对与分形相关的若干科学计算论题的研究
摘要
本文研究与分形相关的若干科学计算的论题。全文分4章。第1章是概述。第2章研究代法，例如Euler迭代E_f~n(x)，在相当广泛的条件下，其收敛球半径的准确值是由Riemann球面上的一个斥性不动点确定的。
Euler迭代的迭代函数为
E_f(x)=x-(I-P_f(x))f′(x)~(-1)f(x)，其中
P_f(x)=-1/2f′(x)~(-1)f″(x)f′(x)~(-1)f(x)。
假设L在区间[0，r_0)有非减并且取正值的导数L′，且L(0)＞0。特别地，L可以使非常数的正系数多项式或具正Madaurin系数的解析函数。令
h(t)=-t+integral from n=0 to((t-u)L(u)du)。
我们证明了
定理2．3．1 对h的Euler迭代在区间(0，r_0)上有唯一的不动点r_E。这是一个斥性的额外不动点。
定理2．3．2 设f在B(x~*，r_E)上满足
‖f′(x~*)~(-1)f″(x~*)‖≤L(0)和
‖f′(x~*)~(-1)(f″(x))‖≤integral from n=τρ(x) to ρ(x)(L′(u)du)，其中x~τ=x~*+τ(x-x~*)，0≤τ≤1。则对所有x_0∈B(x~*，r_E)，Euler迭代序列{E_f~n(x_0)}收敛于x~*且满足
‖E_f~n(x_0)-X~*‖≤q~(3n-1)‖x_0-x~*‖，n=1，2，…这里 。
q=(E_h(t_0)/t_0)~(1/2)＜1，t_0=ρ_0(x_0)。并且收敛球的半径r_E作为仅依赖于L而与f无关的常数，是最好可能的。
对于Smale条件，可以取
浙江大学硕士学位论文：对与分形相关的若干科学计算论题的研究
，，、Zy
L（u）＝MM·
［l—y－）
我们对此绘制了彩图2．4．1．改图还被科学通报第47卷19期（2002）采用为封面．
本章的结果发表于［VfllLW2002］．有待进一步研究的问题是，对于一般三阶的 Euler－Hl
迭代族（2．5．l），讨论其收敛球的准确半径与 iemann球面上的动力行为的关系．
第3章研究分形图形制作的理论和技术问题．为了制作一些迭代法的拟Mandelbrot集，
我们解决了代数曲线的连续跟踪问题．证明了
定理3．2．1 设含复参数t的d次代数方程
x（xl（t））＝＊工（t），t）＝ 0
的 d个根xl＊X xz＊X……，xuO）都是单根，则对每个i＝l，2／··，d，当矿有增量N时，
XO）的增量为
j（X（t），t－at）—。。2、
a。It）一一MM＋Uf面广）
‘’t＼’／ d’”＼——”’
11（xi （）一 xj（t））
jd
，＊
根据这个定理，设xo）的d个值已经求得，我们可以用Weierst：ffiss并行迭代t”＊咖］来求
XO ＋ At）的 d个值．并行迭代的公式是
f《X’””’．t十凸t）
x、－＝X、－‘———，l＝1．二…·．Q·
”’”’d”“’一’’—”
回 回0x”～一x一”且
j二l
j＊
我们取
X。U，＝x、（t）i＝1．2…·．d．
如果仅ti充分小，就有
X。＊十山】＝h血大“’，f＝1．2…·．d．
我们还提出了平面的连续填充算法 这个算法与代数曲线的连续跟踪算法相结合后，我
们能对参数取自一个复平面正方形的代数方程所有根进行正确的编号，从而完成所有拟
Mandelbrot集的制作．
此外，我们还解决了生成SulliVan域的一些理论和技术问题以及表现Julia集和填充Jlllla
集的技术问题，为文献［W1998bl、p 19961、广叮E000］、［WL2001］成功制作了这些高质量
的图形．
本章的部分主要结果发表于［RllWW2000］．有待进一步的工作是尚有大量很有意义的分
形图形需要制作．
第4章研究如何用计算机去搜索规则分形集Hausdoof测度的上方估值问题．通过对典型
例子Sierpinski垫片的剖析，我们实际上提出了下列一套策略：
In this paper, we discuss some scientific computing questions on fractal theory. The paper consists of 4 chapters. Chapter 1 is an Introduction. The relationship between the convergence ball of some kinds of iterations in Banach space and their dynamical behavior on Riemann spheres are
investigated in chapter 2. We find some kinds of iterations such as Euler iteration Enf(x), under
rather extensive circumstances, the radius of whose convergence balls are determined accurately by their exclusive fixed points on Riemann sphere.
The Euler iteration Ef is defined as follows:
Ef(x) = x-(I-Pf(x))f'(x)-1f(x), where
We assume that L has both non-decreasing and positive derivative L' in the interval [0, r0)
and L() > 0. In particular, L may be taken to be polynomials of degree more than or equal to 1
with positive coefficients or to be analytic functions with positive Maclaurin coefficients. Then we have proved the following results.
Theorem 2.3.1 The Euler iteration for h has a unique fixed point rE in the interval
(0, r0), which is an exclusive fixed point.
Theorem 2.3.2 Suppose that f satisfies that
where x , . Then for all , the Euler iteration {Enf(X0)} convergence to x and satisfies that
where
Furthermore, the radius rE of the convergence ball is optimal as a constant that only depends
upon L but not f
For Smale's condition, we can take
We draw the graph 2.4.1 under the condition. This graph was taken as the cover graph by Chinese science bulletin vol. 47, No. 19 (2002).
The result of this chapter published in [WhLW2002]. And the question of further study is the relationship between the convergence ball of the iterations of Euler-Hally family (2.5.1) with 3-rd order in Banach space and its dynamical behavior on Riemann spheres.
In the chapter 3, we discus the theories and the techniques of making fractal graphs. In order to draw some graphs of Mandelbrot sets, we work out the problem of continuous tracing of algebraic curves and then prove the following theorem.
Theorem 3.2.1 Assuming that d degree algebraic equation with complex parameter /
have d roots as x1(t) , X2(t) , ......, xd(t), and they are all single root, then for each
i = 1, 2, ....., d , while t has increment At, the increment of xt (t) is
Thus if we have all xi(t) , we can solve all xi(t + Δt) by Weierstrass parallel iteration.[WH1996]The formula of this iteration is
We select
if |Δt| is small enough, then we have
We also put forward the continuous fill arithmetic of plane. After the combination of this arithmetic and the continuous tracking arithmetic of algebraic curves, we can give the correct number of all roots of algebraic equation whose parameter picked up from a square of complex plane, thus we can draw all of the graphs of Mandelbrotsets.
Furthermore, we also solve the theories and techniques problems of making Sullivan field, the technical questions of how to show the Julia sets and plenary Julia sets which draw the high quality graphs of ref. [W1998b], [WH1996], [WH2000] and [WL2001].
The part of main result was published in [WhWW2000]. And the further study still reeds lots of valuable fractal graphs to be drawn.
In chapter 4, we study how to search the upper estimation of the Hausdorff measure of regular fractal sets. By the analyses of Sierpinski gasket as a typical example, we provide the following set of strategy in fact:
1. Choose an abundant cover sets according to the symmetry of the fractal set;
2. Build the coding arithmetic by the generation rules of the pre-fractal sets in the cover sets;
3. Establish coding arithmetic for the cover sets with arc border in assisting with tracing technique of theories on numerical control;
4. Let the computer choose the optimization automatically.
Using the above methods, we find the upper estimation of Sierpinski gasket is the best so far
The main results of this chapter were published in [Whl999] and [WhW1999]. And the question of further re
Euler迭代的迭代函数为
E_f(x)=x-(I-P_f(x))f′(x)~(-1)f(x)，其中
P_f(x)=-1/2f′(x)~(-1)f″(x)f′(x)~(-1)f(x)。
假设L在区间[0，r_0)有非减并且取正值的导数L′，且L(0)＞0。特别地，L可以使非常数的正系数多项式或具正Madaurin系数的解析函数。令
h(t)=-t+integral from n=0 to((t-u)L(u)du)。
我们证明了
定理2．3．1 对h的Euler迭代在区间(0，r_0)上有唯一的不动点r_E。这是一个斥性的额外不动点。
定理2．3．2 设f在B(x~*，r_E)上满足
‖f′(x~*)~(-1)f″(x~*)‖≤L(0)和
‖f′(x~*)~(-1)(f″(x))‖≤integral from n=τρ(x) to ρ(x)(L′(u)du)，其中x~τ=x~*+τ(x-x~*)，0≤τ≤1。则对所有x_0∈B(x~*，r_E)，Euler迭代序列{E_f~n(x_0)}收敛于x~*且满足
‖E_f~n(x_0)-X~*‖≤q~(3n-1)‖x_0-x~*‖，n=1，2，…这里 。
q=(E_h(t_0)/t_0)~(1/2)＜1，t_0=ρ_0(x_0)。并且收敛球的半径r_E作为仅依赖于L而与f无关的常数，是最好可能的。
对于Smale条件，可以取
浙江大学硕士学位论文：对与分形相关的若干科学计算论题的研究
，，、Zy
L（u）＝MM·
［l—y－）
我们对此绘制了彩图2．4．1．改图还被科学通报第47卷19期（2002）采用为封面．
本章的结果发表于［VfllLW2002］．有待进一步研究的问题是，对于一般三阶的 Euler－Hl
迭代族（2．5．l），讨论其收敛球的准确半径与 iemann球面上的动力行为的关系．
第3章研究分形图形制作的理论和技术问题．为了制作一些迭代法的拟Mandelbrot集，
我们解决了代数曲线的连续跟踪问题．证明了
定理3．2．1 设含复参数t的d次代数方程
x（xl（t））＝＊工（t），t）＝ 0
的 d个根xl＊X xz＊X……，xuO）都是单根，则对每个i＝l，2／··，d，当矿有增量N时，
XO）的增量为
j（X（t），t－at）—。。2、
a。It）一一MM＋Uf面广）
‘’t＼’／ d’”＼——”’
11（xi （）一 xj（t））
jd
，＊
根据这个定理，设xo）的d个值已经求得，我们可以用Weierst：ffiss并行迭代t”＊咖］来求
XO ＋ At）的 d个值．并行迭代的公式是
f《X’””’．t十凸t）
x、－＝X、－‘———，l＝1．二…·．Q·
”’”’d”“’一’’—”
回 回0x”～一x一”且
j二l
j＊
我们取
X。U，＝x、（t）i＝1．2…·．d．
如果仅ti充分小，就有
X。＊十山】＝h血大“’，f＝1．2…·．d．
我们还提出了平面的连续填充算法 这个算法与代数曲线的连续跟踪算法相结合后，我
们能对参数取自一个复平面正方形的代数方程所有根进行正确的编号，从而完成所有拟
Mandelbrot集的制作．
此外，我们还解决了生成SulliVan域的一些理论和技术问题以及表现Julia集和填充Jlllla
集的技术问题，为文献［W1998bl、p 19961、广叮E000］、［WL2001］成功制作了这些高质量
的图形．
本章的部分主要结果发表于［RllWW2000］．有待进一步的工作是尚有大量很有意义的分
形图形需要制作．
第4章研究如何用计算机去搜索规则分形集Hausdoof测度的上方估值问题．通过对典型
例子Sierpinski垫片的剖析，我们实际上提出了下列一套策略：
In this paper, we discuss some scientific computing questions on fractal theory. The paper consists of 4 chapters. Chapter 1 is an Introduction. The relationship between the convergence ball of some kinds of iterations in Banach space and their dynamical behavior on Riemann spheres are
investigated in chapter 2. We find some kinds of iterations such as Euler iteration Enf(x), under
rather extensive circumstances, the radius of whose convergence balls are determined accurately by their exclusive fixed points on Riemann sphere.
The Euler iteration Ef is defined as follows:
Ef(x) = x-(I-Pf(x))f'(x)-1f(x), where
We assume that L has both non-decreasing and positive derivative L' in the interval [0, r0)
and L() > 0. In particular, L may be taken to be polynomials of degree more than or equal to 1
with positive coefficients or to be analytic functions with positive Maclaurin coefficients. Then we have proved the following results.
Theorem 2.3.1 The Euler iteration for h has a unique fixed point rE in the interval
(0, r0), which is an exclusive fixed point.
Theorem 2.3.2 Suppose that f satisfies that
where x , . Then for all , the Euler iteration {Enf(X0)} convergence to x and satisfies that
where
Furthermore, the radius rE of the convergence ball is optimal as a constant that only depends
upon L but not f
For Smale's condition, we can take
We draw the graph 2.4.1 under the condition. This graph was taken as the cover graph by Chinese science bulletin vol. 47, No. 19 (2002).
The result of this chapter published in [WhLW2002]. And the question of further study is the relationship between the convergence ball of the iterations of Euler-Hally family (2.5.1) with 3-rd order in Banach space and its dynamical behavior on Riemann spheres.
In the chapter 3, we discus the theories and the techniques of making fractal graphs. In order to draw some graphs of Mandelbrot sets, we work out the problem of continuous tracing of algebraic curves and then prove the following theorem.
Theorem 3.2.1 Assuming that d degree algebraic equation with complex parameter /
have d roots as x1(t) , X2(t) , ......, xd(t), and they are all single root, then for each
i = 1, 2, ....., d , while t has increment At, the increment of xt (t) is
Thus if we have all xi(t) , we can solve all xi(t + Δt) by Weierstrass parallel iteration.[WH1996]The formula of this iteration is
We select
if |Δt| is small enough, then we have
We also put forward the continuous fill arithmetic of plane. After the combination of this arithmetic and the continuous tracking arithmetic of algebraic curves, we can give the correct number of all roots of algebraic equation whose parameter picked up from a square of complex plane, thus we can draw all of the graphs of Mandelbrotsets.
Furthermore, we also solve the theories and techniques problems of making Sullivan field, the technical questions of how to show the Julia sets and plenary Julia sets which draw the high quality graphs of ref. [W1998b], [WH1996], [WH2000] and [WL2001].
The part of main result was published in [WhWW2000]. And the further study still reeds lots of valuable fractal graphs to be drawn.
In chapter 4, we study how to search the upper estimation of the Hausdorff measure of regular fractal sets. By the analyses of Sierpinski gasket as a typical example, we provide the following set of strategy in fact:
1. Choose an abundant cover sets according to the symmetry of the fractal set;
2. Build the coding arithmetic by the generation rules of the pre-fractal sets in the cover sets;
3. Establish coding arithmetic for the cover sets with arc border in assisting with tracing technique of theories on numerical control;
4. Let the computer choose the optimization automatically.
Using the above methods, we find the upper estimation of Sierpinski gasket is the best so far
The main results of this chapter were published in [Whl999] and [WhW1999]. And the question of further re
引文
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[BCS1998] Blum L, Cucker F, Shub M et al. Complexity and Real Computation, New York: Springer-Verlag, 1998
[CGS1983] Curry H, Garnett L, Sullivan D. On the iteration of a rational function: Computer experiments with Newton's method, Communications in Mathematical Physics, 1983, 91: 267～277
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[B1971] Bruno A D. Analytical from of differential equations. Transactions Moscow Math Soc, 1971, 25: 131～288
[B1972] Bruno A D. Analytical from of differential equations. Transactions Moscow Math Soc, 1972, 26: 199～239
[BCS1998] Blum L, Cucker F, Shub M et al. Complexity and Real Computation, New York: Springer-Verlag, 1998
[CGS1983] Curry H, Garnett L, Sullivan D. On the iteration of a rational function: Computer experiments with Newton's method, Communications in Mathematical Physics, 1983, 91: 267～277
[F1919] Fatou P. Sur les equations fonctionnelles. Bull Soc Math France, 1919, 47: 161～271
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