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Numerical Analysis Zhiping Li LMAM and School of Mathematical Sciences Peking University

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Page 1: LMAM and School of Mathematical Sciences Peking University

Numerical Analysis

Zhiping Li

LMAM and School of Mathematical SciencesPeking University

Page 2: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

��{

��{�S��ª

��{Ä�g��´±��­, � Newton {�«O3u^L­�þü:����O­�, Cq¦)­�� x- ¶��:.

� xk−1, xk �1w¼ê f (x) ":��¥�ü�:, KLü:(xk−1, f (xk−1)), (xk , f (xk)) ���� x-¶�:��I´

xk+1 = xk −xk − xk−1

f (xk)− f (xk−1)f (xk).

ùÒ´��{�S��ª, §��±w�´3 Newton {¥ò���Ç f ′(xk) �����Ç

f (xk )−f (xk−1)xk−xk−1

����,

��{I�kü�Ð��UåÚ, Ïd¡�üÚ{. ù����ØÄ:S�{ØÓ.

2 / 24

Page 3: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

��{

��{�Âñ5ÚÂñ�Ý

½nµ � f (x) 3Ù": x∗ �,���S��ëY��, �f ′(x∗) 6= 0. XJÐ� x0 6= x1 ¿©�C x∗, K��{½Â�S�S� {xk+1}∞k=1 Âñu x∗, �Ù(²þ)Âñ�Ý�±����

r = 1+√

52 ≈ 1.618 �.

y²µ Äk·�F"�Ñ x∗ �����, ¦�±T��¥?Ûü�ØÓ:�Ð�d��{S��ª�Ñ�:Eá3T��¥.

d®�^�§�3 x∗ ��� ∆1 = {x : |x − x∗| ≤ δ1}, ¦�f (x) ∈ C2(∆1), f ′(x) 6= 0, ∀x ∈ ∆1. P M1 =

maxx∈∆1|f ′′(x)|

2 minx∈∆1|f ′(x)| , �

δ2 < 1/M1, - δ = min{δ1, δ2}, ∆ = {x : |x − x∗| ≤ δ}. ·��y²�� ∆ ÷v�¦. 5¿§M = maxx∈∆ |f ′′(x)|

2 minx∈∆ |f ′(x)| ≤ M1, Mδ < 1.

3 / 24

Page 4: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

��{

��{�Âñ5ÚÂñ�Ý�y²£Y¤

é?¿�Ð� x0, x1 ∈ ∆, P {xk}∞k=2 �d��S�{�)�S�. � xk−1, xk ∈ ∆, d x∗ ∈ ∆, f (x∗) = 0, Ú���§

P1(x) = f (xk)x − xk−1

xk − xk−1+ f (xk−1)

x − xkxk−1 − xk

(= f ��5��¼ê)�{��O (5¿§ùp x∗ Ø73 xk−1

Ú xk �m), �3 ξ1 ∈ ∆ ¦�

P1(x∗) = −1

2f ′′(ξ1)(x∗ − xk)(x∗ − xk−1).

,��¡, d���§k

P1(xk+1)−P1(x∗) =f (xk)− f (xk−1)

xk − xk−1(xk+1−x∗) = f ′(ξ2)(xk+1−x∗),

Ù¥ ξ2 3 xk−1, xk �m§Ïd ξ2 ∈ ∆.

4 / 24

Page 5: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

��{

��{�Âñ5ÚÂñ�Ý�y²£Y¤

5¿� P1(xk+1) = 0, ¿P ej = |xj − x∗|, Kd±þüª�

ek+1 =∣∣∣ f ′′(ξ1)

2f ′(ξ2)

∣∣∣ekek−1 ≤maxx∈∆ |f ′′(x)|2 minx∈∆ |f ′(x)|

ekek−1 ≤ (Mδ)δ < δ.

ùÒy² xk+1 ∈ ∆. dþª�k

ek+1 ≤ (Mek−1)ek ≤ (Mδ)ek ≤ · · · ≤ (Mδ)ke1 ≤ (Mδ)kδ ≤ 1

M(Mδ)k+1.

dd9 Mδ < 1 � limk→∞ xk = x∗. Âñ5�y.

��©ÛÂñ�. P E0 = Me0, E1 = Me1, - Ek+1 = EkEk−1,k = 1, 2, · · · , K8B�y Ek+1 ≤ (Mδ)Ek ≤ (Mδ)kE1, ∀k ≥ 1,Ïd, limk→∞ Ek = 0. P yk = lnEk .

5 / 24

Page 6: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

��{

��{�Âñ5ÚÂñ�Ý�y²£Y¤

K yk ÷v�©�§ yk+1 = yk + yk−1, k = 1, 2, · · · , ÙÏ)�L«� Fibonacci ê�yk = C1p

k1 + C2p

k2 , Ù¥ p1, p2 ´A��§

p2 = p + 1

��, p1 = 1+√

52 ≈ 1.618, p2 = 1−

√5

2 ≈ −0.618. u´k

Ek+1

Ep1k

= eC1pk+11 eC2p

k+12

eC1pk+11 eC2p

k2p1

= eC2pk2 (p2−p1). d |p2| < 1, k

limk→∞

Ek+1

Ep1

k

= 1.

dd� Ek Âñu"��Ý� p1 = 1+√

52 ≈ 1.618 ��.

dc©Û�, Mek ≤ Ek , Ïd, ek Âñu"��ÝØ$up1 ≈ 1.618 �.

6 / 24

Page 7: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

��{

��{�Âñ5ÚÂñ�Ý�y²£Y¤

,��¡§�Ä Ek+1 = Ek Ek−1(1 + δ), P yk = ln Ek , K yk ÷v�©�§ yk+1 = yk + yk−1 + ln(1 + δ), k = 1, 2, · · · . T�§���A)� y = − ln(1 + δ), Ïd§�§�Ï)�

yk = C1pk1 + C2p

k2 − ln(1 + δ).

u´k Ek = Ek(1 + δ)−1. q p1 + p2 = 1, ¤±

limk→∞

Ek+1

Ep1

k

= (1 + δ)−p2 .

ù`²§é?¿� δ, Ek Âñu"��Ý�´ p1.

7 / 24

Page 8: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

��{

��{�Âñ5ÚÂñ�Ý�y²£Y¤

é k � 1, M∗ek+1 ≈ (M∗ek)(M∗ek−1), Ù¥ M∗ =∣∣∣ f ′′(x∗)2f ′(x∗)

∣∣∣. Ïd§zk = ln(M∗ek) ìCÂñ��A� Fibonacci ê�. Ïd§ekÂñu"��Ý� p1 ≈ 1.618 �.

5 1µ � Newton {�'§��{Ø^O��ê�§ÏdO�þ�~§¦+Âñ�Ývk����§�E,´��5�"

5 2µ ��{�g��?�Úÿ2�^L­�þn:��g­�� x-¶��:�Ñ#S�:��Ô�{§Ù(²þ)Âñ�Ý´ p3 − p2 − p − 1 = 0 ���¢� ≈ 1.84. ¦+Âñ�ÝEvk����§��Ô�{�`("):��´µ=B�Ñn�¢�Щ� x0, x1, x2, �k�US�� f (x) �E�.

8 / 24

Page 9: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

��5�§|�S�){

��5�§|�¦)— ¯K!(J�]

�Ĺ n > 1 ���þ!n ��§��§|µf1(x1, x2, · · · , xn) = 0,

f2(x1, x2, · · · , xn) = 0,

· · · · · · · · · · · ·fn(x1, x2, · · · , xn) = 0,

Ù¥ fi : Rn → R, i = 1, 2, · · · , n, ¥��k��´��5�.

(J: ØO�þìO�§�æ��´1 "y°()�êÆnØ (�35!��5!· · · );

2 Nõ���¦)g�!�nÚ�{Ã{E��õ�;

] :1 ÛÜ�5z�{!¦)�5�§|�S�{;

2 Ø N��nÚØÄ:S����êÃ'�g���{.

9 / 24

Page 10: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

��5�§|�S�){

��5 Jacobi S�

�¦)�5�§|� Jacobi S�{aq/, 31 i fÚ§3�½ xj , j 6= i �^�e, ò fi �� xi ���5¼ê¦�; é¤k1 ≤ i ≤ n, ѦÑ���§�å�# {xi}ni=1. �{Xeµ

for k = 0, 1, 2, · · ·for i = 1, 2, · · · , n) fi (x

(k)1 , · · · , x (k)

i−1, u, x(k)i+1, · · · , x

(k)n ) = 0 � u;

x(k+1)i = u;

endXJ÷vS�Ê�^�§Kª�Ì�¿ÑÑ x (k+1)

end

10 / 24

Page 11: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

��5�§|�S�){

��5 Gauss-Seidel S�

�¦)�5�§|� Gauss-Seidel S�{aq/, 31 i fÚ§3�½ xj , j 6= i �^�e, ò fi �� xi ���5¼ê¦�, ¦���á=^Ù�# xi . �{Xeµ

for k = 0, 1, 2, · · ·for i = 1, 2, · · · , n) fi (x

(k+1)1 , · · · , x (k+1)

i−1 , u, x(k)i+1, · · · , x

(k)n ) = 0 � u;

x(k+1)i = u;

endXJ÷vS�Ê�^�§Kª�Ì�¿ÑÑ x (k+1)

end

11 / 24

Page 12: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

��5�§|�S�){

��5 SOR S�

�¦)�5�§|� SOR S�{aq/, 31 i fÚ§3�½xj , j 6= i �^�e, ò fi �� xi ���5¼ê¦�, ¦���á=^Ù(ÜtµÏf ω �# xi . �{Xeµ

for k = 0, 1, 2, · · ·for i = 1, 2, · · · , n) fi (x

(k+1)1 , · · · , x (k+1)

i−1 , u, x(k)i+1, · · · , x

(k)n ) = 0 � u;

x(k+1)i = x

(k)i + ω(u − x

(k)i );

endXJ÷vS�Ê�^�§Kª�Ì�¿ÑÑ x (k+1)

end

5µ ù�{3�½^�eUÂñ§���Âñ�ú.

12 / 24

Page 13: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

Newton S�{9ÙU?.�{

ÛÜ�5z���5�§|� Newton S�{

� f : Rn → Rn �1w¼ê�, d Taylor Ðmªk

f(x∗) = f(x(k)) + ∇f(x(k))(x∗ − x(k)) + O(‖x∗ − x(k)‖2),

Ù¥ ∇f(x(k)) =(∂fi (x

(k))

∂x(k)

)´ f � Jacobi Ý.

dd£¡�ÛÜ�5z¤=�Cq�5�§|

∇f(x(k))(x∗ − x(k)) ≈ −f(x(k)).

½Â��5�§|� Newton S�S�µé k = 0, 1, · · ·1 )�5�ê�§| ∇f(x(k))y(k) = −f(x(k));

2 - x(k+1) = x(k) + y(k).

¢SO��, ��±Ú\�5|¢ minλ ‖f(x(k) + λy(k))‖.

13 / 24

Page 14: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

Newton S�{9ÙU?.�{

Newton S�{�C/ Broyden �{

����/aq§Newton S�{���kÛÜÂñ5§é1w¼ê�ü�§Âñ�Ý����;

zÚÑIO� Jacobi Ý, $�þã�, cÙ´p��/.

XÛ;�O� Jacobi Ý, ��±$�þ��CqO�´U? Newton S�{�Ä�Ñu:.

Broyden �{3 ∇f(x(0))−1 �Ä:þ§zÚ�I�é�AgÝ��þ¦{=¼�e�ÚS�¤I�Cq_Ý, l ^é���d¼�S�?��þ y(k).

14 / 24

Page 15: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

Newton S�{9ÙU?.�{

Broyden �{�nØ| — Sherman-Morrison Ún

Únµ � A � n ��_Ý, x, y ∈ Rn. XJyTA−1x 6= −1,KA + xyT ��_, �

(A + xyT )−1 = A−1 − A−1xyTA−1

1 + yTA−1x.

� y(k−1) = x(k) − x(k−1) é��, Cq/k

∇f(x(k))y(k−1) = ∇f(x(k))(x(k)−x(k−1)) ≈ f(x(k))−f(x(k−1)) =: g(k−1).

·�F"é�UCq�O ∇f(x(k)) �Ý A(k), ¦�A(k)y(k−1) = g(k−1), � A(k) = A(k−1) + u(k−1)(y(k−1))T , Ù¥ u(k−1) �½.

15 / 24

Page 16: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

Newton S�{9ÙU?.�{

Broyden �{�g�9�EL§

d^��

g(k−1) = A(k)y(k−1) = A(k−1)y(k−1) + u(k−1)(y(k−1))Ty(k−1).

dd� u(k−1) = g(k−1)−A(k−1)y(k−1)

(y(k−1))T y(k−1) , u´

A(k) = A(k−1) +g(k−1) − A(k−1)y(k−1)

(y(k−1))Ty(k−1)(y(k−1))T .

ØJ�yÚn�^� (y(k−1))T (A(k−1))−1u(k−1) 6= −1 �du (y(k−1))T (A(k−1))−1g(k−1) 6= 0 (Ø�b�o¤á).

16 / 24

Page 17: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

Newton S�{9ÙU?.�{

Broyden �{�g�9�EL§

- A(0) = ∇f(x(0)), � (A(0))−1 ®�. dÚn 4.3.1 �

(A(k))−1

= (A(k−1))−1−[(A(k−1))−1g(k−1)−y(k−1)

](y(k−1))T (A(k−1))−1

(y(k−1))T (A(k−1))−1g(k−1).

k (A(k))−1, Ò�±O�1 k Ú�S�?��þ

y(k) = (A(k))−1f (x(k)).

5 1µ ùÒ´ Broyden �{�Ä�Ú½. dd��§ Broyden�{�O�þ�(é�. �±y² Broyden �{, � f ÷v�½^��, ´ÛÜ��5Âñ�.

5 2µ U?.� Newton S�{�kNõ. ��Ñäk�½^�e�ÛÜ��5Âñ5. éJ'�§��`�.

17 / 24

Page 18: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

����{{0— ÓÔ�{

ÓÔ�{— ���Âñ�{��«kÃ}Á

±þ0���«��5�§Ú��5�§|�S��{Ñ�kÛÜÂñ5§=���Âñ�S�S�§7L�Ñl) x∗ ¿©�C�Ð� x0, ù3¢SA^¥ Ø´��. Ï~�³/²�!�E}Á!±9N$í.

é,AÏa.���5�§½�§|§�±�Eäk�ÛÂñ5��{. ÓÔ{£homotopy¤, �¡�òÿ{£continuation¤,Ò´ù�¡�«kÃ�}Á.

18 / 24

Page 19: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

����{{0— ÓÔ�{

ÓÔ�{�Ä�g�

�¦) F(x) = 0, ��ЩXÚ G(x) = 0, Ù�®�� x0.

½ÂÓÔ¼êH(x, λ) = λF(x) + (1− λ)G(x).

w,k H(x, 0) = G(x), H(x, 1) = F(x).

·�F"H(x, λ) = 0 k) xλ, �§ëY/�6uëêλ ∈ [0, 1].

ù�, �� 0 = λ0 < λ1 < · · · < λn = 1, ¦� λi+1 − λi ¿©�,xi , xλi Ò�±�� xi+1 , xλi+1

�v�S��.

^ÛÜÂñ�S�{�gCq¦) xi , i = 1, 2, · · · , n, �ª��F(x) = 0 �) xn.

19 / 24

Page 20: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

����{{0— ÓÔ�{

ÓÔ�{�¢y

Ú 1: � G(x) = 0 ��®�� x0;Ú 2: é i = 1, 2, · · · , n, ± xi−1 �Ð�,

^ Newton {¦)��5�§| H(x, λi ) = 0,��Cq) xi ;

Ú 3: ��� xn =�¤¦.

5 1µ Ú� λi − λi−1 �>�>½§=g·A�N�.

5 2µ Newton {�±��Ù§?Û�«ÛÜÂñ�S�{.

5 3µ é i = 1, 2, · · · , n − 1, Ø7�Ñp°Ý�ê�). 3Ú�À�·���¹e, xi �°ÝU��

15 |xi+1 − xi | Òv.

5 4µ ý��]Ô5gu G(x) �À�.

20 / 24

Page 21: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

����{{0— ÓÔ�{

ÓÔ�{¢yL§¥�UÑy��¹

éuØÓ�ÓÔ¼ê H(x, λ) ÓÔ�{�U¬�ѱeA«ØÓ�(Jµ

1 �{�l�^n��ÓÔ­�§�ª�ÑF(x) = 0 ����.

2 �{3,� λ0 ?uѧ= limλ→λ0−0 ‖xλ‖ =∞.

3 �{3,� λ0 ?Ñy=ò:, d�, ¤¦�� xλ0 Ø3?Û xλ, λ > λ0, �vC���¥.

4 �{3,� λ0 ?Ñy©�, d�, ¤¦�� xλ0 Ø3?Û­½� xλ, λ > λ0, �vC���¥.

3���/ (2), (3), (4) �§�l��¬�}§d�I���ÓÔ¼ê.

21 / 24

Page 22: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

����{{0— ÓÔ�{

�{ü�ÓÔ¼ê�{

1 H(x, λ) = F(x) + (λ− 1)F(x0).

2 H(x, λ) = λF(x) + (1− λ)A(x− x0), Ù¥A ��_Ý.

5µ ��5`§ÓÔ¼ê�À�vk�½�5. ØL§éõ�ª¼ê®²/¤�@k���{£ë�ë�©z[26]¤.

22 / 24

Page 23: LMAM and School of Mathematical Sciences Peking University

Lecture 11: Numerical Solution for Nonlinear Equations

��5�§�S�){

����{{0— ÓÔ�{

��5�§|¦���`z�{

�¦) F(x) = 0, - G (x) = F(x)TF(x). K¦�¯K�du¦¼ê G (x) ����:. Ïd§�A^�`z�nØÚ�{¦).~^��{k

1 ��eü{�FÝ.�{§ÛÜ�5z�{.

2 Úî{!&6�{§ÛÜ�g%C��{.

3 �[ò»{.

4 · · · · · · · · ·

23 / 24

Page 24: LMAM and School of Mathematical Sciences Peking University

SKoµ3, 6, 8; þÅSKoµ3, 4, 5 (2), (5).

Thank You!