b) does not contain any complex root of f , then v = 0. If the union of the two circumcircles of the equilateral triangles on the edge (a, b) contains exactly one simple root of f , then v = 1. (simplified version) There are certain x-symmetric discs which have (a, b) as a chord s.t. v ≤ the number of roots in their union and v ≥ the number of roots in their intersection (“lens”).

The latter theorem by Obreshkoff in fact implies the two other versions. For detailed formulations and proofs, see e.g. Arno Eigenwillig: Real Root Isolation for Exact and Approximate Polynomials Using Descartes’ Rule of Signs.

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Obreshkoff’s Lens

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Obreshkoff’s Lens

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Obreshkoff’s Lens

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 24

Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Obreshkoff’s Lens

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 25

Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Obreshkoff’s Lens

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 26

Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Obreshkoff’s Lens

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 27

Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Comparison of Subdivision Solvers

Sturm Descartes theoretical complexity O˜( · ) n4L2 n4L2 predicate / tree size ++ + local / adaptive ✪ (✦) ✦ ✦ numerical stability −− − multiple roots ✦ (✪) ✪ ✪ bitstream coefficients ✪ (✦) ✦ ✦ sparse polynomials ✪ ✪ ✦ ✦ complex roots ✪ ✪ ✪ ✦ analytical functions ✪ ✪ ✦ ✦ easy to implement (well) (✪) (✦) ✦ ✦

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

EVAL

Taylor expansion of f at m

f (Z) = f (m) + f ′(m) 1! (Z − m) + f ′′(m) 2! (Z − m)2 + . . . = f (m) +

  • k≥1

f (k)(m) k! (Z − m)k

T f -test on interval (a, b) =: (m − r, m + r) T f (m, r) : |f (m)| >

  • k≥1
  • f (k)(m)

k!

  • rk

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

EVAL

T f -test on interval (a, b) =: (m − r, m + r) T f (m, r) : |f (m)| >

  • k≥1
  • f (k)(m)

k!

  • rk

Theorem If T f (m, r) holds, then (a, b) contains no root of f . ⇒ exclusion predicate If both T f ′(m, r) and sign f (a) = sign f (b) hold, then (a, b) contains exactly one root of f . ⇒ inclusion predicate

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 30

Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Comparison of Subdivision Solvers

Sturm Descartes EVAL theoretical complexity O˜( · ) n4L2 n4L2 n4L2 predicate / tree size ++ + +/− local / adaptive ✪ (✦) (✦) ✦ numerical stability −− − + multiple roots ✦ (✪) (✪) ✪ bitstream coefficients ✪ (✦) ✦ ✦ sparse polynomials ✪ ✪ (✦) ✦ complex roots ✪ ✪ ✪ ✦ analytical functions ✪ ✪ ✦ ✦ easy to implement (well) (✪) (✦) ✦ ✦

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

CEVAL

exclusion predicate for a disc Dr(m) ⊂ C? ⇒ T f -test. ✦ inclusion predicate for a disc Dr(m) ⊂ C? ⇒

  • equivalent for sign change test?
  • r

global reasoning!

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

CEVAL

Sf : simplified T f -test Only consider first derivative: Sf (m, r) : |f (m)| >

  • f ′(m)
  • r

Theorem If Sf (m, r

n) does not hold, then Dr(m) contains a root of f .

⇒ inclusion predicate ⇒ exclusion predicate, as soon as n disjoint discs are found

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

CEVAL

Sf : simplified T f -test Sf (m, r) : |f (m)| >

  • f ′(m)
  • r

Theorem If Sf (m, r

n) does not hold, then Dr(m) contains a root of f .

Proof.

Assume δm := mini{|m − ζi|} > r. To show: Sf (m, r

n) holds.

  • f ′(m)

f (m)

  • =
  • i

1 m − ζi

  • ≤
  • i

1 |m − ζi| ≤

  • i

1 δm < n r ⇒

  • f ′(m)

f (m)

  • r

n < n r r n = 1 ⇔ |f (m)| > |f ′(m)| r n

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 34

Intro Subdivision Numerical Comparison Goals Generic Subdivision Sturm Descartes EVAL CEVAL

Comparison of Subdivision Solvers

Sturm Descartes EVAL CEVAL theoretical complexity O˜( · ) n4L2 n4L2 n4L2 n4L2 predicate / tree size ++ + +/− − local / adaptive ✪ (✦) (✦) (✦) numerical stability −− − + + multiple roots ✦ (✪) (✪) (✪) bitstream coefficients ✪ (✦) ✦ ✦ sparse polynomials ✪ ✪ (✦) (✦) complex roots ✪ ✪ ✪ ✦ analytical functions ✪ ✪ ✦ ✦ easy to implement (well) (✪) (✦) ✦ ✦

Others: homotopy methods, combined numerical & topological ([Pan] O˜(n3L)), . . .

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 35

Intro Subdivision Numerical Comparison Goals Eigenvalues Newton Durand-Kerner Aberth . . .

Eigenvalue Computation

Definition (Companion matrix) W.l.o.g. let f be monic, i.e. an = 1 (ow. consider f/an). Then

Cf :=         1 · · · · · · 1 · · · . . . . . . . . . . . . . . . . . . · · · · · · 1 · · · · · · · · · 1 −a0 −a1 −a2 · · · −an−2 −an−1        

is called the companion matrix of f . Theorem (−1)nf is the characteristic polynomial of Cf . ⇒ The roots of f are exactly the eigenvalues of Cf .

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals Eigenvalues Newton Durand-Kerner Aberth . . .

Eigenvalue Computation

Theorem f is the characteristic polynomial of Cf . ⇒ The roots of f are exactly the eigenvalues of Cf . Numerical solvers for eigenvalues Power iteration QR algorithm Arnoldi iteration Lanzcos iteration

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 37

Intro Subdivision Numerical Comparison Goals Eigenvalues Newton Durand-Kerner Aberth . . .

Newton-Raphson Method

Newton-Raphson method Start with some root guess z(0). Iterate z(k+1) = z(k) − f

  • z(k)

f ′ z(k).

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 38

Intro Subdivision Numerical Comparison Goals Eigenvalues Newton Durand-Kerner Aberth . . .

Newton-Raphson Method

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 39

Intro Subdivision Numerical Comparison Goals Eigenvalues Newton Durand-Kerner Aberth . . .

Newton-Raphson Method

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 40

Intro Subdivision Numerical Comparison Goals Eigenvalues Newton Durand-Kerner Aberth . . .

Newton-Raphson Method

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 41

Intro Subdivision Numerical Comparison Goals Eigenvalues Newton Durand-Kerner Aberth . . .

Newton-Raphson Method

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 42

Intro Subdivision Numerical Comparison Goals Eigenvalues Newton Durand-Kerner Aberth . . .

Newton-Raphson Method

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 43

Intro Subdivision Numerical Comparison Goals Eigenvalues Newton Durand-Kerner Aberth . . .

Newton-Raphson Method

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 44

Intro Subdivision Numerical Comparison Goals Eigenvalues Newton Durand-Kerner Aberth . . .

Newton-Raphson Method

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 45

Intro Subdivision Numerical Comparison Goals Eigenvalues Newton Durand-Kerner Aberth . . .

Newton-Raphson Method

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 46

Intro Subdivision Numerical Comparison Goals Eigenvalues Newton Durand-Kerner Aberth . . .

Newton-Raphson Method

Newton-Raphson method Start with some root guess z(0). Iterate z(k+1) = z(k) − f

  • z(k)

f ′ z(k). Theorem (Quadratic convergence for simple roots) Write ζ − z(k) = ∆(k). Then,

0 = f (ζ) = f

  • z(k)

+ f ′ z(k) ∆(k) + O

  • ∆2

(k)

  • ⇔

z(k) − ζ = f

  • z(k)

f ′ z(k) + 1 f ′ z(k)O

  • ∆2

(k)

  • ⇔

z(k+1) − ζ = 1 f ′ z(k)O

  • ∆2

(k)

  • → O
  • ∆2

(k)

  • .

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 47

Intro Subdivision Numerical Comparison Goals Eigenvalues Newton Durand-Kerner Aberth . . .

Durand-Kerner Method

Global variant of Newton-Raphson Start with some root guesses z(0)

i

for i = 1, . . . , n. Iterate z(k+1)

i

= z(k)

i

− f

  • z(k)

i

  • an
  • j=i
  • z(k)

i

− z(k)

j

. Conjecture (Convergence) Durand-Kerner converges for almost any choice of initial root guesses with at least linear convergence speed, and quadratic if all roots are simple.

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 48

Intro Subdivision Numerical Comparison Goals Eigenvalues Newton Durand-Kerner Aberth . . .

Aberth-Ehrlich Iteration

Refinement of Durand-Kerner Start with some root guesses z(0)

i

for i = 1, . . . , n. Iterate z(k)

i

= −

f

  • z(k)

i

  • f ′
  • z(k)

i

  • 1 −

f

  • z(k)

i

  • f ′
  • z(k)

i

j=i

  • z(k)

i

− z(k)

j

. Conjecture (Convergence) Aberth-Ehrlich converges for almost any choice of initial root guesses with at least linear convergence speed, and cubic if all roots are simple.

Alexander Kobel Certified Complex Numerical Root Finding

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SLIDE 49

Intro Subdivision Numerical Comparison Goals Eigenvalues Newton Durand-Kerner Aberth . . .

Other Methods

Jenkins-Traub’s algorithm Weyl’s algorithm Sch¨

  • nhage’s splitting circle-algorithm / Graeffe’s iteration

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals

Participants

CGAL: Bitstream Descartes [Eigenwillig et al.] CGAL: Descartes [Hemmer et al.] CGAL/MAPLE: RealSolutions library [Rouillier] (Descartes with adaptive precision and interval arithmetic) MPSolve [Bini, Fiorentino] (Aberth’s method) Pari/GP [Gourdon, Cohen, Belabas et al.] (Sch¨

  • nhage splitting-circle algorithm, modified by Pan
  • r Newton-Raphson variant)

Sf -test (verification only!) Untested (since no longer available to the public): Eigensolve [Fortune] “For dense, ill-conditioned [...] faster than MPSolve.” “For well-conditioned or sparse [...] MPSolve is faster.”

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals

Comparison (Timings in [sec])

Input B i t D e s c D e s c R S M P S

  • l

v e G P [ P a n ] G P [ N e w t ] S

f

Resultant of two sparse lemniscates n = 72, L = 90 0.05 0.02 0.01 0.01 0.30 0.60 0.03 n = 72, L = 1206 1.19 4.55 0.21 0.04 8.35 1.68 0.05 n = 136, L = 138 1.31 1.41 0.41 0.02 2.37

  • vf

0.19 n = 272, L = 130 6.00 3.39 0.18 0.43 12.47

  • vf

0.48

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals

Comparison (Timings in [sec])

Input B i t D e s c D e s c R S M P S

  • l

v e G P [ P a n ] G P [ N e w t ] S

f

Random uniform, dense n = 128, L = 128 0.06 0.03 0.01 0.01 1.40 0.22 0.08 n = 128, L = 2048 0.07 0.10 0.02 0.01 1.22 0.39 0.09 n = 256, L = 256 0.58 0.15 0.05 0.04 8.98 1.41 0.42 n = 512, L = 128 5.37 0.59 0.05 0.12 246.00 9.85 1.41

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals

Comparison (Timings in [sec])

Input B i t D e s c D e s c R S M P S

  • l

v e G P [ P a n ] G P [ N e w t ] S

f

Taylor expansion of exp at 0 up to n n = 50, L = 215 0.01 0.00 0.00 0.02 0.17 1.27 0.05 n = 75, L = 364 0.03 0.00 0.00 0.07 0.46 6.67 0.05 n = 100, L = 525 0.04 0.01 0.02 0.17 0.92 1.71 0.10 n = 200, L = 1246 0.24 0.13 0.02 1.82 7.02 18.42 0.62

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals

Goals of Further Work

Multiplicity guessing based on Rouch´ e’s theorem Newton-Raphson as an endgame strategy Partial iterations in Durand-Kerner Bernstein basis Parallelization Approximately sparse polynomials

Alexander Kobel Certified Complex Numerical Root Finding

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Intro Subdivision Numerical Comparison Goals

Multiplicity guessing

Theorem (Rouch´ e) Let f , g and ∆ = g − f be holomorphic functions on a simply connected bounded domain D ⊂ C. If |g(z)| > |∆(z)| for all z ∈ ∂D holds, then f and g have the same number of zeroes in D, counted with multiplicity. Let g(Z) = akZ k for some multiplicity guess k ∈ N. ⇒ ∆(Z) = g(Z) − f (Z) = −

  • i=k aiZ i

⇒ |∆(Z)| ≤

  • i=k |ai|
  • Z i
  • ⇒

If there is a r > 0 such that |ak| rk >

i=k |ai| ri, then

f has k roots in Dr(0) (counted with multiplicity).

Alexander Kobel Certified Complex Numerical Root Finding