dynamical behaviour of iterative methods with memory ... - uji · contrast to newton’s method,...

The problemIterative methods without memory

Iterative methods with memoryMultidimensional real dynamics

ConclusionsReferences

Dynamical behaviour of iterative methods withmemory for solving nonlinear equations

Minisimposio de Dinamica ComplejaIMAC - Universitat Jaume I, May, 14-15, 2015

Beatriz Campos, Alicia Cordero, Juan R. Torregrosa, Pura Vindel

Universitat Jaume I, Castellon, Spain

Universitat Politecnica de Valencia, Valencia, Spain

Beatriz Campos, Alicia Cordero, Juan R. Torregrosa, Pura Vindel Universitat Jaume I, Castellon, Spain Universitat Politecnica de Valencia, Valencia, SpainDynamical behaviour of iterative methods with memory for solving nonlinear equations




Outline

1 The problem

2 Iterative methods without memory

3 Iterative methods with memoryDevelopment and Convergence analysisDynamical analysis

4 Multidimensional real dynamicsSecant methodModified Steffensen method with memoryModified parametric family with memory

5 Conclusions

6 References





The problem

To find a real solution α of a nonlinear equation f (x) = 0, where f is a scalarfunction, f : I ⊆ R→ R.The best known iterative scheme is Newton’s method

xk+1 = xk −f (xk)

f ′(xk), k = 0, 1, . . .

By using f ′(xk) ≈ f [zk, xk] = f (zk)−f (xk)zk−xk

, Steffensen’s method is obtained.

xk+1 = xk −f (xk)

f [zk, xk], zk = xk + f (xk), k = 0, 1, . . .

Both are of second order, require two functional evaluations per step, andtherefore are optimal in the sense of Kung-Traub’s conjecture [KT], but incontrast to Newton’s method, Steffensen’s scheme is derivative-free.





The problem

Newton’s method

xk+1 = xk −f (xk)

f ′(xk), k = 0, 1, . . .

By using f ′(xk) ≈ f [xk−1, xk] = f (xk−1)−f (xk)xk−1−xk

, Secant method is obtained.

xk+1 = xk −f (xk)

f [xk−1, xk], k = 1, 2, . . .

Now, the order of convergence is not preserved, Secant method hasp = 1+

√5

2 < 2 as order of convergence. This is an iterative method withmemory.





Iterative methods without memory

Order of convergence. Sequence {xk}k≥0, generated by an iterativemethod, has order of local convergence p, if there exist constants C andp such that

limk→∞

|xk+1 − α||xk − α|p

= C.

Efficiency index ([Os]), I = p1/d

p, order of convergence, d, number of functional evaluations per step.Kung-Traub’s conjecture ([KT])

The order of convergence of an iterative method without memory, with dfunctional evaluations per step, can not be greater than the bound 2d−1.When this bound is reached, the method is called optimal.

[Os] A.M. Ostrowski, Solution of equations and systems of equations, Prentice-Hall, Englewood Cliffs, NJ,USA, 1964.

[KT] H. T. Kung, J. F. Traub, Optimal order of one-point and multi-point iteration, J. Assoc. Comput. Math.

21, 643–651 (1974).





Development and Convergence analysisDynamical analysis

Iterative methods with memory

Important elements to be taken into account:Number of parameters/accelerators introduced.Role of the accelerators in the error equation.R-order p implies order p (Ortega-Rheinboldt).

A bit of history:

Secant method, order p = 1+√

52 ≈ 1.618.

xk+1 = xk −f (xk)

f [xk−1, xk], given x0, x1, k ≥ 1.

Traub’s scheme with memory, order p = 1 +√

2 ≈ 2.414

Generalized Secant method, order p = 1+√

1+4n2

Inverse interpolation iterative methods, order p ≈ 10.815.Newton’s interpolation approach, order up to 2d + 2d−1 with oneaccelerating parameter.







A bit of history:


52 ≈ 1.618.


2 ≈ 2.414 [Tr]

xk+1 = xk −f (xk)

Γk,

Γk = f [xk + γkf (xk), xk],

γk = − 1Γk−1

, given x0, γ0.


1+4n2


[Tr] J.F. Traub, Iterative methods for the solution of equations, Chelsea Publishing Company, New York 1964.







A bit of history:


52 ≈ 1.618.


2 ≈ 2.414


1+4n2 [GGN]

x(n)k = x(n−1)

k − [xk−1, xk; F]−1F(x(n−1)k ), k > 1,

where x(0)k = xk and, in the last step, the last computed term is

xk+1 = x(n)k .


[GGN] M. Grau-Sanchez, A. Grau, M. Noguera, Frozen divided difference scheme for solving systems of

nonlinear equations, Journal of Computational and Applied Mathematics 235 (2011) 17391743.







A bit of history:


52 ≈ 1.618.


2 ≈ 2.414


1+4n2

Inverse interpolation iterative methods, order p ≈ 10.815 [Ne]Newton’s interpolation approach, order up to 2d + 2d−1 with oneaccelerating parameter [PDP].

Higher order by using more parameters.Specific expression of the error equation.

[Ne] B. Neta, A new family of high order methods for solving equations, Int. J. Comput. Math. 14, (1983)191-195.

[PDP] M.S. Petkovic, J. Dzunic, L.D. Petkovic, A family of two-point methods with memory for solving

nonlinear equations, Appl. Anal. Discrete Math. 5 (2011) 298317.






Modified Steffensen method

xk+1 = xk −f (xk)

f [zk, xk], k = 0, 1, 2 . . . , zk = xk + γf (xk).

Order of convergence 2, for any nonzero value of parameter γ. The errorequation of this family is

ek+1 = (1 + γf ′(α))c2e2k + O(e3

k), where c2 =f ′′(α)

2f ′(α).

If we take γ = −1/f ′(α), then the order increases, but ...If we take γk = −1/f ′(xk), then the order increases, but ...

We propose

γk =−1

f ′(xk)≈ −1

N′1(xk−1, xk)= − 1

f [xk−1, xk].






Steffensen method with memory

x0, x1 initial guesses,

γk = − 1f [xk−1, xk]

, zk = xk + γkf (xk),

xk+1 = xk −f (xk)

f [zk, xk], k = 1, 2, . . .

(1)

Theorem

Let α be a simple zero of a sufficiently differentiable function f : D ⊆ R→ R in anopen interval D. If x0 and x1 are close enough to α, then the R-order of convergenceof the method with memory (3) is at least 1 +

√2, being its error equation

ek+1 = c22ek−1e2

k + O4(ek−1ek),

where ek−1 = xk−1 − α, ek = xk − α, ck =1k!

f (k)(α)

f ′(α), k ≥ 2 and O4(ek−1ek)

denotes all the terms in which the sum of the exponents of ek−1 and ek is at least 4.






How to determine the R-order?

Theorem (Ortega-Rheinboldt)

Let ψ be an iterative method with memory that generates a sequence {xk} ofapproximations of root α, and let this sequence converges to α. If there exist anonzero constant η and nonnegative numbers ti, i = 0, 1, . . . ,m such that theinequality

|ek+1| ≤ ηm∏

i=0

|ek−i|ti

holds, then the R-order of convergence of the iterative method ψ satisfies theinequality OR(ψ, α) ≥ s∗, where s∗ is the unique positive root of the equation

sm+1 −m∑

i=0

tism−i = 0.

In the case of Steffensen’s method with memory

s2 − 2s− 1 = 0.






A known iterative method without memory

Steffensen-type iterative method with fourth-order of convergence presentedby Zheng et al. in [ZLH], obtained by composition of Steffensen’ andNewton’s scheme and approximating the derivative in a particular way.{

yk = xk − f (xk)f [xk,zk]

, zk = xk + f (xk),

xk+1 = yk − f (yk)f [xk,yk]+(yk−xk)f [xk,zk,yk]

,

where f [x, y] = f (x)−f (y)x−y and f [x, z, y] = f [x,z]−f [z,y]

x−y are the divideddifferences of order 1 and 2, respectively.

Error equation:

ek+1 = (1 + f ′(α))2c2(c22 − c3)e4

k + O(e5k).

[ZLH] Q. Zheng, J. Li, F. Huang, An optimal Steffensen-type family for solving nonlinear equations, Applied

Mathematics and Computation 217 (2011) 9592-9597.Beatriz Campos, Alicia Cordero, Juan R. Torregrosa, Pura Vindel Universitat Jaume I, Castellon, Spain Universitat Politecnica de Valencia, Valencia, SpainDynamical behaviour of iterative methods with memory for solving nonlinear equations





New iterative method with memory

Does the iterative method without memory hold the order of convergencewhen the accelerating parameters λ and γ are introduced?

Theorem

Let α ∈ I be a simple root of a sufficiently differentiable functionf : I ⊆ R→ R in an open interval I. If x0 is close enough to α, then theorder of convergence of the class of two-step methods{

yk = xk − f (xk)f [xk,zk]+λf (zk)

, zk = xk + γf (xk),


,

where λ and γ 6= 0, is at least four and its error equation is given by

ek+1 = (1 + γf ′(α))2(λ+ c2) (c2(λ+ c2)− c3) e4k + O(e5

k).






Development and analysis of convergence

A key element: the error equation


k).

Order of convergence of the family: four when

γ 6= −1/f ′(α) and λ 6= −c2.

To improve the order of convergence,

γ = −1/f ′(α) and λ = −c2 = −f ′′(α)/(2f ′(α)),

gives order 7, being ek+1 = −c22c2

3e7k + O(e8

k) its error equation, but...The idea: calculation of the parameters

γk = −1/f ′(α) and λk = −c2 = −f ′′(α)/(2f ′(α)),

for k = 1, 2, . . ., where f ′ and c2 are approximations to f ′(α) and c2,respectively.








k).

We consider the accelerators:

γk =−1

f ′(α)=−1

N′3(xk), λk = − N′′4 (zk)

2N′4(zk), (2)

where

N3(t) = N3(t; xk, yk−1, xk−1, zk−1) and N4(t) = N4(t; zk, xk, yk−1, zk−1, xk−1).

are Newton’s interpolating polynomials of third and fourth degree.







Theorem

If an initial estimation x0 is close enough to a simple root α of f (x) = 0,being f a real sufficiently differentiable function, then the R-order ofconvergence of the two-point method with memory

x0, γ0, λ0 are given, then z0 = x0 + γ0f (x0),

γk = − 1N′3(xk)

, zk = xk + γkf (xk), λk = − N′′4 (zk)2N′4(zk)

, k = 1, 2, . . .

yk = xk − f (xk)f [zk,xk]+λk f (zk)

,


.

is at least 7. This method is denoted by M7.






Dynamical analysis

Aims:Compare the stability of the seventh-order method with memory withoptimal eighth-order schemes without memory. Methods of Zheng etal. ZLH8, Soleymani S8 and Thukral T8.γ0 = −0.01 for M7.Different kind of functions.

Tools:Software presented in [CCT], implemented in Matlab R2011a.A mesh of 400× 400 points,Error estimation lower than 10−3,Maximum number of 40 iterations.

[CCT] F.I. Chicharro, A. Cordero and J.R. Torregrosa, Drawing dynamical and parameters planes of iterative

families and methods, The Scientific World Journal Volume 2013, Article ID 780153, 11 pages.






Results on f1(x) = 13x4 − x2 − 1

3x + 1

Re{z}

Im{z

}

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

(a) M7

Re{z}

Im{z

}

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

(b) ZLH8

Re{z}

Im{z

}

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

(c) T8

Re{z}

Im{z

}

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

(d) S8

Dynamical planes of the proposed methods on f1(x)






Results on f2(x) = (x− 2)(x6 + x3 + 1)e−x2

Re{z}

Im{z

}

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(e) M7

Re{z}

Im{z

}

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(f) ZLH8

Re{z}

Im{z

}

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(g) T8

Re{z}

Im{z

}

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(h) S8







Results on f3(x) = (x− 1)3 − 1

Re{z}

Im{z

}

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(i) M7

Re{z}

Im{z

}

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(j) ZLH8

Re{z}

Im{z

}

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(k) T8

Re{z}

Im{z

}

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(l) S8







On the dynamics

By using small values of γ0 the seventh-order method with memory M7performs better than optimal eighth-order schemes without memory.Which is the role of parameters γ0?It seems that schemes with memory are more stable than optimalhigher-order methods. Can this be confirmed by the analysis?Can a standard analysis on the complex plane be made for betterunderstanding the behavior of M7 or other schemes with memory?





Secant methodModified Steffensen method with memoryModified parametric family with memory

Can we manage with Complex Dynamics tools?

Let p(x) = (x− a)(x− b), defined on C.

x0, γ0, λ0, are given, then z0 = x0 + γ0p(x0),

y0 = x0 − p(x0)p[z0,x0]+λ0p(z0)

,

x1 = y− p(y0)f [x0,y0]+(y0−x0)f [x0,z0,y0]

γ1 = − 1N′3(z) , z1 = x0 + γ1p(x0), λ1 = − N′′4 (z0)

2N′4(z0), k = 1, 2, . . .

y1 = x1 − p(x1)p[z0,x1]+λ1p(z1)

,

z2 = y1 − p(y1)p[x1,y1]+(y1−x1)p[x1,z0,y1]

.

Two iterations are needed to initialize the process.Has a double fixed point iteration function sense?






Dynamics of methods with memory

Let us redefine the problem:

Let us consider the problem of finding a simple zero of a functionf : I ⊆ R −→ R, that is, a solution α ∈ I of the nonlinear equation f (x) = 0.If an iterative method with memory is employed,

xk+1 = g(xk−1, xk), k ≥ 1

where x0, x1 are the initial estimations.

A fixed point of g will be obtained if xk+1 = xk, that is, g(xk−1, xk) = xk.






Dynamics of methods with memory:second round

Now, this solution can be obtained as a fixed point of G : R2 −→ R2 as

G (xk−1, xk) = (xk, xk+1),

= (xk, g(xk−1, xk)), k = 1, 2, . . . ,

So, we state that (xk−1, xk) is a fixed point of G if

G (xk−1, xk) = (xk−1, xk).

So, not only xk+1 = xk, but also xk−1 = xk by definition of G.







Let G(z) be the vectorial fixed-point function associated to an iterativemethod with memory on a scalar polynomial p(z).

Orbit of a point x∗ ∈ R2: {x∗,G(x∗), . . . ,Gm(x∗), . . .}.A point (z, x) ∈ R2 is a fixed point of G if G(z, x) = (z, x). If a fixedpoint is not a zero of p(z), it is called strange fixed point.A point xc ∈ R2 is a critical point of G if det(G′(xc)) = 0. Indeed, if acritical point is not a zero of p(z) it will be called free critical point.






Stability of the fixed points

Theorem [Ro]

Let G from Rn to Rn be C2. Assume x∗ is a period-k point. Letλ1, λ2, . . . , λn be the eigenvalues of G′(x∗).

a) If all λj have |λj| < 1, then x∗ is attracting.b) If one λj0 has |λj0 | > 1, then x∗ is unstable (repelling or saddle).c) If all λj have |λj| > 1, then x∗ is repelling.

A fixed point is called hyperbolic if all λj have |λj| 6= 1.Moreover, if ∃λi, λj s.t. |λi| < 1 and |λj| > 1, the hyperbolic point iscalled saddle point.If x∗ is an attracting fixed point of G, its basin of attraction A(x∗) isdefined as

A(x∗) ={

x(0) ∈ Rn : Gm(x(0))→ x∗,m→∞}.

[Ro] R.C. Robinson, An Introduction to Dynamical Systems, Continous and Discrete, American Mathematical Society, Providence, 2012.






Return to origins: Secant method

Secant

Secant method is a derivative-free superlinear scheme, p =1 +√

52

,

xk+1 = xk −f (xk)(xk − xk−1)

f (xk)− f (xk−1), k ≥ 1.

so it is not possible to establish a Scaling Theorem (see [CCGT]). Then, wewill work with p(z) = z2 − 1.

Its associate fixed point operator (xk ≡ x and xk−1 ≡ z) on p(z) is

Sp (z, x) =(

x, x− p(x)(x− z)p(x)− p(z)

)=

(x,

1 + xzx + z

).

Note that it is not defined on z = −x.[CCGT] F. Chicharro, A. Cordero, J.M. Gutierrez, J.R. Torregrosa, Complex dynamics of derivative-free methods for nonlinear equations, Applied

Mathematics and Computation 219 (2013) 7023–7035.







TheoremThe only fixed points of the operator associated to Secant iterative methodon quadratic polynomial p(z) correspond to its roots, x = z = ±1, beingboth attracting.

Proof: By solving the equation

Sp (z, x) = (z, x) ⇔ x = z = ±1,

we find that that the only fixed points are (z, x) = (−1,−1) and (z, x) = (1, 1). Tostudy their behavior,

Let us consider (z, x) = (1, 1). Then,

S′p(1, 1) =(

0 10 0

)and its eigenvalues are {λ1 = λ2 = 0}.If (z, x) = (−1,−1), the eigenvalues of the associated Jacobian matrix are also{λ1 = λ2 = 0}. 2







TheoremThe free critical points of the operator associated to Secant iterative methodon quadratic polynomial p(x) are those belonging to the lines x = ±1.

Proof: By definition, (z, x) is a critical point if det(S′p(z, x)) = 0 ⇔ 1−x2

(x+z)2 = 0 2

800× 800 points in (x,z)-plane.

Maximum number of iterations:40(black points).

Stopping criterium: absolute errorlower than 10−3.

Basins of attraction painted in thecolor assigned to each root (whitestars placed in x = z).

Color used is brighter when thenumber of iterations is lower. Dynamical plane of Secant on p(z) = z2 − 1






Modified Steffensen method with memory

MSTGiven x0, x1 initial estimations,

γk =−1

f [xk−1, xk]

wk = xk + γkf (xk)

xk+1 = xk −f (xk)

f [xk,wk], k = 1, 2, . . . .

It is not possible to establish a Scaling Theorem.Associate fixed point operator of MST on p(z) = z2 − 1:

Stp (z, x) =

(x,

z + x(2 + xz)1 + x2 + 2xz

).







Theorem

The fixed points of operator Stp (z, x) are (z, x) = (1, 1) and(z, x) = (−1,−1), being both attracting, and (z, x) = (0, 0), which is asaddle point.


Stp (z, x) = (z, x) ⇔ z = x and (−1 + x2)2x = 0.

So, (z, x) = (1, 1), (z, x) = (−1,−1) and (z, x) = (0, 0) are fixed points ofStp(z, x). To study their behavior,

If z = x = 1 or z = x = −1, the eigenvalues of St′p(1, 1) (resp.St′p(−1,−1)) are {0, 0} and then (z, x) = (1, 1) and (z, x) = (−1,−1)are attracting.

Also, the eigenvalues of St′p(0, 0) are{

1−√

2, 1 +√

2}

and then it isa saddle point. 2







TheoremPoints satisfying x = 1 or x = −1 are free critical points of the fixed pointoperator Stp(z, x).

Dynamical plane of MST on p(z) = z2 − 1Beatriz Campos, Alicia Cordero, Juan R. Torregrosa, Pura Vindel Universitat Jaume I, Castellon, Spain Universitat Politecnica de Valencia, Valencia, SpainDynamical behaviour of iterative methods with memory for solving nonlinear equations





Modified parametric family with memory

HMT fourth-order family

Given x0 initial estimation,

yk = xk − θf (xk)

f ′(xk),

tk = xk −f (yk) + θf (xk)

f ′(xk),

xk+1 = xk −f (tk) + f (yk) + θf (xk)

f ′(xk), k = 1, 2, . . .

Error equation: ek+1 = −2((−1 + θ2

)c2

2

)e3

k + O(e4k).

First step: to avoid the derivatives, introducing parameter γ[HMT] J.L. Hueso, E. Martınez, J.R. Torregrosa, New modifications of Potra-Ptak’s method with optimalfourth and eighth orders of convergence, Journal of Computational and Applied Mathematics 234 (2010)2969–2976.







MHMT third-order family



f [xk,wk],


f [xk,wk],

xk+1 = xk −f (tk) + f (yk) + θf (xk)

f [xk,wk], k = 1, 2, . . .

Error equation: ek+1 = −(2 + f ′(α)γ)(θ− 1)(θ + f ′(α)γ + 1)c22e3

k + O(e4k).

It is not possible to establish a Scaling Theorem.







TheoremLet α be a simple zero of a sufficiently differentiable function f : I ⊂ R→ R in anopen interval I. If x0 and x1 are sufficiently close to α, then the order of convergence

of method with memory MHMT is at least12

(3 +√

13)

, by using

γk = −2

f [xk, xk−1]. The error equation is

ek+1 =(−2((−1 + θ)2c3

2

))ek−1e3

k + O5(ek−1ek),

where cj =1j!

f (j)(α)

f ′(α), j = 2, 3, . . .. However, if θ = 1, the error equation is

ek+1 = −4c52e2

k−1e4k + O7(ek−1ek),

being in this case the local error 2 +√

6.







Associate fixed point operator of MHMT on p(z) = z2 − 1:

Hθp (z, x) =

x, x −

(−1 + x2

)(x + z)

64(1 + xz)6(2 + 2xz)

((−1 + x2)3

q1(z, x, θ) + 16(1 + xz)4q2(z, x) +(−1 + x2)2

q3(z, x, θ)) ,

whereq1(z, x, θ) = θ4(x + z)6 − 8θ3(x + z)4(1 + xz),

q2(z, x) = 8 + x4 + 10xz− 2x3z− z2 + 5x2(−1 + z2

)and

q3(z, x, θ) = −32θ(1+xz)3(−2 + x2 + z2

)+8θ2(x+z)2(1+xz)2

(−4 + 3x2 + z2

).







TheoremThe fixed points of the operator associated to MHMT on quadraticpolynomial p(z) are:

a) The roots z = x = ±1, being both attracting,b) The origin (z, x) = (0, 0), which is an attracting fixed point for−2.34315 ≈ 4(−2 +

√2) ≤ θ < −2

c) The real roots of polynomialm(x) = 2 + θ +

(9− 2θ2

)x2 +

(17− 3θ + 2θ2 + 2θ3

)x4 +(

18 + 4θ2 − 4θ3 − θ4)

x6 +(12 + 3θ − 4θ2 + 3θ4

)x8 +(

5− 2θ2 + 4θ3 − 3θ4)

x10 +(1− θ + 2θ2 − 2θ3 + θ4

)x12, whose

number varies depending on the range of parameter θ: they are two ifθ < −2, none if −2 ≤ θ < 6.66633, two if θ = 6.66633 and four ifθ > 6.66633. All of these fixed points are saddle points.







Sketch of proof:

a) The eigenvalues of H′pθ(±1,±1) are {0, 0}.

b) The eigenvalues of H′pθ(0, 0) are{ 1

4

(−√θ2 + 16θ + 32 + θ + 4

), 1

4

(√θ2 + 16θ + 32 + θ + 4

)}, satisfying

|λi| < 1, i = 1, 2 if −2.34315 ≈ 4(−2 +√

2) ≤ θ < −2.

c) Real roots ri of m(x), when exist, are saddle points:

(m) A.v. eig. of H′p(ri, ri),i = 1, 2, θ < 6.66633

(n) A.v. eig. of H′p(ri, ri),i = 1, 4, θ > 6.66633

(o) A.v. eig. of H′p(ri, ri),i = 2, 3, θ > 6.66633







(p) θ = −5 (q) θ = −2.1 (r) θ = −1

(s) θ = 1 (t) θ = 7 (u) θ = 12







Theorem Critical points

The critical points of Hθp (z, x) are:

x = −1 ∧ z− 1 6= 0,x = 1 ∧ z + 1 6= 0,if θ = 0, the points of the curve 3x3 − 2xz + 3z2 = 8if θ 6= 0, the points of the curve

0 = 7a4(−1 + x2

)2(x + z)6

+ 16(1 + xz)4(−8 + 3x2 − 2xz + 3z2

)−8a3

(−1 + x2

)(x + z)4

(−5− 4xz + 6x3z + x2

(6 + z2

))−32a(1 + xz)3

(2 + 4x4

+ 3x3z− 3z2+ xz

(−8 + z2

)+ x2

(−9 + 2z2

))+8a2

(x + z + x2z + xz2

)2 (12 + 15x4 − 10xz + 6x3z− 5z2

+ x2(−29 + 3z2

))





Conclusions

The tools from multidimensional real dynamics have shown to beuseful for analyzing the behavior of iterative schemes with memory.It is very easy to extend this kind of analysis to schemes with memorythat use more than two previous iterations.The obtained results are consistent with our idea about the stability ofmethods with memory.In this context, what is the role of free critical point? Does exist acritical point inside any immediate basin of attraction?What about iterative methods with memory satisfying ScalingTheorem? How we deal with it? Suggestions are welcome!





References

[ZLH] Q. Zheng, J. Li, F. Huang, An optimal Steffensen-type family for solving nonlinear equations,Applied Mathematics and Computation 217 (2011) 9592-9597

[KT] H. T. Kung, J. F. Traub, Optimal order of one-point and multi-point iteration, J. Assoc. Comput.Math. 21, 643–651 (1974).

[PNPD] M.S. Petkovic, B. Neta, L.D. Petkovic, J. Dzunic, Multipoint methods for solving nonlinearequations, Ed. Elsevier (2013)

[OR] J. M. Ortega, W. G. Rheinboldt, Iterative solutions of nonlinear equations in several variables,Ed. Academic Press, New York (1970).

[Tr] J. F. Traub, Iterative Methods for the Solution of Equations, Prentice Hall, New York, 1964.

[GGN] M. Grau-Sanchez, A. Grau, M. Noguera, Frozen divided difference scheme for solving systemsof nonlinear equations, Journal of Computational and Applied Mathematics 235 (2011). 17391743.

[Ne] B. Neta, A new family of high order methods for solving equations, Int. J. Comput. Math. 14,

(1983) 191-195.





References

[PDP] M.S. Petkovic, J. Dzunic, L.D. Petkovic, A family of two-point methods with memory forsolving nonlinear equations, Appl. Anal. Discrete Math. 5 (2011) 298317.

[Ja] I.O. Jay, A note on Q-order of convergence, BIT Nunerical Mathematics 41, 422–429 (2001).

[CCT] F.I. Chicharro, A. Cordero and J.R. Torregrosa, Drawing dynamical and parameters planes ofiterative families and methods, The Scientific World Journal Volume 2013, Article ID 780153, 11pages.

[Ro] R.C. Robinson, An Introduction to Dynamical Systems, Continous and Discrete, AmericanMathematical Society, Providence, 2012.

[ZLH] Q. Zheng, J. Li, F. Huang, An optimal Steffensen-type family for solving nonlinear equations.Appl. Math. Comput. 217 (2011) 95929597

[T] R. Thukral, Eighth-order iterativemethods without derivatives for solving nonlinear equations,ISRN Appl. Math. 2011(693787) (2011) 12.

[S] F. Soleymani, S. Shateyi, Two optimal eighth-order derivative-free classes of iterative methods.Abstr. Appl. Anal. 2012(318165)(2012) 14.


Quasiconformal maps and Holomorphic Dynamics

Toni Garijo –Universitat Rovira i Virgili–

Xavier Jarque –Universitat de Barcelona–

Universitat Jaume I – Castelló – May 14-15 2015

Plan of the talks

1 Holomorphic Dynamics2 The Quadratic Family and The Newton’s Method3 Quasiconformal Mappings4 Quasiconformal Surgery

Holomorphic Dynamics

Introduction

We study those dynamical systems generated by the iteration of anholomorphic map on:

The plane C,The Riemann sphere C = C ∪∞,The punctured plane C∗ = C \ 0

History:Newton’s Method (1900) – Schröder, Cayley.Local Theory (1890-1910) – Böttcher, Leau, Koenigs.Global Theory (1910-1920) – Fatou, Julia, Lattès.Quasiconformal surgery (1985 - ) – Sullivan, Douady, Hubbard,Shishikura.

Introduction

Holomorphic iteration

Polynomial case P : C→ C is a polynomial.The quadratic family Pc(z) = z2 + c.

Rational case R : C→ C, where P and Q are polynomials.The Newton’s method R(z) = z − p(z)

p′(z) .

Entire case f : C→ C, where ∞ is an essential singularity.The Exponential map Eλ(z) = λez.

Holomorphic in C∗ case f : C∗ → C∗ where ∞ and 0 are essentialsingularities.

The standard family Eλ(z) = eiαzeβ(z+1/z).

Introduction

As in all dynamical systems, we try to understand:

Dynamical Plane: Asymptotic behavior of orbits in the phase space fora given map or a class of maps;Parameter Plane: Stability issues and bifurcations for holomorphicfamilies of maps.

Tools: Complex Analysis, Hyperbolic Geometry, Dynamical Systems,Covering Theory, Differential Geometry, etc...

Montel’s Theorem (today)Local Theory of fixed points (today)Quasiconformal surgery (tomorrow)

Rational case

We consider the case where R : C→ C is a rational map acting on theRiemann sphere C

R(z) = P (z)Q(z) where P and Q are polynomials without common factors.

given an initial value z0 ∈ C, we construct

z1 = R(z0)z2 = R(z1) = R ◦R(z0) = R2(z0)· · · = · · ·zn = R(zn−1) = Rn(z0)

The main goal in the Dynamical Plane is to understand the sequence{Rn(z0)}n≥0 depending on the initial condition z0.

Rational case

Given U an open neighbordood of z0 we would like to study the sequenceof functions

Rn : U 7→ Cz 7→ Rn(z)

Normal family

Definition

Let z0 ∈ C and U be an open neighborhood of z0. The sequence ofholomorphic functions {fn}n≥0

fn : U 7→ C

is normal in z0 , if and only if, there exists a subsequence which converge,uniformly on compacts subsets, to some limit function.

Dichothomy in the Dynamcal plane

Given R : C→ C a rational map, we can classify ervery point of z0depending on the character of the sequence {Rn}n≥0.

Fatou set or stable set,

F(R) = {z0 ∈ C , ∃U open nhb. of z0 ; Rn : U 7→ C is normal }

Julia set of chaotic set,

J (R) = C \ F(R)

.

Montel’s Theorem

Theorem (Montel, 1916)

Given z0 ∈ C and U an open neighborhood of z0. Let a, b and c threedifferent points in C. Assume that the holomorphic sequence of functions{fn}n≥0 verifies that

fn : U 7→ C \ {a, b, c}

then, the sequence {fn}n≥0 is a normal family in z0.

IMPORTANT: If the sequence {fn}n≥0 covers the Riemann sphere exceptthree points, then it is a normal family.

Proof of Montel’s Theorem

We use the following Analysis result.

Theorem (Arzelà-Ascoli, 1882-1884)A sequence of functions (not necessary holomorphic) {fn}n≥0 is normal, ifand only if, it is equicontinuous and uniformly bounded.

Assuming that the sequence of functions is holomorphic,then Uniformly bounded ⇒ Equicontinuous.

|fn(x)| < M , independent of x and n. (Uniformly bounded)|fn(x)− fy(y)| ≤ |f ′n(c)| · |x− y|. (Mean value Theorem)

f ′n(c) =1

2πi

∫γfn(w)(w−c)2dw (Cauchy’s formula)

|f ′n(c)| ≤ Mr (Cauchy Inequality).

Proof of Montel’s Theorem

The universal covering of C \ {a, b, c} is the unit disk D.We can lift the sequence from fn : C→ C \ {a, b, c} to fn : C→ D.

D

��

fn : C

88

// C \ {a, b, c}

Thus, the lift sequence fn is uniformly bounded (|fn| ≤ 1) and thesequence fn is also normal.

Properties of Julia and Fatou sets (from Montel’s Theorem)

Given R : C→ C a rational function, we denote by F(R) and J (R) theFatou and Julia set, respectively.

F(R) is an open set and J (R) is closed.F(R) and J (R) are completely invariant set.R(F(R)) = R−1(F(R)) = F(R), the same for the Julia set.F(Rn) = F(R) and J (Rn) = J (R) for all n > 0.J (R) 6= ∅ if degree(R) ≥ 2.Take z0 ∈ J (R), then J (R) = O−(z0)

The interior of J (R) = ∅ or J (R) = C.Take z0 ∈ J (R), for all ε > 0 the set⋃

n≥0Rn(D(z0, ε))

covers C except at most two points.

Local Theory

Let p be a periodic point of R with minimal period k. We can classify paccording to their multiplier λ

λ = (Rk)′(p) = R′(p) ·R′(R(p)) ·R′(R2(p)) · · ·R′(Rk−1(p))

Why is important λ? Using the local development of Rk at p, we have

Rk(z) = Rk(p) +(Rk)′(p)(z − p) +O((z − p)2)Rk(z) = p +λ(z − p) +O((z − p)2)

Local Theory. Main Question

Is Rk, near p, conformally conjugate (c.c.) to the linear part?

Rk : U −→ Rk(U)ϕ ↓ ≡ ↓ ϕ

λ· : V −→ C

Where U is an open nhb. of p and V an open nhb. of 0.

ϕ(Rk(z)

)= λ · ϕ(z)

Local Theory

p is a superattractor fixed point (f.p.) ⇐⇒ λ = 0.p is an attractor f.p. ⇐⇒ |λ| < 1.p is an repelling f.p. ⇐⇒ |λ| > 1.p is a parabolic f.p. ⇐⇒ λq = 1 for some q ∈ Z, q 6= 0.

The case |λ| = 1 and λq 6= 1 for all q ∈ Z splits into two sub cases:

1 p is a linearizable or a Siegel f.p. if Rk is c.c. to z 7→ λz.

2 p is a not linearizable or a Cremer f.p. if Rk is not c.c. to z 7→ λz.

Local Theory

ClassificationIf p is a superattractor fixed points (f.p.), then Rk(z) is c.c. toz 7→ zn for n > 1. (Böttcher, 1904)If p is an atractor f.p., then Rk(z) is c.c. to z 7→ λz. (Koenigs, 1884)If p is a repelling f.p., then Rk(z) es c.c. a z 7→ λz. (Koenigs, 1884)If p is a parabolic f.p., then , Rk(z) es c.c. az 7→ z + azn+1 + bz2n−1 +O(zN ). (Leau, 1897)If p is an indifferent f.p., |λ| = 1 but λq 6= 1 for all q ∈ Z, q 6= 0.

1 If p is a linearizable f.p., then Rk is c.c. a z 7→ λz. (Siegel, 1942)2 If p is a not linearizable f.p., then Si Rk is not c.c. a z 7→ λz (Cremer,

1927)

Local Theory

Example: Pλ(z) = λz + z2 (Uniparametric family)

Polinomy of degree 2.P (0) = 0, the origin is a fixed point (k = 1).P ′(0) = λ. The multiplier coincides with the parameter. !!For z big enough the orbit escapes to ∞. The point of ∞ is asuperattracting fixed point.

Local Theory

(a) λ = 0 superattracting case (b) λ = 0.5 attracting case

Figure : Two dynamical plane of λz + z2.

Local Theory

(a) λ = −1 parabolic case (b) λ = (−0.737369,−0.67549) lin-earizable case

Figure : Two dynamical planes λz + z2.

Local Theory

Superattracting case

TheoremSuppose f has a superattracting fixed point at z0,

f(z) = z0 + ap(z − z0)p + · · · , ap 6= 0, p ≥ 2.

Then there is a conformal map ξ = ϕ(z) of a nhb. of z0 onto a nhb. of 0which conjugates f(z) to ξp.

Proof: Two normalizations:Assume that z0 = 0. Change of variables ω = z − z0.Assume that ap = 1. Change of variables w = cω, where cp−1 = 1/ap.

Local Theory

After that f(w) = wp + · · · , we wish to find ϕ(z) = z + · · · such that

ϕ(f(w)) = (ϕ(w))p

Idea: Find a sequence {ϕn}n≥0 such that

ϕn(f(w)) = (ϕn+1(w))p , then

ϕ(w) = limn→∞

ϕn(w) is the conjugacy

We define:

ϕn(w) =pn√fn(w) = pn

√wpn + · · · = w pn

√1 + · · ·

ϕn(f(w)) =pn√fn(f(w)) =

(pn+1√

fn+1(w))p

= (ϕn+1(w))p

Local Theory

For the convergence of {ϕn}n≥0 we use that

If∞∏n=1

ϕn+1

ϕn<∞⇒ {ϕn} converges

and

ϕn+1(w)

ϕn(w)= pn

√ϕ1 ◦ fnfn

= 1 +O(p−n)

Then,∞∏n=1

ϕn+1

ϕn<∞

Local Theory

Periodic orbits: Fatou or Julia?Superattracting , attracting and linearizable (Siegel) periodic orbitsbelong to the Fatou set.repelling, parabolic and no linearizable (Cremer) periodic orbits belongto the Julia set.

Global Theory

Dynamical properties of the Julia set J (R):

R|J (R) has sensible dependence on initial conditions.J (R) is selfsimilar.For all z ∈ J (R) the set of all preimages of z is dense in J (R).R is transitive in J (R).Repelling periodic orbits are dense in J (R).

Local Theory (Self Similar)

z2 -1

Global Theory (Julia set)

The Fatou set is an open set. Let U be a connected component of F(R).We have only three possibilities:

Periodic component Rk(U) = U for some k ≥ 0.Preperiodic component Rk+l(U) = Rl(U) for l ≥ 1 and k ≥ 0.Wandering component Rn(U) ∩Rm(U) = ∅ for all n 6= m.

Theorem (Sullivan, 1985)Let R be a rational map of degree d ≥ 2, The Fatou set does NOT havewandering domains.

New tool: Quasiconformal Surgery based on Quasiconformal mappings(Ahlfors-Bers,1960 )

Global Theory (Fatou set)

Classification of Fatou components. Let U be a fixed connectedcomponent of F(R), i.e, R(U) = U , then

U contains an attracting or a superattracting fixed points p ∈ U .∂U contains a parabolic fixed point p ∈ ∂U .U contains a linearizable (Siegel) fixed point p ∈ U .U is a ring (doubly connected) and R : U 7→ U is c.c. to z 7→ λz with|λ| = 1 and λq 6= 1 for all q ∈ Z, q 6= 0. This kind of components arecalled Herman ring.

Example of a Herman Ring

Global Theory

Definition

Given R : C 7→ C a rational map.We say that c is a critical point ⇐⇒ R′(c) = 0.

TheoremLet R be a rational map of degree d ≥ 2. Each fixed Fatou component

superattracting,attracting,parabolic,Siegel disc, andHerman ring

“needs” a critical point associated to it.

Important: Following critical points we can detect Fatou components.

Idea Proof: Without critical points we can extend the local conjugacy.

Global Theory

Definition

Given R : C 7→ C a rational map.We say that c is a critical point ⇐⇒ R′(c) = 0.

TheoremLet R be a rational map of degree d ≥ 2. Each fixed Fatou component

superattracting,attracting,parabolic,Siegel disc, andHerman ring

“needs” a critical point associated to it.

Important: Following critical points we can detect Fatou components.Idea Proof: Without critical points we can extend the local conjugacy.

Holomorphic dynamics: Polynomial iteration andNewton’s method

Antonio Garijo (U. Rovira i Virgili)Xavier Jarque (U. de Barcelona)

U. Jaume I, Castello de la Plana, May 2015

Quadratic family and Newton’s method

Summary: Two parts

1 Bottcher coordinates and polynomial iteration (as an exampleof rational iteration). Parameter plane.

2 Newton’s method (applied to polynomials and transcendentalentire functions).


Polynomial iterationParameter plane


Polynomial iteration

Let

P(z) = zd + ad−1zd−1 + . . . a1z + a0, aj ∈ C, d ≥ 2

be a polynomial of degree d .

We extend the map to the whole sphere by definingP(∞) =∞ and using the local chart

w =1

z

to study the new map at z =∞.

Hence abusing notation we write

P : C 7→ C, C = C ∪ {∞}



Definition: Assume P : C→ C is a polynomial and assume p0 ∈ Cis a superattracting fixed point (P(p0) = p0 and P ′(p0) = 0). Thebasin of attraction of p0 is the set

A(p0) = {z ∈ C | Pn(z)→ p0}.

Moreover we denoted by A?(p0) ⊂ A(p0) the immediate attractingbasin which is the connected component of A(p0) containing p0.

Remark: Notice that if p0 is a superattracting fixed point of P then

P(z) = bm(z − p0)m + bm+1(z − p0)m+1 + . . . bd(z − p0)d ,

with bm 6= 0, 2 ≤ m < d .



Lemma Let

P(z) = adzd + ad−1z

d−1 + . . . a1z + a0, aj ∈ C, d ≥ 2

be a polynomial of degree d . The point z =∞ is a superattracingfixed point of P.

Proof: Let ε > 0. Let C \ D1/ε a disc centered at z =∞. Themap w = φ(z) = 1/z conformally conjugates P(z) in C \ D1/ε to

f (w) =1

P(

1w

)in Dε (that is near w = 0).

f (w) =1

ad1wd + ad−1

1wd−1 + . . . a1

1w + a0

=wd(ad + ad−1w

d + . . . a1wd−1 + a0w

d)−1

=Awd (1 + O(w))



Lemma The iterates of P are bounded on all bounded Fatoucomponents. In particular

A(∞) = A?(∞),

or equivalently, A(∞) is connected.

Proof: Assume, by contradiction, that V is a bounded Fatoucomponent in A(∞). Hence, as n tends to ∞, we have

|Pn(z)| → ∞ for all z ∈ V , and

|Pn(w)| bounded for all w ∈ ∂V .

A contradiction with the maximum modulus principle (a non constantholomorphic function cannot attain its maximum absolute value at any interior pointof its region of definition).



Lemma J (P) = ∂A(∞).

Proof: We prove the two inclusions.

J (P) ⊂ ∂A(∞) = ∂J (P).

Let z0 ∈ ∂A(∞) and U a neighborhood z0. By construction,there exist M > 0 such that |Pn(z0)| < M for all n ≥ 0 andthere exists w ∈ (U ∩ A(∞)) such that |Pn(z0)| → ∞ asn→∞. Hence no subsequence of {Pn|U} might converge.

∂A(∞) ⊂ J (P)...



Definition: The filled-in Julia set of P, is given by

K(P) := C \ A(∞),

that is the union of the Julia set and the bounded components ofF(P).

Lemma: The following statements hold for K(P).

It is compact (be definition).

It has connected complement (previous lemma).

∂K(P) = J (P) (previous lemma).

It is the union of all bounded components of the Fatou settogether with J (P) (previous lemma).



Theorem (Bottcher coordinates) Let

f (w) = adwd + ad+1w

d+1 + . . . , a 6= 0

a holomorphic map defined on some neighborhood U of w = 0.There exists a local holomorphic change of coordinate u = ϕ(w)with ϕ(0) = 0 which conjugates f to u 7→ ud throughout someneighborhood of zero.Furthermore, ϕ is unique up to multiplication by an (d − 1)st rootof unity.

1

ϕw = 0 u = 0

f u 7→ ud

(ϕ ◦ f ) (w) = (ϕ(w))d

U



Definition: We denote by ψε the (local) well defined inverse of theabove map ϕ. By construction ψε maps a ε-disc around the originto a small ε neighborhood around p0.

1

ϕw = p0 u = 0

(ϕ ◦ ψε) (u) = u

Dεψε

A?(p0)



Theorem (Critical points in the basin) There exist a unique opendisc Dr of maximal radius 0 < r ≤ 1 such that ψε extendsholomorphically to a map ψ from the disc Dr into the immediatebasin A(p0).

(a) If r = 1 then ψ maps the unit disc D1 biholomorphically ontoA(p0) and p0 is the only critical point in this basin.

(b) If r < 1 then there is at least one other critical point inA(p0), lying on the boundary of ψ(Dr )

Remark: If P is a quadratic polynomial and p0 ∈ C then we arealways in case (a) above. If p0 =∞ then both cases are available.



1ϕ

w = p0Dε

ψε(u)

A?(p0)

ud

P−1

r



Corollary (Bottcher coordinates at infinity) Let P(z) be apolynomial of degree d ≥ 2. Using the local chart w = 1/z theexpression of P in a neighborhood af z =∞ is given by

f (w) = Awd (1 + O(w))

in a neighborhood of w = 0. Hence locally around z =∞ theaction of the polynomial is holomorphically conjugated to u 7→ ud .

1ϕ

(w = 0)

u = 0

fu 7→ ud

z =∞

ϕ ◦ f ◦ ϕ−1 : u 7→ ud

(ϕ ◦ f ) (w) = (ϕ(w))d

Figure: The holomorphic conjugacy near the point at infinity



Theorem (Connected K(P) ⇐⇒ Bounded critical orbits) Let P apolynomial of degree d ≥ 2.

If K(P) contains all the (finite) critical points of P then bothK(P) and J (P) are connected and the complement of K(P)(i.e., A(∞)) is conformally isomorphic to the exterior of theclosed unit disc under an isomorphism

φ : C \ K(P) 7→ C \ D,

which conjugates P on C \ K(P) to u → ud on C \ D.

If at least one of the critical points of P belongs to C \ K(P),then both K(P) and J (P) have uncountably many connectedcomponents.

Idea of the proof: Blackboard...


Polynomial iteration: The quadratic family

Corollary (The dichotomy): Let Qc(z) = z2 + c, c ∈ C. Then,either

z = 0 has bounded orbit and K(Qc) is connected, or

z = 0 has orbit tending to infinity and K(Qc) is a Cantor set(perfect and totally disconnected).

Proof: There is only only critical point. So, either it escape or ithas bounded orbit.



Definition: Let Qc(z) = z2 + c , c ∈ C. The Mandelbrot set is thedefined as the set of parameters for which K(Qc) is connected, orequivalently,

M := {c ∈ C | Qnc (0)→∞}.

(a) (b)

Figure: The mandelbrot setQuadratic family and Newton’s method

Newton’s method


Newton’s method

General scenario:

Notation: P is a polynomial and F is a transcendental entirefunction.

P(z) = 0 or F (z) = 0.

H(z) = 0 when H is either P or F .

Definition: The map

NH(z) := z − H(z)

H ′(z)

is called the Newton’s map associated to H.

NP is rational, and

NF is transcendental meromorphic.


Newton’s method

Newton’s method: It is a dynamical system defined in C or in C to(numerically) compute the zeros of H.

If we want to solve H(z) = 0 we consider an initial conditionz0 ∈ C and consider its NH -orbit, that is,

NnH (z0) , n ≥ 0.

We want to justify that, generally (or, more precisely, in somecases), this iteration converges, as n 7→ ∞, to a point α such thatH(α) = 0.

This is called

The Newton’s method to compute the zeros of H.


Newton’s method

Lemma: H(α) = 0 ⇐⇒ NH(α) = α.

Proof: α is a zero of H if and only if

H(z) = (z − α)k(a +O((z − α)), a 6= 0

ThereforeH ′(z) = (z − α)k−1(ka +O((z − α)),

and so

NH(z) = z − (z − α)k(a +O((z − α))

(z − α)k−1(ka +O((z − α))= z − (z − α)L(z),

where L(α) = 1/k 6= 0. Clearly H(α) = 0 ⇐⇒ NH(α) = α.


Newton’s method

(a)

Figure: Newton’s method applied to p(z) = z3 − 1.


Newton’s method

Lemma: Let α a simple zero of H. Then N ′H(α) = 0 (α is asuperattracting fixed point of NH).

Proof: A direct computation shows that

N ′H(z) =H(z)H ′′(z)

(H ′(z))2→ N ′H(α) = 0

N ′′H(z) =H ′(z)H ′′(z) + H(z)H ′′′(z)

(H ′(z))2− 2H(z) (H ′′(z))2

(H ′(z))3

From this one can show that, locally, Newton’s method is, at least,quadratic.


Newton’s method

Definition: Let α ∈ C be a (superattracting) fixed point of NH .Then, we define the basin of attraction of α as

A(α) := {z0 ∈ C | limn→∞

NnH (z0) = α}

Main question: Let αj , j = 1, . . . the roots of H. What can besaid about

C \⋃

αj , j=1,...

A(αj) ?


Newton’s method

Lemma: Let T (z) = rz + s, r , s ∈ C. Then, if L(z) = H (T (z))we have

T ◦ NL ◦ T−1 = NH .

Proof: It is equivalent to see that

(T ◦ NL)(z) = (NH ◦ T )(z).

Doing some computations we have

(T ◦ NL)(z) = T

(z − L(z)

L′(z)

)= T

(z − H (T (z))

rH ′ (T (z))

)=rz − H (T (z))

H ′ (T (z))+ s = T (z)− H (T (z))

H ′ (T (z))= (NH ◦ T )(z)


Newton’s method

Corollary:

(a) If H(z) = A (z − α1) (z − α2), then NH is always conformally(linearly) conjugated to

NL(z) =z2 − 1

2z(w 7→ w2).

(b) If H(z) = A (z − α1) (z − α2) (z − α3), then NH isconformally (linearly) conjugated to

L(z) = z(z − 1)(z − a),

for some a ∈ C.

Proof: (a) We choose r and s so that T (α1) = i and T (α2) = −i .(b) We choose r and s so that T (α1) = 0 and T (α2) = 1.


Newton’s method

(a)

Figure: Newton’s method applid to p(z) = z2 + 1.


Newton’s method

(a)

Figure: Choosing a = 0.906 + 0.422i .


Newton’s method

Lemma: If H = P then z =∞ is a repelling fixed point of NP .

Proof: To study the point at infinity we use the local chart (localcoordinate) w = 1/z .

f (w) =1

NP (1/w).

If we write P(z) = zd + ad−1zd−1 + . . . a1z + a0, aj ∈ C, d ≥ 2.

Simple computations show

f (w) =wP ′ (1/w)

P ′ (1/w)− wP (1/w)=

dw +O(w2)

(d − 1) +O(w)=

d

d − 1w+O

(w2).

Hence w = 0 is a repelling fixed point for f and z =∞ is arepelling fixed point for NP .


Newton’s method

Theorem (Przytychi,Schleicher,BFJK): Let NH be a Newton mapand let U be an immediate basin of attraction. Then U isunbounded.

Proof: Assume U is bounded (ζ the attracting fixed point).

U contains finitely many critical points and all of them areattracted to ζ.

Let z0, z1 ∈ U \ {ζ} such that N(z1) = z0. Let γ0 ⊂ U be acurve not touching the critical orbits.

Denote by h the local branch of N−1 mapping z0 to z1. Thisbranch can be extended along γ0 unless γ0 contains asingularity of the inverse (see above and remember that U isbounded).

Repeating the argument, we can define inductivelyγn = h(γn−1) where now h denotes the extension of the initialbranch along the curve

⋃n−1j=0 γj . Set γ =

⋃∞n=0 γn.


Newton’s method

Theorem (Przytychi,Schleicher,BFJK): Let NH be a Newton mapand let U be an immediate basin of attraction. Then U isunbounded.

z0

z1

γ

U

zn

γn

Figure: Summary of the proof

One can see that the diameters of the γn tends to 0.

Hence |zn − zn+1| = |zn − N(zn)| → 0 and therefore there is afinite fixed point in ∂U, which is a contradiction.


Newton’s method

Lemma: If H = P ◦ exp then z =∞ is a parabolic fixed point ofNH with a unique attracting (and repelling) petal.

Proof: First we notice that

NH(z) = z − P(z)

P(z) + P ′(z)

is a rational map. Again doing some computations we can showthat

f (w) =1

NH (1/w)= . . . = w(1 +O(w)).

Hence f (0) = 0 and f ′(0) = 1.


Newton’s method

(a) (b)

Figure: Newton’s method applied to P ◦ exp.


Newton’s method

Lemma: Consider the following examples.

If H1(z) = exp(z), then NH1(z) = z − 1.If H2(z) = exp (exp(−z)), then NH2(z) = z + exp(−z).

In particular, there are no basins of attraction of fixed points.

(a)

Figure: Dynamical plane for NH2 . There are infinitely many domains atinfinity having infinitely many accesses to infinity.


Newton’s method

Remark: Three natural continuations:

About the topology of the Fatou components and Julia set.(Baranski, Fagella, J., Karpinska, 2015)

About the (global) goodness of the method. Are there badinitial conditions? (McMullen, 1985)

Which are the good initial conditions to catch all roots(H = P)? (Hubbard, Schleicher, Sutherland, 2000)


Newton’s method

About the topology

Theorem (BFJK) Let H be a polynomial or an entire function, andlet NH its Newton’s map. Then J (NH) is connected in C, orequivalently, all its Fatou components are simply connecteddomains in C.

Remark: The proof does not work for the Traub’s method

TH(z) := NH(z)− H (NH(z))

H ′(z).

However we believe the result is still true.


Newton’s method

About the initial conditions

Remark: They use

If T (z) = rz + s and Q(z) = P (T (z)) then

T ◦ NQ ◦ T−1 = NP

So they might assume all roots belong to the unit disc.

Connectivity and unboundedness of basins of attraction.

The existence of channels to infinity.

A delicate study of the width of the channels to infinity.


Newton’s method


(a)

Figure: Newton’s map applied to p(z) = (z3 − 1)z(z + 1)(z2 + 14 ).


Newton’s method: Initial conditions when H = P


Fix d ≥ 2 (we do think that d is large).

We take [s = 0.26632 log(d)] circles where [x ] is the minimalentire number bigger than x > 0.

In each circle we take N = [8.32547d log(d)] points.

r` = (1 +√

2)

(d − 1

d

) 2`−14s

and θj =2πj

N

with 1 ≤ ` ≤ s and 0 ≤ j ≤ N − 1.

ThenSd = {r`e iθj , 1 ≤ ` ≤ s , 0 ≤ j ≤ N − 1}




Theorem (HSS, 2001) Fix d ≥ 2. Let P be any polynomial ofdegree d . Let {α1, . . . αd} all roots of P (counting multiplicity).Then there exists at least one initial condition z0 ∈ Sd ∩ A(αj) forj = 1, . . . , d .

Remark: Notice that #Sd ≈ 1.11d log2 d .



Example: Let P(z) = z10 − iz + 1. So d = 10 and s = 1, r = 2.35and N = 191.


Quasiconformal maps and Holomorphic Dynamics

Toni Garijo –Universitat Rovira i Virgili–

Xavier Jarque –Universitat de Barcelona–

Universitat Jaume I – Castelló – May 14-15 2015

Quasiconformal Mappings

Introduction. Why quasiconformal mappings?

(a) Qc1(z) = z2 + c1 (b) Qc2(z) = z2 + c2

Figure : Conjugate dynamics


Global conjugacy between two holomorphic maps, f, g : C→ C

f = φ−1 ◦ g ◦ φ

Regularity of φ?

φ Homeomorphism ? (Yes)φ Conformal ? (No)

There are not so many conformal mappings. In fact,

φ(z) = az + b

The multipliers are preserved at periodic orbits. If p is a fixed point,

f ′(p) = (φ−1)′(g(φ(p))) · g′(φ(p)) · φ′(p) = g′(φ(p))


Quasiconformal maps are more than homeos and less than conformal.

Nice properties of a quasiconformal map φ : C→ C,φ is an homeomorphism.φ is Hölder continuous |φ(z)− φ(w)| < C|z − w|α, with 0 < α < 1.The composition of quasiconformal mappings is a quasiconformalmapping.There exists ∂φ

∂z a.e. and ∂φ∂z a.e. (... perhaps in the distributional sense).

If ∂φ∂z = 0 a.e. then φ is conformal.If E is a set with measure zero, then φ(E) also has measure zero.


Quasiconformal maps are more than homeos and less than conformal.Nice properties of a quasiconformal map φ : C→ C,

φ is an homeomorphism.φ is Hölder continuous |φ(z)− φ(w)| < C|z − w|α, with 0 < α < 1.The composition of quasiconformal mappings is a quasiconformalmapping.There exists ∂φ

∂z a.e. and ∂φ∂z a.e. (... perhaps in the distributional sense).

If ∂φ∂z = 0 a.e. then φ is conformal.If E is a set with measure zero, then φ(E) also has measure zero.


The maps Qc1 and Qc2 are conjugate under a Quasiconformal Mapping φ.

(a) Qc1(z) = z2 + c1 (b) Qc2(z) = z2 + c2

Figure : Conjugate dynamics under a quasiconformal map.

Definition of a Quasiconformal map

Definition ( K-Quasiconformal mapping)

Let U and V be domains in C, and let K ≥ 1 be given. Set k := K−1K+1 .

Then φ : U → V is K−quasiconformal if and only if:φ is an homeomorphism;the partial derivatives ∂φ and ∂φ exists in the sense of distributionsand belong to L2

loc (i.e. are locally square integrable);and satisfy |∂φ| ≤ k|∂φ| in L2

loc.

Note: If φ is a C1 homeomorphism then φ is quasiconformal it if satisfies|∂φ| ≤ k|∂φ|.Geometric meaning of |∂φ||∂φ| ≤ k < 1?

Quasiconformal maps

If φ is conformal at z0, it preserves angles between curves crossing atz0, because

Dφ(z0) : C→ C is a complex linear map z 7→ φ′(z0) z.

In general , if φ : U → C is differentiable at z0:

Dφ(z0) : C→ C is a linear map z 7→ a z + b z.

with a = ∂zφ(z0) and b = ∂zφ(z0)It defines an ellipse in the tangent space at z0

Ez0 = (Dφz0)−1 (S1) Dφz0

φ(z0)z0

Quasiconformal maps





with a = ∂zφ(z0) and b = ∂zφ(z0)

It defines an ellipse in the tangent space at z0

Ez0 = (Dφz0)−1 (S1) Dφz0

φ(z0)z0

Quasiconformal maps





with a = ∂zφ(z0) and b = ∂zφ(z0)It defines an ellipse in the tangent space at z0

Ez0 = (Dφz0)−1 (S1) Dφz0

φ(z0)z0

Quasiconformal maps

Linear Algebra:Consider L : R2 → R2 a linear transformation given by(

xy

)7→(a bc d

)(xy

)Preimage of x2 + y2 = 1 is the conic (ax+ by)2 + (cx+ dy)2 = 1, or

(a2 + c2)︸︷︷︸A

x2 + 2(ab+ cd)︸︷︷︸B

xy + (b2 + d2)︸︷︷︸C

y2 = 1

B2 − 4AC < 0 ⇒ the conic is an ellipse.A = C and B = 0 ⇒ the conic is an circle.

Simple computations B2 − 4AC = −|detL|2 < 0.

Quasiconformal maps

Tangent space:Consider f : R2 → R2, φ = (u, v), then Dφ(p) : Tp → Tφ(p) where

Dφ(p) =

(ux(p) uy(p)vx(p) vy(p)

)The preimage of a circle is an ellipse if detDφ(p) 6= 0.Cauchy-Riemann eq. ux = vy and uy = −vx implies that thepreimage is a circle.

IDEA: We can measure the “distortion" of φ using the properties of thisellipse.

Quasiconformal maps

If φ is orientation preserving (|b| < |a|), the dilatation of the ellipse Ez0 isgiven by

K(z0) =|major axis of Ez0 ||minor axis of Ez0 |

=|a|+ |b||a| − |b|

∈ [1,∞).

The Beltrami coeffcient at z0 is the quantity

µ(z0) =b

a=∂zφ

∂zφ(z0) ∈ D

which codes the information of the ellipse, up to scaling.The quantities that measure the angle distortion at z0 are (assume o.p.):

µ(z0) = ba = ∂zφ

∂zφ(z0) ∈ D and

K(z0) = |a|+|b||a|−|b| ∈ [1,∞)

Note: K(z0) = 1+|µ(z0)|1−|µ(z0)|

Quasiconformal maps



=|a|+ |b||a| − |b|

∈ [1,∞).


µ(z0) =b

a=∂zφ

∂zφ(z0) ∈ D

which codes the information of the ellipse, up to scaling.

The quantities that measure the angle distortion at z0 are (assume o.p.):µ(z0) = b

a = ∂zφ∂zφ

(z0) ∈ D and

K(z0) = |a|+|b||a|−|b| ∈ [1,∞)

Note: K(z0) = 1+|µ(z0)|1−|µ(z0)|

Quasiconformal maps



=|a|+ |b||a| − |b|

∈ [1,∞).


µ(z0) =b

a=∂zφ

∂zφ(z0) ∈ D

which codes the information of the ellipse, up to scaling.The quantities that measure the angle distortion at z0 are (assume o.p.):

µ(z0) = ba = ∂zφ

∂zφ(z0) ∈ D and

K(z0) = |a|+|b||a|−|b| ∈ [1,∞)

Note: K(z0) = 1+|µ(z0)|1−|µ(z0)|

Quasiconformal maps

A map φ which is a orientation preserving and differentiable almosteverywhere defines a field of ellipses σ (up to scaling). Two ways to controlthe distortion of this field of ellipses

The dilatation of the field of ellipses, or the dilatation of the map is

Kφ = ess supz∈U

Kφ(z).

The Beltrami coefficient of ellipses, or the Beltrami coefficient of themap is

||µ||∞ = supz∈U|µ(z)|

We equivalently say that φ has Bounded Distortion

∃K > 1 such that Kφ < K ⇔ ∃k < 1 such that ||µ||∞ < k

Definition of a Quasiconformal map

Definition ( K-Quasiconformal mapping)

Let U and V be domains in C, and let K ≥ 1 be given. Set k := K−1K+1 .

Then φ : U → V is K−quasiconformal if and only if:φ is an homeomorphism;the partial derivatives ∂φ and ∂φ exists in the sense of distributionsand belong to L2

loc (i.e. are locally square integrable);and satisfy |∂φ| ≤ k|∂φ| in L2

loc.and φ has bounded dilatation/distortion Kφ < K, or ||µ||∞ < k.

Example of a Quasiconformal map

Example: Let α > 0 and consider that map φ : C \ {0} → C \ {0}

φ(z) := |z|αz

φ is a K-quasiconformal mappingProof:

φ is an homeo. The inverse is φ−1(z) = |z|−αα+1 z.

∂φ = ∂φ∂z = (α2 + 1)zα/2zα/2. (Existence of partial derivatives)

∂φ = ∂φ∂z = α

2 zα/2+1zα/2−1. (Existence of partial derivatives)

µ(z) =∂φ

∂φ=

α

α+ 2· z/z ( Beltrami coefficient )

||µ||∞ = αα+2 < 1, since α > 0. So, φ has bounded distortion.

Quasiconformal Maps

Given φ a K−quasiconformal mapping we construct the Beltramicoefficient µ(z)

µ(z) =∂φ

∂φ⇔ ∂φ

∂zµ(z) =

∂φ

∂z

We can think this equation as a Partial Differential Equation

∂φ

∂zµ(z) =

∂φ

∂zBeltrami Equation

Given φ a K−q.c. map ; then ∃µ(z) is a measurable map s.t.||µ||∞ ≤ k < 1 verifying the Beltrami EquationGiven µ a measurable map s.t. ||µ||∞ ≤ k < 1; then∃φ a K−q.c.map verifying the Beltrami Equation

Almost complex structures

An almost complex structure (or conformal structure) σ on U is ameasurable field of ellipses (Ez)z defined almost everywhere, up to scaling,with bounded dilatation

K = ess supz∈U

K(Ez) <∞.

Any measurable function µ : U → D defines an almost complex structurewith dilatation

K := K(σ) =1 + k

1− k< 1 where k = ‖µ‖∞.

The standard complex structure σ0 is defined by circles at every point, orby µ0 ≡ 0.

µ0µ

Pullbacks

Quasiconformal (or quasiregular) maps φ : U → V can be used to pull backalmost complex structures in V to almost complex structures in U .

Given σ and a.c.s. on V , we may define a new a.c.s. σ′ = φ∗σ on U , bythe field of ellipses

E′z = (Dφz)−1Eφ(z)

defined almost everywhere.

E′z

zDφz φ(z)

Eφ(z)

Pullbacks

Quasiconformal (or quasiregular) maps φ : U → V can be used to pull backalmost complex structures in V to almost complex structures in U .Given σ and a.c.s. on V , we may define a new a.c.s. σ′ = φ∗σ on U , bythe field of ellipses

E′z = (Dφz)−1Eφ(z)

defined almost everywhere.

E′z

zDφz φ(z)

Eφ(z)

Pullbacks

We say that φ transports σ to σ′, and write it as a pullback

σ′ = φ∗(σ) or µ′ = φ∗µ or (U, µ′)φ−→ (V, µ).

Finally, µ is φ−invariant if φ : U → U transports µ to itself, i.e.

φ∗µ = µ.

Ez

Dφz

φ

Eφ(z)

The Measurable Riemann Mapping Theorem

Theorem (Morrey, Ahlfors, Bers, Bojarski)

Let U ' D (resp. C or C via charts) and µ be a measurable map on Uwith dilatation ||µ||∞ ≤ k < 1. Then, there exists a quasiconformalhomeomorphism φ : U → D (resp. C or C) such that

µ = φ∗µ0.

Moreover,φ is unique up to post-composition with conformal self maps of D(resp. C or C).If µ depends continuosly on a parameter, so does φ. (If U ' C or C,we also have holomorphic dependence).

Note: φ is the solution of the Beltrami Equation.

Quasiconformal Surgery: Application to Dynamics

Theorem (Weyl’s Lemma)If F is a quasiconformal mapping and F ∗µ0 = µ0 then F is holomorphic.

Assume that you have the following diagram:

(C, µ)f−−−−→ (C, µ)

φ

y yφ(C, µ0)

F :=φfφ−1

−−−−−−−→ (C, µ0)

Assume that

f is the model and is µ−invariant, i.e., f∗µ = µ.φ comes from the Measurable Riemann Mapping Theorem.

Then F is holomorphic.Proof:

F ∗µ0 = (φ ◦ f ◦ φ−1)∗(µ0) = (φ ◦ f)∗(µ) = φ∗(µ) = µ0




Dynamical behaviour of iterative methods withmemory for solving nonlinear equations

Minisimposio de Dinamica ComplejaIMAC - Universitat Jaume I, May, 14-15, 2015

Beatriz Campos, Alicia Cordero, Juan R. Torregrosa, Pura Vindel

Universitat Jaume I, Castellon, Spain

Universitat Politecnica de Valencia, Valencia, Spain





Outline

1 The problem

2 Iterative methods without memory

3 Iterative methods with memoryDevelopment and Convergence analysisDynamical analysis

4 Multidimensional real dynamicsSecant methodModified Steffensen method with memoryModified parametric family with memory

5 Conclusions

6 References





The problem

To find a real solution α of a nonlinear equation f (x) = 0, where f is a scalarfunction, f : I ⊆ R→ R.The best known iterative scheme is Newton’s method

xk+1 = xk −f (xk)

f ′(xk), k = 0, 1, . . .

By using f ′(xk) ≈ f [zk, xk] = f (zk)−f (xk)zk−xk

, Steffensen’s method is obtained.

xk+1 = xk −f (xk)

f [zk, xk], zk = xk + f (xk), k = 0, 1, . . .

Both are of second order, require two functional evaluations per step, andtherefore are optimal in the sense of Kung-Traub’s conjecture [KT], but incontrast to Newton’s method, Steffensen’s scheme is derivative-free.





The problem

Newton’s method

xk+1 = xk −f (xk)

f ′(xk), k = 0, 1, . . .

By using f ′(xk) ≈ f [xk−1, xk] = f (xk−1)−f (xk)xk−1−xk

, Secant method is obtained.

xk+1 = xk −f (xk)

f [xk−1, xk], k = 1, 2, . . .

Now, the order of convergence is not preserved, Secant method hasp = 1+

√5

2 < 2 as order of convergence. This is an iterative method withmemory.





Iterative methods without memory

Order of convergence. Sequence {xk}k≥0, generated by an iterativemethod, has order of local convergence p, if there exist constants C andp such that

limk→∞

|xk+1 − α||xk − α|p

= C.

Efficiency index ([Os]), I = p1/d

p, order of convergence, d, number of functional evaluations per step.Kung-Traub’s conjecture ([KT])

The order of convergence of an iterative method without memory, with dfunctional evaluations per step, can not be greater than the bound 2d−1.When this bound is reached, the method is called optimal.

[Os] A.M. Ostrowski, Solution of equations and systems of equations, Prentice-Hall, Englewood Cliffs, NJ,USA, 1964.

[KT] H. T. Kung, J. F. Traub, Optimal order of one-point and multi-point iteration, J. Assoc. Comput. Math.

21, 643–651 (1974).







Important elements to be taken into account:Number of parameters/accelerators introduced.Role of the accelerators in the error equation.R-order p implies order p (Ortega-Rheinboldt).

A bit of history:


52 ≈ 1.618.

xk+1 = xk −f (xk)

f [xk−1, xk], given x0, x1, k ≥ 1.


2 ≈ 2.414


1+4n2








A bit of history:


52 ≈ 1.618.


2 ≈ 2.414 [Tr]

xk+1 = xk −f (xk)

Γk,

Γk = f [xk + γkf (xk), xk],

γk = − 1Γk−1

, given x0, γ0.


1+4n2


[Tr] J.F. Traub, Iterative methods for the solution of equations, Chelsea Publishing Company, New York 1964.







A bit of history:


52 ≈ 1.618.


2 ≈ 2.414


1+4n2 [GGN]

x(n)k = x(n−1)

k − [xk−1, xk; F]−1F(x(n−1)k ), k > 1,

where x(0)k = xk and, in the last step, the last computed term is

xk+1 = x(n)k .


[GGN] M. Grau-Sanchez, A. Grau, M. Noguera, Frozen divided difference scheme for solving systems of

nonlinear equations, Journal of Computational and Applied Mathematics 235 (2011) 17391743.







A bit of history:


52 ≈ 1.618.


2 ≈ 2.414


1+4n2

Inverse interpolation iterative methods, order p ≈ 10.815 [Ne]Newton’s interpolation approach, order up to 2d + 2d−1 with oneaccelerating parameter [PDP].

Higher order by using more parameters.Specific expression of the error equation.

[Ne] B. Neta, A new family of high order methods for solving equations, Int. J. Comput. Math. 14, (1983)191-195.

[PDP] M.S. Petkovic, J. Dzunic, L.D. Petkovic, A family of two-point methods with memory for solving

nonlinear equations, Appl. Anal. Discrete Math. 5 (2011) 298317.






Modified Steffensen method

xk+1 = xk −f (xk)

f [zk, xk], k = 0, 1, 2 . . . , zk = xk + γf (xk).

Order of convergence 2, for any nonzero value of parameter γ. The errorequation of this family is

ek+1 = (1 + γf ′(α))c2e2k + O(e3

k), where c2 =f ′′(α)

2f ′(α).

If we take γ = −1/f ′(α), then the order increases, but ...If we take γk = −1/f ′(xk), then the order increases, but ...

We propose

γk =−1

f ′(xk)≈ −1

N′1(xk−1, xk)= − 1

f [xk−1, xk].






Steffensen method with memory

x0, x1 initial guesses,

γk = − 1f [xk−1, xk]

, zk = xk + γkf (xk),

xk+1 = xk −f (xk)

f [zk, xk], k = 1, 2, . . .

(1)

Theorem

Let α be a simple zero of a sufficiently differentiable function f : D ⊆ R→ R in anopen interval D. If x0 and x1 are close enough to α, then the R-order of convergenceof the method with memory (3) is at least 1 +

√2, being its error equation

ek+1 = c22ek−1e2

k + O4(ek−1ek),

where ek−1 = xk−1 − α, ek = xk − α, ck =1k!

f (k)(α)

f ′(α), k ≥ 2 and O4(ek−1ek)

denotes all the terms in which the sum of the exponents of ek−1 and ek is at least 4.






How to determine the R-order?

Theorem (Ortega-Rheinboldt)

Let ψ be an iterative method with memory that generates a sequence {xk} ofapproximations of root α, and let this sequence converges to α. If there exist anonzero constant η and nonnegative numbers ti, i = 0, 1, . . . ,m such that theinequality

|ek+1| ≤ ηm∏

i=0

|ek−i|ti

holds, then the R-order of convergence of the iterative method ψ satisfies theinequality OR(ψ, α) ≥ s∗, where s∗ is the unique positive root of the equation

sm+1 −m∑

i=0

tism−i = 0.

In the case of Steffensen’s method with memory

s2 − 2s− 1 = 0.






A known iterative method without memory

Steffensen-type iterative method with fourth-order of convergence presentedby Zheng et al. in [ZLH], obtained by composition of Steffensen’ andNewton’s scheme and approximating the derivative in a particular way.{

yk = xk − f (xk)f [xk,zk]

, zk = xk + f (xk),


,

where f [x, y] = f (x)−f (y)x−y and f [x, z, y] = f [x,z]−f [z,y]

x−y are the divideddifferences of order 1 and 2, respectively.

Error equation:

ek+1 = (1 + f ′(α))2c2(c22 − c3)e4

k + O(e5k).

[ZLH] Q. Zheng, J. Li, F. Huang, An optimal Steffensen-type family for solving nonlinear equations, Applied

Mathematics and Computation 217 (2011) 9592-9597.Beatriz Campos, Alicia Cordero, Juan R. Torregrosa, Pura Vindel Universitat Jaume I, Castellon, Spain Universitat Politecnica de Valencia, Valencia, SpainDynamical behaviour of iterative methods with memory for solving nonlinear equations





New iterative method with memory

Does the iterative method without memory hold the order of convergencewhen the accelerating parameters λ and γ are introduced?

Theorem

Let α ∈ I be a simple root of a sufficiently differentiable functionf : I ⊆ R→ R in an open interval I. If x0 is close enough to α, then theorder of convergence of the class of two-step methods{

yk = xk − f (xk)f [xk,zk]+λf (zk)

, zk = xk + γf (xk),


,

where λ and γ 6= 0, is at least four and its error equation is given by


k).







A key element: the error equation


k).

Order of convergence of the family: four when

γ 6= −1/f ′(α) and λ 6= −c2.

To improve the order of convergence,

γ = −1/f ′(α) and λ = −c2 = −f ′′(α)/(2f ′(α)),

gives order 7, being ek+1 = −c22c2

3e7k + O(e8

k) its error equation, but...The idea: calculation of the parameters

γk = −1/f ′(α) and λk = −c2 = −f ′′(α)/(2f ′(α)),

for k = 1, 2, . . ., where f ′ and c2 are approximations to f ′(α) and c2,respectively.








k).

We consider the accelerators:

γk =−1

f ′(α)=−1

N′3(xk), λk = − N′′4 (zk)

2N′4(zk), (2)

where

N3(t) = N3(t; xk, yk−1, xk−1, zk−1) and N4(t) = N4(t; zk, xk, yk−1, zk−1, xk−1).

are Newton’s interpolating polynomials of third and fourth degree.







Theorem

If an initial estimation x0 is close enough to a simple root α of f (x) = 0,being f a real sufficiently differentiable function, then the R-order ofconvergence of the two-point method with memory

x0, γ0, λ0 are given, then z0 = x0 + γ0f (x0),

γk = − 1N′3(xk)

, zk = xk + γkf (xk), λk = − N′′4 (zk)2N′4(zk)

, k = 1, 2, . . .

yk = xk − f (xk)f [zk,xk]+λk f (zk)

,


.

is at least 7. This method is denoted by M7.






Dynamical analysis

Aims:Compare the stability of the seventh-order method with memory withoptimal eighth-order schemes without memory. Methods of Zheng etal. ZLH8, Soleymani S8 and Thukral T8.γ0 = −0.01 for M7.Different kind of functions.

Tools:Software presented in [CCT], implemented in Matlab R2011a.A mesh of 400× 400 points,Error estimation lower than 10−3,Maximum number of 40 iterations.

[CCT] F.I. Chicharro, A. Cordero and J.R. Torregrosa, Drawing dynamical and parameters planes of iterative

families and methods, The Scientific World Journal Volume 2013, Article ID 780153, 11 pages.






Results on f1(x) = 13x4 − x2 − 1

3x + 1

Re{z}

Im{z

}

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

(a) M7

Re{z}

Im{z

}

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

(b) ZLH8

Re{z}

Im{z

}

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

(c) T8

Re{z}

Im{z

}

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

(d) S8







Results on f2(x) = (x− 2)(x6 + x3 + 1)e−x2

Re{z}

Im{z

}

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(e) M7

Re{z}

Im{z

}

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(f) ZLH8

Re{z}

Im{z

}

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(g) T8

Re{z}

Im{z

}

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(h) S8







Results on f3(x) = (x− 1)3 − 1

Re{z}

Im{z

}

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(i) M7

Re{z}

Im{z

}

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(j) ZLH8

Re{z}

Im{z

}

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(k) T8

Re{z}

Im{z

}

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

(l) S8







On the dynamics

By using small values of γ0 the seventh-order method with memory M7performs better than optimal eighth-order schemes without memory.Which is the role of parameters γ0?It seems that schemes with memory are more stable than optimalhigher-order methods. Can this be confirmed by the analysis?Can a standard analysis on the complex plane be made for betterunderstanding the behavior of M7 or other schemes with memory?






Can we manage with Complex Dynamics tools?

Let p(x) = (x− a)(x− b), defined on C.

x0, γ0, λ0, are given, then z0 = x0 + γ0p(x0),

y0 = x0 − p(x0)p[z0,x0]+λ0p(z0)

,

x1 = y− p(y0)f [x0,y0]+(y0−x0)f [x0,z0,y0]

γ1 = − 1N′3(z) , z1 = x0 + γ1p(x0), λ1 = − N′′4 (z0)

2N′4(z0), k = 1, 2, . . .

y1 = x1 − p(x1)p[z0,x1]+λ1p(z1)

,

z2 = y1 − p(y1)p[x1,y1]+(y1−x1)p[x1,z0,y1]

.

Two iterations are needed to initialize the process.Has a double fixed point iteration function sense?






Dynamics of methods with memory

Let us redefine the problem:

Let us consider the problem of finding a simple zero of a functionf : I ⊆ R −→ R, that is, a solution α ∈ I of the nonlinear equation f (x) = 0.If an iterative method with memory is employed,

xk+1 = g(xk−1, xk), k ≥ 1

where x0, x1 are the initial estimations.

A fixed point of g will be obtained if xk+1 = xk, that is, g(xk−1, xk) = xk.







Now, this solution can be obtained as a fixed point of G : R2 −→ R2 as

G (xk−1, xk) = (xk, xk+1),

= (xk, g(xk−1, xk)), k = 1, 2, . . . ,

So, we state that (xk−1, xk) is a fixed point of G if

G (xk−1, xk) = (xk−1, xk).

So, not only xk+1 = xk, but also xk−1 = xk by definition of G.







Let G(z) be the vectorial fixed-point function associated to an iterativemethod with memory on a scalar polynomial p(z).

Orbit of a point x∗ ∈ R2: {x∗,G(x∗), . . . ,Gm(x∗), . . .}.A point (z, x) ∈ R2 is a fixed point of G if G(z, x) = (z, x). If a fixedpoint is not a zero of p(z), it is called strange fixed point.A point xc ∈ R2 is a critical point of G if det(G′(xc)) = 0. Indeed, if acritical point is not a zero of p(z) it will be called free critical point.






Stability of the fixed points

Theorem [Ro]

Let G from Rn to Rn be C2. Assume x∗ is a period-k point. Letλ1, λ2, . . . , λn be the eigenvalues of G′(x∗).

a) If all λj have |λj| < 1, then x∗ is attracting.b) If one λj0 has |λj0 | > 1, then x∗ is unstable (repelling or saddle).c) If all λj have |λj| > 1, then x∗ is repelling.

A fixed point is called hyperbolic if all λj have |λj| 6= 1.Moreover, if ∃λi, λj s.t. |λi| < 1 and |λj| > 1, the hyperbolic point iscalled saddle point.If x∗ is an attracting fixed point of G, its basin of attraction A(x∗) isdefined as

A(x∗) ={

x(0) ∈ Rn : Gm(x(0))→ x∗,m→∞}.

[Ro] R.C. Robinson, An Introduction to Dynamical Systems, Continous and Discrete, American Mathematical Society, Providence, 2012.







Secant

Secant method is a derivative-free superlinear scheme, p =1 +√

52

,

xk+1 = xk −f (xk)(xk − xk−1)

f (xk)− f (xk−1), k ≥ 1.

so it is not possible to establish a Scaling Theorem (see [CCGT]). Then, wewill work with p(z) = z2 − 1.

Its associate fixed point operator (xk ≡ x and xk−1 ≡ z) on p(z) is

Sp (z, x) =(

x, x− p(x)(x− z)p(x)− p(z)

)=

(x,

1 + xzx + z

).

Note that it is not defined on z = −x.[CCGT] F. Chicharro, A. Cordero, J.M. Gutierrez, J.R. Torregrosa, Complex dynamics of derivative-free methods for nonlinear equations, Applied

Mathematics and Computation 219 (2013) 7023–7035.







TheoremThe only fixed points of the operator associated to Secant iterative methodon quadratic polynomial p(z) correspond to its roots, x = z = ±1, beingboth attracting.


Sp (z, x) = (z, x) ⇔ x = z = ±1,

we find that that the only fixed points are (z, x) = (−1,−1) and (z, x) = (1, 1). Tostudy their behavior,

Let us consider (z, x) = (1, 1). Then,

S′p(1, 1) =(

0 10 0

)and its eigenvalues are {λ1 = λ2 = 0}.If (z, x) = (−1,−1), the eigenvalues of the associated Jacobian matrix are also{λ1 = λ2 = 0}. 2







TheoremThe free critical points of the operator associated to Secant iterative methodon quadratic polynomial p(x) are those belonging to the lines x = ±1.

Proof: By definition, (z, x) is a critical point if det(S′p(z, x)) = 0 ⇔ 1−x2

(x+z)2 = 0 2

800× 800 points in (x,z)-plane.

Maximum number of iterations:40(black points).

Stopping criterium: absolute errorlower than 10−3.

Basins of attraction painted in thecolor assigned to each root (whitestars placed in x = z).

Color used is brighter when thenumber of iterations is lower. Dynamical plane of Secant on p(z) = z2 − 1







MSTGiven x0, x1 initial estimations,

γk =−1

f [xk−1, xk]

wk = xk + γkf (xk)

xk+1 = xk −f (xk)

f [xk,wk], k = 1, 2, . . . .

It is not possible to establish a Scaling Theorem.Associate fixed point operator of MST on p(z) = z2 − 1:

Stp (z, x) =

(x,

z + x(2 + xz)1 + x2 + 2xz

).







Theorem

The fixed points of operator Stp (z, x) are (z, x) = (1, 1) and(z, x) = (−1,−1), being both attracting, and (z, x) = (0, 0), which is asaddle point.


Stp (z, x) = (z, x) ⇔ z = x and (−1 + x2)2x = 0.

So, (z, x) = (1, 1), (z, x) = (−1,−1) and (z, x) = (0, 0) are fixed points ofStp(z, x). To study their behavior,

If z = x = 1 or z = x = −1, the eigenvalues of St′p(1, 1) (resp.St′p(−1,−1)) are {0, 0} and then (z, x) = (1, 1) and (z, x) = (−1,−1)are attracting.

Also, the eigenvalues of St′p(0, 0) are{

1−√

2, 1 +√

2}

and then it isa saddle point. 2







TheoremPoints satisfying x = 1 or x = −1 are free critical points of the fixed pointoperator Stp(z, x).

Dynamical plane of MST on p(z) = z2 − 1Beatriz Campos, Alicia Cordero, Juan R. Torregrosa, Pura Vindel Universitat Jaume I, Castellon, Spain Universitat Politecnica de Valencia, Valencia, SpainDynamical behaviour of iterative methods with memory for solving nonlinear equations






HMT fourth-order family



f ′(xk),


f ′(xk),

xk+1 = xk −f (tk) + f (yk) + θf (xk)

f ′(xk), k = 1, 2, . . .

Error equation: ek+1 = −2((−1 + θ2

)c2

2

)e3

k + O(e4k).

First step: to avoid the derivatives, introducing parameter γ[HMT] J.L. Hueso, E. Martınez, J.R. Torregrosa, New modifications of Potra-Ptak’s method with optimalfourth and eighth orders of convergence, Journal of Computational and Applied Mathematics 234 (2010)2969–2976.
