第三章 线性方程组的数值解法(3)

迭代法

Ax = b x = Bx+d

dBxx kk )()1(

nTn Rxxxx ),,,( )0()0(

2)0(

1)0(

迭代公式

Jacobi 迭代法是最简单的一种迭代法

Jacobi 迭代法

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

n n

n n

n n nn n n

a x a x a x b

a x a x a x b

a x a x a x b

0iia 1,

1 n

i i ij j

ii j j i

x b a xa

11 1 1 12 2 13 3 14 4 1

22 2 2 21 1 23 3 24 4 1

1 1 2 2 3 3 ( 1) 1

( ... )

( ... )

( ... )

n n

n n

nn n n n n n n n n

a x b a x a x a x a x

a x b a x a x a x a x

a x b a x a x a x a x

( 1) ( )

1

1 nk k

i i ij j

ii jj i

x b a xa

Jacobi 迭代公式

( 1) ( ) ( ) ( ) ( )

1 1 12 2 13 3 14 4 1

11

( 1) ( ) ( ) ( ) ( )

2 2 21 1 23 3 24 4 2

22

( 1) ( ) ( )

1 1 2 2

1( ... )

1( ... )

1(

k k k k k

n n

k k k k k

n n

k k k

n n n n

nn

x b a x a x a x a xa

x b a x a x a x a xa

x b a x a xa

( ) ( )

3 3 ( 1) 1... )k k

n n n na x a x

上一步结果

一般地，x（k+1）比 x(k) 更接近准确解

( 1) ( ) ( ) ( ) ( )

1 1 12 2 13 3 14 4 1

11

( 1) ( ) ( ) ( ) ( )

2 2 21 1 23 3 24 4 2

22

( 1) ( ) ( ) ( ) ( )

3 3 31 1 32 2 34 4 3

33

1( ... )

1( ... )

1( ... )

k k k k k

n n

k k k k k

n n

k k k k k

n n

x b a x a x a x a xa

x b a x a x a x a xa

x b a x a x a x a xa

( 1) ( ) ( ) ( ) ( )

1 1 2 2 3 3 ( 1) 1

1( ... )k k k k k

n n n n n n n n

nn

x b a x a x a x a xa

( 1)

1

kx

( 1)

1

kx ( 1)

2

kx

( 1)

1

kx ( 1)

2

kx ( 1)

3

kx ( 1)

4

kx

( 1) ( ) ( ) ( ) ( )

1 1 12 2 13 3 14 4 1

11

( 1) ( 1) ( ) ( ) ( )

2 2 21 1 23 3 24 4 2

22

( 1) ( 1) ( 1) ( ) ( )

3 3 31 1 32 2 34 4 3

33

1( ... )

1( ... )

1( ... )

k k k k k

n n

k k k k k

n n

k k k k k

n n

x b a x a x a x a xa

x b a x a x a x a xa

x b a x a x a x a xa

( 1) ( 1) ( 1) ( 1) ( 1)

1 1 2 2 3 3 ( 1) 1

1( ... )k k k k k

n n n n n n n n

nn

x b a x a x a x a xa

1( 1) ( 1) ( )

1 1

1 i nk k k

i i ij j ij j

ii j j i

x b a x a xa

Gauss-Seidel迭代

德国数学家Philipp Ludwig von

Seidel (1821--1896)

Philipp von Seidel's

Seidel completed his school studies in the autumn of 1839, he did not enter university

immediately but received private coaching in mathematics before beginning his

university career. He was coached by L C Schnürlein who was a mathematics teacher.

This was valuable coaching for Seidel, particularly since Schnürlein was a good

mathematician who had studied under Gauss.

Seidel entered the University of Berlin in 1840 and studied under Dirichlet and Encke.

Seidel moved to Königsberg in 1842 where he studied under Bessel, Jacobi and Franz

Neumann. In the autumn of 1843 Jacobi left Königsberg on the grounds of ill health and

set off for Italy with Borchardt, Dirichlet, Schläfli and Steiner.. He obtained his doctorate

from Munich in 1846. Six months later he submitted his habilitation dissertation

Untersuchungen über die Konvergenz und Divergenz der Kettenbrüche and qualified to

become a lecturer at Munich. It is worth noting that these two theses, submitted only six

months apart, were on two completely different topics - the first was on astronomy while

the second was on mathematical analysis. （将概率论引入天文学研究）

He was appointed as an extraordinary (卓越的) professor in Munich in 1847. He was

elected to the Bavarian Academy of Sciences. Other academies also honoured him, for

example he was elected to the academies of Göttingen and of Berlin.

An interesting aspect of Seidel's astronomical work involved, as we mentioned above,

the use of probability theory. However, he did not restrict his use of this mathematical

discipline to astronomy, for he also applied his skills in this area to study the frequency of

certain diseases and also looked at certain questions relating to the climate. He lectured

on probability theory, and also on the method of least squares.

Problems with his eyesight forced Seidel into early retirement. Since he had never

married he had no immediate family to help him when he became ill, but he had an

unmarried sister Lucie Seidel who looked after him until 1889.

GS迭代方法的矩阵形式

1( 1) ( 1) ( )

1 1

1 i nk k k

i i ij j ij j

ii j j i

x b a x a xa

( 1) ( ) ( ) ( ) ( )

11 1 1 12 2 13 3 14 4 1

( 1) ( 1) ( ) ( ) ( )

21 1 22 2 2 23 3 24 4 2

( 1) ( 1) ( 1) ( ) ( )

31 1 32 2 33 3 3 34 4 3

( ... )

( ... )

( ... )

k k k k k

n n

k k k k k

n n

k k k k k

n n

a x b a x a x a x a x

a x a x b a x a x a x

a x a x a x b a x a x

( 1) ( 1) ( 1) ( 1) ( 1)

1 1 2 2 3 3 ( 1) 1

...k k k k k

n n n n n n nn n na x a x a x a x a x b

A L D U

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

0 0 00 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

a a a aa a a aa a a aa a a a

a a a aa a a aa a a aa a a a

L D U

( 1) ( ) ( ) ( ) ( )

11 1 1 12 2 13 3 14 4 1

( 1) ( 1) ( ) ( ) ( )

21 1 22 2 2 23 3 24 4 2

( 1) ( 1) ( 1) ( ) ( )

31 1 32 2 33 3 3 34 4 3

( ... )

( ... )

( ... )

k k k k k

n n

k k k k k

n n

k k k k k

n n

a x b a x a x a x a x

a x a x b a x a x a x

a x a x a x b a x a x

( 1) ( 1) ( 1) ( 1) ( 1)

1 1 2 2 3 3 ( 1) 1

...k k k k k

n n n n n n nn n na x a x a x a x a x b

( 1) ( )( ) k kL D b U x x

( 1) 1 ( ) 1( ) ( )k kL D U L D b x x

迭代矩阵

B

Jacobi迭代矩阵

( 1) 1 ( ) 1( ) ( )k kL D U L D b x x

GS迭代矩阵

( 1) 1 ( ) 1( )k kx D L U x D b

由于利用了最新的信息(或者说，迭代矩阵更接近原矩阵的逆)

GS迭代比Jacobi迭代收敛更快（一般）

• 例T=1000C T=0CT = ?

1 12 0i i iT T T

i i+1i-10 N

( 1) ( ) ( )1 1

1

2

k k ki i iT T T

Jacobi迭代：

i=1, 2, …, N-1

i i+1i-10 N

GS迭代：

( 1) ( 1) ( )1 1

1

2

k k ki i iT T T

i=1, 2, …, N-1

0 1000T 0NT

边界条件

1 12 0i i iT T T

1000 0 0 0 0 0 0 00 0 0 0 0

1000 0 0 0 0 0 0 0500 0 0 0 0

( 1) ( ) ( )1 1

1

2

k k ki i iT T T

1000 0 0 0 0 0 0 0500 250 0 0 0

1000 0 0 0 0 0 0 0625 250 125 0 0

1000 0 0 0 0 0 0 0625 375 125 0 0

1000 0 0 0 0 0 0 00 0 0 0 0

1000 500 250 125 62.5 31.25

( 1) ( 1) ( )1 1

1

2

k k ki i iT T T

1000 625 437.5 250

1000 0718.75 484.375

2 500

3 250

4 125

5 125

6.0000 93.7500

7.0000 78.1250

8.0000 70.3125

9.0000 54.6875

10.0000 54.6875

11.0000 46.8750

12.0000 43.9453

13.0000 40.2832

14.0000 36.2549

15.0000 34.9121

16.0000 30.5481

17.0000 30.5481

18.0000 27.7710

19.0000 26.9775

20.0000 25.2895

21.0000 24.0259

22.0000 23.0937

23.0000 21.5578

24.0000 21.1526

25.0000 19.4696

26.0000 19.4312

27.0000 18.1114

28.0000 17.8969

29.0000 16.9070

30.0000 16.5212

2 500

3 125

4.0000 78.1250

5.0000 54.6875

6.0000 43.9453

7.0000 36.2549

8.0000 30.5481

9.0000 26.9775

10.0000 24.0259

11.0000 21.5578

12.0000 19.4696

13.0000 17.8969

14.0000 16.5212

15.0000 15.2809

16.0000 14.1564

17.0000 13.1323

18.0000 12.1955

19.0000 11.3357

20.0000 10.5442

21.0000 9.8137

22.0000 9.1382

23.0000 8.5125

24.0000 7.9320

25.0000 7.3930

26.0000 6.8921

27.0000 6.4261

28.0000 5.9924

29.0000 5.5886

30.0000 5.2124

迭代法的加速

Jacobi迭代法和G—S迭代法

都涉及到收敛速度问题

如何加快迭代法的速度呢？

)()1()(1

kkk xxrA

加速)()()( kkk xxrA 11

)()(1

)1( kkk xrAx

加速法主要思想

通过加权平均优化 ҧ𝐴

)()*()( )( kkk xxx 111

)()1( )()1()()1( kkkk UxLxbDxDx

bxUDxLD kk )()1( ))1(()(

上式为逐次超松弛法(SOR迭代法)的矩阵形式

))1(()( 1 UDLDB

bLDf 1)(

令

fxBx kk )()1(

法的迭代矩阵为SORB

𝐷 + 𝐿 𝑥 𝑘+1 = 𝐷 + 𝐿 𝑥(𝑘) + (𝑏 − 𝐿 + 𝐷 + 𝑈 𝑥(𝑘))

𝐷𝑥 𝑘+1 = 𝐷𝑥(𝑘) + (𝑏 − 𝐿 + 𝐷 + 𝑈 𝑥(𝑘))

Jacobi

G-S

SOR

𝜔−1𝐷 + 𝐿 𝑥 𝑘+1 = 𝜔−1𝐷 + 𝐿 𝑥(𝑘) + (𝑏 − 𝐿 + 𝐷 + 𝑈 𝑥(𝑘))

残差 𝑟

,1时当 SOR法化为

bLDUxLDx kk 1)(1)1( )()( G-S迭代法

G-S法为SOR法的特例, SOR法为G-S法的加速

例 用G-S法和SOR法求下列方程组的解, 45.1取

321

242

124

3

2

1

x

x

x

3

2

0

要求精度1e-6

解: (1)G-S迭代法

GB ULD 1)(

1

321

042

004

000

200

120

5.03/10

625.025.00

25.05.00

f bLD 1)(

1

321

042

004

3

2

0

3/2

5.0

0

)',0,0()0( ０取初值 x

[x,k]=gauss_seidel(a,b,[1,1,1]',1e-6)

1 1 1

0.7500000 0.3750000 1.5000000

0.5625000 0.5312500 1.5416667

0.6510417 0.5963542 1.6145833

0.7018229 0.6582031 1.6727431

……………………………………….

0.9999933 0.9999923 1.9999926

0.9999943 0.9999935 1.9999937

0.9999952 0.9999944 1.9999946

k = 71

x=

0.999995

0.999994

1.999995

满足精度的解

迭代次数为71次

(1)SOR迭代法

1 1 1

0.6375000 0.0121875 1.3199063

0.2004270 0.3717572 1.3122805

0.6550335 0.5340119 1.6922848

0.7058468 0.7733401 1.7771932

………………………………………..

0.9999990 0.9999976 1.9999991

0.9999984 0.9999993 1.9999989

0.9999998 0.9999994 1.9999998

0.9999996 0.9999998 1.9999997

k = 24

x=

1.000000

1.000000

2.000000

满足精度的解

迭代次数为24次

bLDxUDLDx kk 1)(1)1( )()-)1(()(

SOR法的收敛速度比G-S法要快得多,选取适当的

迭代法的收敛性

1 ( )k kx x | ( ) | 1x L

非线性方程迭代求解的收敛性

线性方程组迭代求解的收敛性?

( 1) ( )k kB f x x || || 1B

向量和矩阵的范数

范数：一种描述向量或矩阵”大小”的度量

• 常用的向量范数

• 常用的矩阵范数

• 矩阵的谱半径

• 例：范数在误差估计中的应用

pxxx pp

n

p

p1

1

1 ，　

px

AxA

p

p

p1sup ，　

nA ,,max)( 1

• 在数值计算中，常用的向量范数有三种。设 ，规定nT

n Rxxxx ),,( 21

ini

n

i

i

n

i

i

xx

xx

xx

1

2/1

1

2

2

11

max"")3(

)("2")2(

"1")1(

范数向量的

范数向量的

范数向量的

(1)矩阵的列范数：

(2)矩阵的行范数：

(3)矩阵的谱范数：

n

i

ijnj

aA1

11max

n

j

ijni

aA1

1max

𝐴 2 = 𝜌(𝐴𝑇𝐴)

13

12A

5}2,5max{1

A

max{3,4} 4A

25

513

13

12

11

32AAT

)22115(2

1

8643.3)22115(2

1max2

A

例：

• 性质

小 结

• 本章主要介绍了解线性方程组的直接法和迭代法

• 直接法的基础是Gauss消去法及其矩阵形式的LU

分解。

• 选取主元素是保证消去法计算稳定性及提高精度

的有效方法，列主元比较常用。

• 利用对称正定矩阵的矩阵形式的特殊性，可以简

化LU分解，得到追赶法及平方根法，这两个方法

• 对于大型稀疏方程组可采用迭代法求解，

比较简便有效的迭代法是 Jacobi迭代和

Gauss-Seidel方法;

• 当选取合适的松弛因子，SOR方法可获得较

快的收敛速度，被广泛应用.

• Jacobi迭代：Ã = D

定理：A行对角优、或A列对角优，Jacobi迭

代收敛。

• Gauss-Seidel迭代：Ã = D + L

定理：A行对角优、或A列对角优、或A正定，

Gauss-Seidel迭代收敛。

• 松弛迭代： Ã = w-1D + L

定理：松弛迭代收敛 0<w<2

定理：A正定且0<w<2 松弛迭代收敛