giáo trình phương pháp tính

8/2/2019 Gio trnh phng php tnh

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PHNG PHP TNHB MN TON NG

DNG

H.BCH KHOATP.H CH MINHGing vin:

TS.L Th Qunh H


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2

GII THIU MN HC

MSMH: 006023 S TN CH: 2 S tit: 45 tit

Gio trnh Phng php tnh L Thi Thanh Numerical Analysis Burden & Faires

My tnh b ti Gia hc k: Trc nghim (20%)

Bi tp ln (20%) Cui hc k: Trc nghim (60%)


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NI DUNG MN HC

M u: S gn ng v sai s. Chng 1: Gii phng trnh phi tuyn Chng 2: Gii h phng trnh i s tuyn tnh Chng 3: Ni suy v bnh phng cc tiu Chng 4: Tnh gn ng o hm, tch phn

Chng 5: Gii gn ng phng trnh vi phnthng


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Gii thiu: Khi nim v sai s

1/ SAI S GI THUYT: Chp nhn khi xy dngm hnh

2/ SAI S S LIU BAN U: Cc hng s vtl, o lng3/ SAI S PHNG PHP: phng php gii xp

x sai s (gii hn yu cu)

4/ SAI S TNH TON: ch yu do lm trn strong tnh ton


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Sai s tuyt i & sai s tng i

A: gi tr chnh xc; a: gi tr gn ng. Vit:A a Sai s tuyt i: a = A a (phi thc t: A khng tnh

c!) Thc t: Tm s dng a, cng b cng tt tha

A a a A a a a a A a + a. VitA = a a V dA = , a = 3.14

3.14 0.01 < < 3.14 + 0.01 c th chn a = 0.013.14 0.002 < < 3.14 + 0.002 c th chn a = 0.002 Sai s tng i a

a

A a a

A a


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V d v sai s

A = e; a = 2,7a 0,019 < e < a + 0,019

c th chn a = 0,019Sai s tng i a a/a = 0,019/2,7 0,007


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Cng thc tng qut ca sai s

Gi s phi tm i lngy theo cng thcy = f (x

1

, x2

,, xn

)yx

i, - gi tr chnh xc; xi, y gi tr gn ng

Nufl hm kh vi lin tc th

n

i

i

i

nnxx

x

fxxxfxxxfyy

1

2121,...,,,...,,

n

iin

i

xxxxx

fy

121,...,,

n

ii

i

yx

x

f

y

y

1

ln


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n

i

in

i

xxxxx

fy

1

21,...,,

n

i

i

i

yx

x

f

y

y

1

ln

Sai s ca tng, hiu: nn xxxxxxf ...,...,, 2121

n

i

i

i

xyx

f

1

1

Sai s ca tch, thng

n

i

i

ii

xyxx

f

1

1ln

112

1

121.....,...,, nn xxxxxxf

Cng thc tng qut ca sai s (tt)


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V d tm sai s ca tng v hiu

Chox = 2.51 0.01;y = 2.50 0.01. Tm sai s tuyt i v sai s tng i ca tng

v hiu ca 2 s : S1 =x + y; S2 =x y. So snh sai s tuyt i v sai s tng i ca 2

i lng ny.


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V d tm sai s ca tch v thng

Chox = 2.50 0.01;y = 0.10 0.01. Tm sai s tuyt i v sai s tng i ca tch

v thng ca 2 s : S3 =x y; S4 =x / y. So snh sai s tuyt i v sai s tng i ca 2

i lng ny.


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lm trn s thp phn a thnh a n ch s th ksaudu chm thp phn, ta xt ch s th k+1 l k+1. Nu k+1 5 ta tng k ln mt n v Nu k+1 < 5 ta gi nguyn k

Sai s lm trn: a = a a Lm trn s trong bt ng thc

a x b

Quy trn s v sai s quy trn

Vit s dng thp phn:90,1010 121

i

mma


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Ch s c ngha

Trong cch vit thp phn ca s a, ch s cngha l tt c cc ch s bt u t mt ch s

khc khng tnh t tri sang V d:

10,20003 c 7 ch s c ngha0,010203 c 5 ch s c ngha

10,20300 c 7 ch s c ngha


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V d v ch s c ngha

Trong cch vit thp phn ca mt s, cc ch skhng bn tri khng phi l ch s c ngha!

Tm cc ch s c ngha ca cc s sau0,03456; 10,1110; 0,00456700


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Ch s ng tin

Cho a A vi sai s tuyt i a. Trong cch vit thpphn ca s a, ch s kgi l ng tin, nu

1102

ka

1= 0.001 10

2

ka

k log (2a)

V d: a = 12,3456 vi a = 0,001

vy a c 4 ch s ng tina = 0,0044 a = 0,0054

log 2k a 2k


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V d - ch s ng tin

Cho gi tr h = 6,626176 0,000036 Xc nh s ch s ng tin ca h


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V d

A c gi tr gn ng l a = 12.7 vi sai s tngi a = 0.012%. Trong cch vit thp phn ca a

c bao nhiu ch s ng tin?

Vy a c 2 ch s ng tin sau du thp phn nntng cng a c 3 ch s ng tin

aaam 2log2log m - 2

51,2%)012.07.122log(


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Chng 1: Phng trnh phituyn


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NI DUNG

1. Khi nim tng qut. Cng thc sai s2. Phng php chia i3. Phng php lp n4. Phng php Newton (tip tuyn)5. H phng trnh phi tuyn. Phng php

Newton Raphson.


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Khi nim tng qut

Phng trnh f(x) = 0 (1),f: hm s lin tc, c o hm

Khong cch ly nghim: on [a,b] (hoc khong), trn phngtrnh (1) c nghim duy nht

Nu hm f(x) lin tc trn on [a,b] v gi tr ca hm tri du trn

hai u mt th phng trnh (1) cnghim trn [a, b]. Nu f(x) niu th nghim l duy nht

Tm KCLN: lp bng bin thin


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Cng thc sai s

Cng thc sai s tng qut: Phng trnh f(x) = 0 (1)vi nghim chnh xc trn khong cch ly nghim [a, b]

Nu x* l nghim gn ng ca nghim chnh xc trong[a, b] v , , 0x a b f x m

**

f xx x

m

*

*

f x f xf c

x x

0f x 0f c m

th ta c cng thc nh gi sai s tng qut

Chng minh


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Cng thc sai s (tt)

V d: Phng trnh f(x) = x 2 = 0 c khongcch ly nghim [1,2]

Nu chn nghim gn ng

Gii

x 0.00595 Sai s lun lm trn ln

*

*/0.005?

1.41/0.006?x

ax

b

* 0.006x


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Phng php chia i

tng: Lin tc chiai khong cch ly

nghim f(x) = 0 trn khong cch

ly nghim [a, b]. K hiu:a0 = a , b0 = b

f(a0). f(b0)


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Tm nghim gn ng ca phng trnhf(x) =

0 trn on [a, b] bng phng php chia i

1. f(a). f(b) < 0, a0 = a , b0 = b;2. k = 03. Chia i: ck= (ak+ bk)/2 ; k= (bkak)/2k+1

4. Nu k < th nghim gn ng x* = ck; nukhng:

Nuf(ck)f(ak) < 0 th ak+1 = ak; bk+1 = ck; nukhng ak+1 = ck; bk+1 = bk

k = k+ 1

5. Quay tr li 3


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Phng php chia i (tt)

c im ca phng php chia i: n gin

chnh xc khng cao Tnh tay: lp bng cha mi kt qu trung gian cn thit

n an bn cn n0 a0

(Duf(a0 ))

b0 +

(Duf(b0))

(a0 + b0)/2

Du ?

(b0a0)/2

Nghim xp x


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V d phng php chia i

Xp x nghim ca phng trnh f(x) = x cosx = 0 trnkhong cch ly nghim [0, 1] vi sai s 0.1

12log

log

abn

log 1 0 0.1 1 3log2

n n

n an bn cn n0 0 1 + 0.5

123

0.5

0.5 1 + 0.75 0.25+0.5 0.75 + 0.625 0.125

0.625 0.75 + 0.6875 0.0625


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Bi tp phng php chia i

x = tg (x) trong [4,4.5] visai s nh hn 0.01

log 4.5 4 0.011 5

log2n n

n an bn cn n0 4 4.5 + 4.25 0.251 4.25 4.5 + 4.375 0.125

2 4.375 4.5 + 4.4375 0.06253 4.4375 4.5 + 4.46875 0.031254 4.46875 4.5 + 4.484375 0.0156255 4.484375 4.5 4.4921875 0.0078


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e x sinx = 0 trn [0, 1] vi = 0,1


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n an bn cn n

0 0 + 1 0.5 + 0.51 0.5 + 1 0.75 0.25

2 0.5 + 0.75 0.625 0.125

3 0.5 + 0.625 0.5625 + 0.0625

4 0.5625 + 0.59375 0.578125 + 0.0156255 0.578125 + 0.59375 0.5859375 + 0.0078125

e x sinx = 0 trn [0, 1] vi = 0,1


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3x ln (2x + 1) = 0 trn [ 0.4, 0.2]; = 0,01


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3x ln (2x + 1) = 0 trn [ 0.4, 0.2]; = 0,01

n an bn cn n0 0.4 + 0.2 0.3 + 0.1

1 0.3 + 0.2 0.25 0.052 0.3 + 0.25 0.275 0.0253 0.3 + 0.275 0.2875 0.01254 0.3 + 0.2875 0.29375 + 0.00625

5 0.29375 + 0.2875 0.290625 0.0031256 0.29375 + 0.290625 0.2921875 + 0.00156257 0.2921875 + 0.290625 0.29140625 + 0.00078125


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3. Phng php lp n

Chuyn (1) v dng tng ng trong [a, b] x =g(x) Nghim cn gi l im bt ng ca hmg(x)

Gi tr ban ux0 ty thuc [a, b]xn+1 = g(xn ), n = 0, 1, 2, 3

Dy {xn } l dy lp n, hmg(x) hm lp Dy lp n xn+1 = g(xn) hi t v l nghim ca

phng trnhx =g(x)


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3. Phng php lp n (tt)

V d: Kim tra nhngdy sau c l lp n?

Nu c, vit ra hm lpg.Tnh 5 s hng u cady (x0 bt k). T ,on tnh hi t.

1

1

/ cos

10

/ 115

nn

nn

xa x

zb z


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Dy lp n hi t


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Dy lp n phn k


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Hm co

Hmy =g(x) co trn [a, b] vi h s co q nu nh tn timt s q, 0 q < 1 gi l h s co sao cho:

1 2 1 2 1 2, , :x x a b g x g x q x x

VD: Hmy = x co trn [1/4, 1/4] vi q =1/2 v

2 2

1 2 1 2 1 2 1 2 1 2

1, 1 4,1 4 :

2x x x x x x x x x x

x (a, b), |g(x)| q < 1 thg(x) co trn [a, b] vi h s co q


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V d iu kin co

Trong nhng hm sau y, hm no tho iu kin co?Xc nh hng s q vi cc hm co

2

/ cos , 0,11

/ 1 , , , 0

/ arccos , , , 0

a g x x x

b g x x a b a bx

c g x x x a b a b


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Phng php lp n

Phng trnhf(x) = 0. Xc nh khong cch ly nghim [a, b] a v dng lp nx =g(x), sao cho:

gco trn [a, b] x (a, b),g(x) (a, b),

Lyx0 bt k [a, b]. Dy lpxn+1 =g(xn)

Ch : Nhiu cch chn hm gcng n gin cng tt


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c lng sai s

q: h s co ca hm lp ng(x)

1 01

n

n

q

x x xq 11n n nq

x x x xq

nx x

q

xxqn

log

1log 01

S ln lp ti thiu:


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V d phng php lp n

f(x) =x3 +x 1000 = 0 vi sai s 10-8 Khong cch ly nghim: [9, 10] Lp n:x = 1000 x3 =g(x): Kim tra iu kin co? Xy dng hm lp mi: Dy lp

n xn n0 10

1

2

3

3 1000x x g x

0

31

9,10

1000n n n

x

x g x x

Sai s

1

1n n n

qx x x

q

10.0035

n

n n nx x

9.966554934 1.2 104

9,966667166 4 107

9,966666789 109


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V d phng php lp n (tt) Xp x nghim ca phng trnhf(x) =x cosx = 0 trn [0,

1] vi phng php lp n,x0 = 0 vi sai s 10-8

Dng lpx = cosx =g(x) q = 0.85 x (0,1),g(x) (0,1), x0 = 0 x1 = 1 c lng sai s

n xn n

0 0123

80.85 1 0 101 0.85

n

nx x n

1 5.66670.5403023059 2.60500.8575332158 1.7978


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4. Phng php Newton (tip tuyn)

f(x) = 0 lp n

Cng thc lp Newton:

0

'

f xf x x x g x

f x

n

nnnn

xf

fxxgx

'1


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4. Phng php Newton (tt)


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iu kin lp Newton sai s

iu kin hi t:1. o hm f, f khng

i du trn [a, b]2. Gi tr lp ban u

tho: f(x0) .f(x0) > 0(iu kin Fourier)

c lng sai s: Cng

thc tng qut

**

f xx x

m


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V d phng php lp Newton (tip tuyn)

Gii xp xf(x) =x cosx = 0 trn [0,1] sai s 10-8

1/ Kim tra iu kin hi t

2/ Xy dng dy lp:

n xn n

0 1123

0

1

1

cos

1 sin

n n

n n

n

x

xx x

x

Sai s:

1

n

n n

f xx x

0.460.750363868 0.0190.739112891 4.7 10 5

0.739085133 2.9 10 10


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Bi tp

sinx e x = 0 trn [0,1]

iu kin hi t:1. o hmf,f khng i du trn [a, b]2. Gi tr lp ban u tho: f(x0) .f(x0) > 0 (iu

kin Fourier)

c lng sai s : Cng thc tng qut

** f xx xm


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V d

Hi t chm gn im cc tr, im un Gii phng trnhf(x) =x = 0 trn [-1, 1] x 2x 3 = 0 vix0 = 0.9999 (1.0001) sin x =0 vix0 /2


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5. H PHNG TRNH PHI TUYN.

PHNG PHP NEWTON RAPHSON.

Xt h n gin gm 2 phng trnh phi tuynF(x,y) = 0, G (x,y) = 0

F(x,y), G (x,y) lin tc v c o hm ring theox vy lin tctrong ln cn nghim ,x y

, 0x y

x y

F FJ x y

G G


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g=

0

g=

0

g=0


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5. H PHNG TRNH PHI TUYN.PHNG PHP NEWTON RAPHSON.

Xt h n gin gm 2 phng trnh phi tuynF(x,y) = 0, G (x,y) = 0 F(x,y), G (x,y) lin tc v c o hm ring theox vy lin tc

trong ln cn nghim ,x y

, 0x y

x y

F FJ x y

G G

1

1

, ,1

, ,,

, ,1

, ,,

n n y n n

n n

n n y n nn n

x n n n n

n n

x n n n nn n

F x y F x yx x

G x y G x yJ x y

x y F x yy y

G x y G x yJ x y

Nu chn (x0,y0) gn nghim th

s hi t v nghim ca h


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V d lp Newton Raphson vi h phi tuyn

Tm nghim gn ngx1, y1 ca h phi tuyn sau vi 3 ch s l:

2

2

, 3ln 0

, 2 5 1 0

F x y x x y

G x y x xy x

0 01.5, 1.5x y

Gii: Ma trnJ = f(x)

31 2

,

4 5

x y

x y

F F yJ x y x

G G

x y x

b nh:x(k) gn nghim

n x(n) Ma trn JacobianJ Vect f(x(n) ) Vect h0 1.5 3 3

1.5 2.5 1.51

0.466 0.25

1.379 1.535

0.121

0.0350.013

0.0253.175 3.072.051 -1.379


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2 2

3

1 0

0

x y

x y

chnx0 = 0.8,y0 = 0.5

n x(n) Ma trn JacobianJ Vect f(x(n) ) Vect h0 0.8

0.51

22 2

,3 1

x y

x y

F F x yJ x y

G G x

V d lp Newton Raphson vi h phi

tuyn (tt)

1.6 11.92 1

0.110 0.012

0.0280.065

0.8280.565

4.81 103

2.66 103


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2

3

0.5 0

5 0

x x y

x xy y

chnx0 = 1.2,y0 = 0.1

n x(n) Ma trn JacobianJ Vect f(x(n) ) Vecth0 1.2 1.4 1 0.16 0.00675

0.1 3.82 7 1.028 0.150541 1.20675 3.56 105

0.25054 4.91 10 3

22 1 1

,3 5 5 1

x y

x y

F F xJ x y

G G x y x


tuyn (tt)


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tuyn (tt)

2 1,2 2

x y

x y

F F xJ x yG G x y

2

2 2

1

5

y x

y x

chnx0 =y0 = 1.75

n x(n) Ma trn JacobianJ Vect f(x(n) ) Vecth0 1.75

1.751

3.5 1

3.5 3.5

0.3125

1.125

0.1409

0.18061.60911.5696

0.0196 0.0528


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tuyn (tt)

2 1,2 2

x y

x y

F F xJ x yG G x y

2

2 2

1

5

y x

y x

chnx0 =y0 = 1.5

n x(n) Ma trn JacobianJ Vect f(x(n) ) Vecth0 1.5

1.51

0.25

0.5

3 1

3 3

0.1042

0.06251.60421.5625

0.0110 0.0149


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Bi tp n chng 1

f(x) = sinx e x = 0 trn [0,1] - Phng php Newton1/ Kim tra iu kin hi t

f(x) = cos(x) + e x f(x) >0 x [0,1]f(x) =sin (x) e x f(x)


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Bi tp n chng 1 (tt)

f(x) = ln (x) + cosx =0 trn [0.1,1]vi chnh xc 102


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Li gii bng phng php lp nXy dng hm lp: Dy lp

n xn n0 0,1

1

2

3

cosxx e g x

0

cos

1

0.1,1

nx

n n

x

x g x e

Sai s

11

n n n

qx x x

q

10.5

n

n n nx x

0,369722 0,269723

0,393597 0,023876

0,397113 0,003517


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f(x) = ln (x) + cosx =0 trn [0.1,1] vi chnh xc 102 bng phng php Newton

1/ Kim tra iu kin hi tf(x) = 1/x sinx f(x) >0 x [0.1, 1]

f(x) = 1/x cosx f(x) < 0 x [0.1, 1]2/ Xy dng dy lp:

nn

nnnn

xx

xxxx

x

sin1

cosln

1,0

1

0

Sai s:

15,0

xf

m

xfxxn

n xn n0 0.1 8.717211 0.232077 3.24998

2 0.351593 0.709713 0.394179 0.050934 0.397727 0.000315 0.397748 1.0710

8


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Bi tp n chng 1 (tt)

f(x) = x + 2x 1 = 0 trn [0,1] Phng php Newton vi chnh xc 105


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Bi tp n chng 1 (tt)

f(x) = (x 2) lnx = 0 trn [1,2] Phng php Newton vi chnh xc 105


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f(x) = x + 2x 1 = 0 trn [0,1]

Phng php lp n vi chnh xc 105

n xn n0 01 0.5 0.52 0.444444 0.055556

3 0.455056 0.0106124 0.453088 0.0019685 0.453455 0.000367

6 0.453387 6.844 105

7 0.453400 1.276 105

8 0.453397 2.379 106

Xy dng hm lp: Dy lp

2

1

2x

x

11n n n

q

x x xq

10.5

n

n n nx x

Sai s

2

11,0

21

0

n

nnx

xgx

x

q = 0,5


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Bi tp n chng 1 (tt)

3 3

sin 1

3 1

x y

x xy y

chn x0 =0.2, y0 = 1.2

n x(n) Ma trn Jacobian Vect f(x(n) ) Vecth

0 0.2 0.17 0.17 0.0146 0.0466

1.2 3.48 3.72 0.016 0.0393

1 0.2466 0.0036

1.2393 0.0016


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Phng php Newton Raphson cho h m

phng trnh phi tuyn

Xt h gm m phng trnh

0,,

0,,

21

211

mm

m

xxxf

xxxf

f1,,fm l cc hmlin tc v c ohm ring theo x1,

x2, xm v lin tctrong ln cnnghim mxxx ,, 21

m

mmm

m

m

m

x

f

x

f

x

f

x

f

x

f

x

f

x

f

x

f

x

f

xxJ

21

2

2

2

1

2

1

2

1

1

1

1 ,,


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Phng php Newton Raphson cho h m

phng trnh phi tuyn (tt)

Gi s chn (x10,x20,,xm0) gn nghim th

m

m

k

mm

k

mmm

mkk

mkk

nmnn

nknk

x

f

x

ff

x

f

x

f

x

f

x

f

x

ff

x

f

x

f

x

fx

f

x

ff

x

f

x

f

x

f

xxxJxx

1121

2

1

22

1

2

2

2

1

2

1

1

11

1

1

2

1

1

1

,21

,1,,...,,

1

s hi t v nghim ca h

ct th ktrong ma trn Jacobianbin th k

V d h h N t R h h


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V d phng php Newton Raphson choh m phng trnh phi tuyn

Gii h ba phng

trnh sau

0,,

0202,,

0364369,,

222

3

222

222

1

zyxzyxf

zyxzyxf

zyxzyxf

zyxfz

zyxfy

zyxfx

zyxfz

zyxfy

zyxfx

zyxf

z

zyxf

y

zyxf

xzyxJ

,,,,,,

,,,,,,

,,,,,,

,,

333

222

111

zyx

yx

zyx

zyxJ

222

2042

87218

,,

V d h h N t R h h


68/480

V d phng php Newton Raphson choh m phng trnh phi tuyn (tt)

zyx

yx

zyx

zyxJ

222

2042

87218

,,

nnnnn

nnnn

nnnnn

nnn

nn

zyzyxf

yzyxf

zyzyxf

zyxJxx

22,,

204,,

872,,

,,

1

3

2

1

1

nnnnn

nnnn

nnnnn

nnn

nn

zzyxfx

zyxfx

zzyxfx

zyxJyy

2,,2

20,,2

8,,18

,,

1

3

2

1

1

nnnnn

nnnnn

nnnnn

nnn

nn

zyxfyx

zyxfyx

zyxfyx

zyxJzz

,,22

,,42

,,7218

,,

1

3

2

1

1


69/480

69

V d phng php Newton Raphson cho

h m phng trnh phi tuyn (tt)

0,1,1 000 zyx

n xn yn zn

0 1.0 1.0 0.01 0.9 0.9 0.04

2 0.8936507937 0.8945436508 0.04008928571

3 0.8936282347 0.8945270105 0.04008928614

4 0.8936282345 0.8945270104 0.04008928616

Kt qu nhn c vi


70/480

70

Bi tp phng trnh phi tuyne x sinx = 0trn [0, 1] vi = 0,1

3x ln (2x + 1) = 0trn [ 0.4, 0.2]; = 0,01


71/480

71

Phng php lp n e x sinx= 0

trn[0, 1]Xy dng hm lp: Dy lp

n xn n0 0

1

2

34

5

Sai s

11

n n n

qx x x

q

0,5 0,610,563553 0,07642

0,397113 0,02104

2

sin xexxg

x

nn xgx

x

1

0 1,0

n

nnn

xxx

1

25.1

0,586254 0,006260,587836 0,00191


72/480

72

Phng php lp n 3x ln (2x+ 1)

= 0 trn [ 0.4, 0.2]Xy dng hm lp:

x =g(x) = (ex 1) /2

Dy lp

n xn n0 0

1234

567

Sai s

11

n n n

qx x x

q

0

1

0.4, 0.2

n n

x

x g x

14,9

n

n n nx x

q = 0,83

-0.3494 0.2471-0.3247 0.1206-0.3112 0.0658

-0.3035 0.0381

-0.2988 0.0227-0.2960 0.0138-0.2943 0.0085


73/480

73

Phng php Newtonf(x) = sinx e x= 0 trn [0,1]

1/ Kim tra iu kin hi tf(x) = cos(x) + e x f(x) >0 x [0,1]

f(x) =sin (x) e x

f(x)


74/480

74

Phng php Newtonf(x) = 3x ln (2x+ 1) = 0 trn [ 0.4, 0.2]

1/ Kim tra iu kin hi tf(x) = 3 2/(2x + 1) f(x) 0

x

[0.4, 0.2]2/ Xy dng dy lp:

0

1

0,4,

3 ln(2 1)

3 2 /(2 1)

n nn n

n

x

x xx x

x

n xn n0 -0,412345

Sai s:

1/3

n n

n

f x f xx x

m

1.2284-0.3415 0.3732-0.3039 0.0731

-0.2923 0.0051-0.291411 3 105

-0.291406 1.5 109


75/480

2. H phng trnh tuyn tnh

Mn hc:Phng php tnh


76/480

76

Ni dung

A. Cc phng php gii ng1/ Phng php kh Gauss (phn t tr).2/ Phn tch nhn tLU3/ Phn tch Cholesky

B. Cc phng php lp

1/ Lp Jacobi2/ Lp Gauss Seidel

C. S iu kin v h iu kin xu


77/480

77

Tng quanH n phng trnh tuyn tnh, n n DngAx = b

11 12 1

21 22 2

1 2

...

...

... ... ... ...

...

n

n

n n nn

a a a

a a aA

a a a

1

2

...

n

b

bb

b

1

2

1 2...

...

T

n

n

x

xx x x

x


78/480

78

Tng quan (tt) n gin nht: Ma trn ng cho

H tng ng vi n phng trnh bc nht

11

22

0 ... 0

0 ... 0

... ... ... ...

0 0 ...nn

a

aA

a

nibxa iiii ,1ii

ii

a

bx


79/480

79

Tng quan (tt) n gin: Ma trn tam gic trn

11 12 1

22 2

...

0 ...... ... ... ...

0 0 ...

n

n

nn

a a a

a aA

a

1,...,1,1

1

nkxaba

x

a

bx

n

kj

jkjk

kk

k

nn

nn


80/480

80

Tng quan (tt) n gin: Ma trn tam gic di

11

21 22

1 2

0 ... 0

... 0... ... ... ...

...n n nn

a

a aA

a a a

nkxaba

x

a

bx

k

j

jkjk

kk

k ,...,2,1 1

1

11

11


81/480

81

Phng php kh Gauss S dng cc php bin i s cp theo hng

chuyn v mt h phng trnh mi tng ng

c ma trn h s dng tam gic Cc php bin i s cp hay s dng:

Nhn mt hng vi mt s khc khng; Hon chuyn hai hng cho nhau;

Cng mt hng cho mt hng khc nhn vi mt skhc khng.


82/480

82

Phng php kh Gauss (tt)

nnnnnn

nn

nn

bxaxaxa

bxaxaxa

bxaxaxa

2211

22222121

11212111

11

2

1

2

12122122

11212111

nnnnn

nn

nn

bxaxa

bxaxa

bxaxaxa

111 1

11

0 kka

a ma


83/480

83


143084

51476

1322

321

321

321

xxx

xxx

xxx

Ma trn h s m rng

2 2 3 1

6 7 14 5

4 8 30 14

A A b

32

6

11

2121

a

am

jj

kj

kja

am 2

2

4

11

3131

a

am

2 2 3 1

6 7 14 5

4 8 30 14

2 2 1

3 3 1

3

2

h h h

h h h

2 2 3 1

0 1 5 2

0 4 24 12


84/480

84

Phng php kh Gauss (tt)4

1

4

22

3232

a

am

122440

2510

1322

44

25

1322

3

32

321

x

xx

xxx

233 4hhh

4400

2510

1322

22/321

31/52

14/4

321

32

3

xxx

xx

x

1

3

2

x


85/480

85

Phng php kh Gauss (tt) Kh ct 1: a(1)11 0 Kh ct 2: a(2)22 0 Gii li: a(3)33 0

Phn t tr (pivot) akk0


86/480

86

Kh Gauss vi lnh Maple> with(linalg); # Khi ng gi lnh i s tuyn tnh> A := matrix(2,3,[2, 3, 4, 1, 2, 3]); # Nhp ma trn

> m21 := A[2,1]/A[1,1]; # Tnh h s kh> A := addrow(A,1,2,m21) ; # Cng hng h2 h2 m21h1> A := swaprow(A,1,2) ; # Nu cn thit, i hng h2 h1> AA := gausselim(A); # Lnh gp kh Gauss ton ma trn> x := backsub(AA) ; # H dng tam gic trn: Gii li


87/480

87

Phng php kh Gauss (tt) VD: Gii h

434

2

203322

82

4321

321

4321

4321

xxxx

xxx

xxxx

xxxx


88/480


(0 )

1 1 2 1 8

2 2 3 3 20

1 1 1 0 2

1 1 4 3 4

A

(3)

1 1 2 1 8

0 2 1 1 6

0 0 1 1 4

0 0 0 2 4

A

7 3 2 2T

x

Phn t(1)

22 0a do tip tc ta i ch hng th 2 v th 3

2 2 1

3 3 1

4 4 1

2h h h

h h h

h h h

(1)

1 1 2 1 8

0 0 1 1 4

0 2 1 1 6

0 0 2 4 12

A

(2 )

1 1 2 1 8

0 2 1 1 6

0 0 1 1 4

0 0 2 4 12

A

4 4 32h h h

Th h l


89/480

Thc t tnh ton: vn lm trn s

Quy tc lm trn trn my tnh: Lm trn ch s c ngha

35,1210235,110234567,134567,1211

Tr kh: a11 = 0.003 0 176467,1763

11

2121

a

am

Bin i ct mt: (E2) (E2) m21(E1)

1 2

2

0.003 59.14 59.17

104300 104400

x x

x

VD: Gii h trn my tnh vi php lm trn 4 ch c ngha

)(E78.46130.6291.5

)(E17.5914.59003.0

221

121

xx

xx Nghim chnh xc: [10, 1]T

2

1

1.001

10

x

x

Ti sao?

Thc t tnh ton: vn lm trn s (tt)


90/480

Thc t tnh ton: vn lm trn s (tt)

i hng

VD: Gii h trn my tnh vi php lm trn 4 ch c ngha

)(E78.46130.6291.5

)(E17.5914.59003.0

221

121

xx

xx Nghim chnh xc: [10, 1]T

1 2 2

1 2 1

5.291 6.130 46.78 (E )

0.003 59.14 59.17 (E )

x x

x x

Tr kh: a11 = 5.291 0 421

21

11

5.6700 10a

ma

Bin i ct mt: (E1) (E1) m21(E2)

1 2 2 2

2 1

5.291 6.130 46.78 (E ) 1.00059.14 59.14 10.00

x x xx x


91/480

91

V d

8232

4223

183242

6232

4321

4321

4321

4321

xxxx

xxxx

xxxx

xxxx

81232

42123

183242

62321

205470

1481040

61800

62321

205470

141800

641041

62321


92/480

92

V d (tt)

20547061800

1481040

62321

5,495,130061800

1481040

62321

8117

16117000

61800

1481040

62321

2

1

2

1

x


93/480

93

Phng php Gauss Jordan

Ci tin phng php Gauss Ti mi bc, khi chn phn t bin i, ta chn phn

t c gi tr tuyt i ln nht, sao cho khng cng hngv ct vi cc phn t chn trc

Phn t ny gi l phn t chnh hoc phn t tri Bin i sao cho tt c cc phn t trn cng ct ca phn

t tri bng 0 Qua n bc nh vy ta s tm c nghim

Phn t (1)22 0a do tip tc ta i ch hng th 2 v th 3


94/480

94

Phng php Gauss Jordan (tt)(0)

43 4a Chn phn t chnh l phn t

(0)

1 1 2 1 8

2 2 3 3 201 1 1 0 2

1 1 4 3 4

A

1 1 4

2 2 4

3 3 4

2

4 34

h h h

h h hh h h

(1)

1 1 0 5 20

5 5 0 21 923 5 0 3 12

1 1 4 3 4

A

(1)

24 21a Phn t chnh tip theo (khng nm trn hng th t v ct thba) l phn t

1 1 2

3 3 2

4 4 2

21 5

7

7

h h h

h h h

h h h

(2 )

4 4 0 0 40

5 5 0 21 92

16 40 0 0 8

12 12 28 0 64

A


95/480

95

Phng php Gauss Jordan (tt)

(2)

32 40a

Phn t chnh tip theo (khng nmtrn hng th hai, th t v ct thba, th t) l phn t

(2 )

4 4 0 0 40

5 5 0 21 92

16 40 0 0 8

12 12 28 0 64

A

1 1 3

2 2 3

4 4 3

10

8

10 3

h h h

h h h

h h h

(3)

56 0 0 0 392

56 0 0 168 728

16 40 0 0 8

168 0 280 0 616

A

(4 )

56 0 0 0 3920 0 0 168 336

0 280 0 0 840

0 0 280 0 560

A

Phn t chnh cuicng l phn t

(3)

11 56a

2 2 1

3 3 1

4 4 1

7 2

3

h h h

h h h

h h h

1

4

2

3

56 392168 336

280 840

280 560

xx

x

x

73

2

2

x


96/480

96

V d Phng php Gauss - Jordan

8232

4223

183242

6232

4321

4321

4321

4321

xxxx

xxxx

xxxx

xxxx

81232

42123

183242

62321

244

233

211

34

2

2

hhh

hhh

hhh

2252014

107004

183242

61800

81232

42123

183242

62321

61800

61800

11hh


97/480

2252014

107004

183242

2252014

11459400

1042616280

728098082611459400

43120100816520

468046800

826000826

11859000

33040016520

10100

826826

11859

33041652

1

1

4

2

3

x

x

x

x

1

2

2

1

1

4

2

3

x

x

x

x

433

422

11

27

7

hhh

hhh

344

322

311

559

2659

59

hhh

hhh

hhh

154

133

22

11

98

4

1008

468/

hhh

hhh

hh

hh


98/480

98


2.43 3.45 -6.21 1.45

0.43 4.24 -5.05 2.23

2.67 -1.13 3.27 3.21

A b

1 2 3

1 2 3

1 2 3

2.43 3.45 6.21 1.45

0.43 4.24 5.05 2.23

2.67 1.13 + 3.27 3.21

x x x

x x x

x x x

2.43 3.45 6.21 1.45

0 3.630 3.951 1.973

0 0 4.737 4.292

A b

0.7401.530

0.906

x


99/480

99

Phn tch nhn tLU Phn tch ma trn h s A thnh tch ca hai ma

trnL v U, trong L l ma trn tam gic di

cn Ul ma trn tam gic trn Khi vic gii h phng trnhAx = b s a v

vic gii h phng trnhLy = b v U x = y

Nu A l ma trn khng suy bin th bao gi cng

tn ti mt ma trn Pkhng suy bin sao cho matrn PA phn tch c thnh tch ca ma trn tamgic diL v ma trn tam gic trn U


100/480

100

Phn tch nhn tLU(tt)To ma trn khL, ma trn kt qu Uv xt tchL.U

143084

51476

1322

321

321

321

xxx

xxx

xxx

142

013

001

400

510

322 2 2 3

6 7 14

4 8 30

. =


101/480

101

Phn tch nhn tLU(tt)Kt qu: Nu qu trnh kh Gauss din ra bnhthng (khng i hng), ma trn A ca h Ax = b

phn tch c thnh tchLU, tcA = LUviL (lower): ma trn tam gic di, ng chochnh bng 1, cha cc h s kh v tr kh U(upper): ma trn tam gic trn, cng l ma trn

kt qu nhn c sau qu trnh kh Gauss


102/480

102

Phn tch nhn tLU(tt)Ma trn A phn tch c thnh dng LU, ma trn L cng cho chnh bng 1 (phng php Doolittle)Cc h s caL v U

1 1

1

1

11

1

,1

1

,

1,

(1 )

(2 )

(2 )

1(1 )

j j

j

j

j

i j ij ik kjk

j

i j ij ik kj

ki j

u a j n

al i n

a

u a l u i j

l a l u j iu

Gii h ( h b 1)


103/480

Gii thut tmLU(ng choL bng 1)

2 2 3

6 7 14

4 8 30

A

2 2 3

0 1 5

0 4 24

2 2 3

0 1 5

0 0 4

1 0 0

1 0

1

L

H s kh:truPT

cotCung 2 0

6 , 1

4 4

A

2Cot 1: Cot 2 :

0

; 1

4

1

He so kh cot 1: 3 hso kh cot 2 :

2

1 0 0

3 1 0

2 4 1

L

Phn tch LU vi ng cho chnh L bng 1 Kh Gauss

(khng i hng). Cc h s kh toL, ma trn kt qu: U


104/480

104

Phn tch nhn tLU(tt)HAx = b (LU)x = b

(1)

(2)

Ux y

Ly b

Gii h u Gii 2 h tam gic:Ly = b (2) tmy;Ux = y (1) tmx


105/480

105

Phn tch nhn tLU(tt)

(2)Ly b

1 0 0 2 2 3

3 1 0 0 1 5

2 4 1 0 0 4

A

2 2 3 1

6 7 14 5

4 8 30 14

A A b

1 0 0 1

3 1 0 5

2 4 1 14

1

2

4

y

(1)Ux y2 2 3 1

0 1 5 2

0 0 4 4

2

3

1

x


106/480

106

Phn tch nhn tLU(tt) Gi s ma trnA phn tch c thnh dngLUnh sau: S dng phn tchLUtrn gii hAx = b = (9 5 7 11)T

3 7 2 2 1 0 0 0 3 7 2 2

3 5 1 1 1 1 0 0 0 2 1 1

6 4 0 0 2 5 1 0 0 0 1 1

9 5 5 4 3 8 3 1 0 0 0 1

A


107/480

107

Phn tch nhn tLU(tt)

1 0 0 0 9

1 1 0 0 5

2 5 1 0 7

3 8 3 1 11

9

4

5

1

y

GiiLy = b tmy

Gii U x = y tmx3 7 2 2 9

0 2 1 1 4

0 0 1 1 5

0 0 0 1 1

4.17

4.5

6

1

x


108/480

108

Phn tch nhn tLU(tt)

S dng phng php nhn t gii h

1 2 3

1 2 3

1 2 3

2 5 4 1

3 3 9 03 6 5 4.1

x x x

x x xx x x

2 5 4

3 3 93 6 5

A

1 0 0

1.5 1 0

1.5 1.286 1

L

2.674

0.124

0.932

x

2 5 4

0 10.5 3

0 0 4.858

U


109/480

109

Phn tch nhn tLUtheo phng php Doolittle,phn t u33 s l bao nhiu

2 1 2

6 5 9

4 4 4

A


110/480

110

Phn tch nhn tLU(tt)

1 2 3

1 2 3

1 2 3

2 2 3 1

6 7 14 6

4 8 30 18

x x x

x x x

x x x

(2)Ly b

1 0 0 2 2 3

3 1 0 0 1 5

2 4 1 0 0 4

A

1 0 0 1

3 1 0 6

2 4 1 18

1

3

4

y

(1)Ux y2 2 3 1

0 1 5 3

0 0 4 4

1

2

1

x


111/480


112/480

112

Bi tp phng php phn tch nhntLU

S dng phng php nhn t gii h

1 2 3

1 2 3

1 2 3

2.2 0.3 0.2 1.5

0.3 3.4 0.2 2.40.2 0.2 0.2 4.1

x x x

x x xx x x

2.2 0.3 0.2 1.5

0.3 3.4 0.2 2.40.2 0.2 4.1 3.2

2.2 0.3 0.2

0 3.359 0.173

0 0 0.173

U

1 0 0

0.136 1 0

0.091 0.051 1

L

y = (1.5, 2.195, 3.851)T x = (1.275, 0.491, 22.267)T


113/480

113

Phn tch nhn tLU(tt) Trnghp ma trn ba ng cho

11 12

21 22 23

32 33

1, 1 1,

, 1 ,

0 0 0

0 0

0 0 0

0 0 0

0 0 0

n n n n

n n n n

a a

a a a

a aA

a a

a a

Trng hp ma trn ba ng cho (tt)


114/480

Trng hp ma trn ba ng cho (tt) Phn rLUcho ta

21

32

1 0 0 0

1 0 0

0 1 0

0 0 0 1

l

L l

11 12

22 23

33

0 0

0 0

0 0 0

0 0 0 nn

u u

u u

U u

u

2111 11 12 12 21

11

, , , 1 1,

1,

, 1 , 1 1,

,

, ,

2,3,...,

, 2,3,..., 1

i i i i i i i i

i i

i i i i i i

i i

au a u a l

u

u a l u i n

au a l i n

u



115/480

2 1 0 01 10 5 0

0 1 5 2

0 0 1 4

A

5

18

40

27

b

S dng phng php nhn t gii h phng trnhAx = b vi

1 0 0 0

0.5 1 0 0

0 0.11 1 0

0 0 0.22 1

L

2 1 0 0

0 9.5 5 0

0 0 4.47 2

0 0 0 4.45

U

5

15.5

38.37

35.58

y

3

1

5

8

x


116/480

116


2 1 0 0 0 0 0 0 0 0

1 2 1 0 0 0 0 0 0 0

0 1 2 1 0 0 0 0 0 0

0 0 1 2 1 0 0 0 0 0

0 0 0 1 2 1 0 0 0 0

0 0 0 0 1 2 1 0 0 0

0 0 0 0 0 1 2 1 0 0

0 0 0 0 0 0 1 2 1 00 0 0 0 0 0 0 1 2 1

0 0 0 0 0 0 0 0 1 2

A

2

1

1

1

1

1

1

11

2

b

S dng phng php nhn t gii h 10 phng trnhAx = b vi

x = [6, 10, 13, 15, 16, 16, 15, 13, 10, 6]T


117/480

11

7

Gii h phng trnh ba ng cho

H phng trnh dng ny gp rt nhiu trong thc tingii quyt cc h phng trnh ton-l


118/480

118

Phng php truy ui

gii h phng trnh dngAx = F Phng trnh tng qut ca h

Akxk-1+ Ckxk+Bkxk+1=Fk (1) Gi s cc n lin h vi nhau theo cng thc

xk= k+1xk+1+ k+1 (2) Vi k= 1, 2,, n-1


119/480

119

Phng php truy ui (tt)

Biu din xk-1 v xkqua xk+1 v thay vo phngtrnh (1):

xk= k+1xk+1+ k+1 xk-1= kxk+ k= k(k+1xk+1+ k+1) + k

Ak[k(k+1xk+1+ k+1) + k] ++ Ck(k+1xk+1+ k+1) +Bkxk+1 = Fk


120/480

120


Ak[k (k+1xk+1+ k+1) + k] + Ck (k+1xk+1+k+1) +Bkxk+1 = Fk

(Akkk+1 + Ckk+1 +Bk)xk+1 ++ (Akk+1 +Akk+ Ckk+1 Fk) = 0

kkk

kkk

k

kkk

k

kAC

AF

AC

B

11 ,

ng thc ny s khng ph thuc vo nghimca h nu c hai du ngoc bng khng. T


121/480

121


T phng trnh u tin C1x1+B1x2 =F1 T ta c

1

1

2

1

1

2,

C

F

C

B

T phng trnh cui cngAnxn-1+ Cnxn =Fnm xn-1= nxn+ n t ta c

nnn

nnn

nCA

AFx


122/480

122

Phn tch Cholesky

Trng hp c bit ca phng php nhn tLU c dng cho ma trn A i xng v xc nh

dng nh ngha:

Ma trn vungA i xng nuAT= A tc aij=aji Ma trn vung A xc nh dng nu vi mi

vctx 0 ta lun cxTAx > 0


123/480

123

Phn tch Cholesky (tt)

VD: ma trnA sau y i xng v xc nh dng

1 2 3, ,T

x x x x 1 1 0

1 5 2

0 2 2

A

2 2 2

1 2 3 1 2 2 35 2 2 4 0T

x Ax x x x x x x x

nh l: Mt ma trn l xc nh dng khi v chkhi tt c cc nh thc con chnh ca n u dng

nh thc con chnh cp k, 1 k n ca ma trn lnh thc con thu c t k hng v kct u tinca ma trn


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124


nh l: Ma trn A i xng xc nh dng tn tima trn tam gic diB tha mn:A =BBT

ijbbab

b

nibab

nib

ab

ab

j

k

jkikij

jj

ii

i

k

ikiiii

ii

11

1

1

1

1

1

1

2

11

11

1111

Minh ha gii thut Cholesky


125/480

1 1 0

1 5 2

0 2 2

A

11

21 22

31 32 33

0 0

0

b

B b b

b b b

11 11b a

11

11

ii

ab

b i=2 2121

11

ab

b 31

31

11

ab

bi=3

12

1

i

ii ii ik

k

b a b

i=21

2 2

22 22 2 22 21

1

k

k

b a b a b

i=32

2 2 2

33 33 3 33 31 32

1

k

k

b a b a b b

j=31

1

1 i

ij ij ik jk

kjj

b a b bb

1

32 31 2132 32 3

122 22

1k jk

k

a b bb a b b

b b

Minh ha gii thut Cholesky


126/480

2 1 0

1 2 1

0 1 2

A

11

21 22

31 32 33

0 0

0

b

B b b

b b b

11 11b a

11

11

ii

ab

b i=2 2121

11

ab

b 31

31

11

ab

bi=3

12

1

i

ii ii ik

k

b a b

i=21

2 2

22 22 2 22 21

1

k

k

b a b a b

i=32

2 2 2

33 33 3 33 31 32

1

k

k

b a b a b b

j=31

1

1 i

ij ij ik jk

kjj

b a b bb

1

32 31 2132 32 3

122 22

1k jk

k

a b bb a b b

b b

Minh ha gii thut Cholesky (tt)


127/480

1 1 0

1 5 2

0 2 2

A

11

21 22

31 32 33

0 0

0

b

B b b

b b b

11 11b a 11 1b

i=2 212111

ab

b 3131

11

ab

bi=3

i=2 222 22 21

b a b

i=3 2 233 33 31 32b a b b

j=3 32 31 213222

a b bb

b

21

11

1b 31

00

1b

2

22 5 1 2b

32

2 0 11

2

b

22

33 2 0 1 1b

0 0

0B

1

0

2

1 1

1



128/480

1 1 0

1 5 2

0 2 2

A

S dng phng php Cholesky gii h phng trnhAx = b vi1

3

2

b

1 1 0 1 0 0 1 1 0

1 5 2 1 2 0 0 2 1

0 2 2 0 1 1 0 0 1

TA BB

1 0 0

1 2 0

0 1 1

B

By b1

2

3

1 0 0 1

1 2 0 3

0 1 1 2

y

y

y

1

1

1

y

TB x y1

2

3

1 1 0 1

0 2 1 1

0 0 1 1

x

x

x

1

0

1

x


129/480

129


A khng phi xc nh dng (ch i xng). Khi cc phn t ca ma trn tam gic B sao cho

A = BBT

c th cha s phc

2 h BT

x = y &By = b: phc. Nhng nghimx s l nghim thc!



130/480

1 2 0

2 3 2

0 2 3

A

11 11b a 11 1b

i=2 212111

ab

b 3131

11

ab

bi=3

i=2 222 22 21

b a b

i=3 2 233 33 31 32b a b b

j=3 32 31 213222

a b bb

b

1 0 0

2 0

0 2 1

B i

i

212b 31 0b

22b i

32 2b i

22

33 3 0 2 1b i

det 1A



131/480

1 2 0

2 3 2

0 2 3

A

S dng phng php Cholesky gii h phng trnhAx = b vi5

11

1

b

1 2 0 1 0 0 1 2 0

2 3 2 2 0 0 2

0 2 3 0 2 1 0 0 1

TA i i i BB

i

1 0 0

2 0

0 2 1

B i

i

By b1

2

3

1 0 0 5

2 0 11

0 2 1 1

y

i y

i y

5

1

y i

TB x y1

2

3

1 2 0 5

0 2

0 0 1 1

x

i i x i

x

3

1

1

x


132/480

132

Bi tp gii thut Cholesky

S dng phng php Cholesky gii h phng trnhAx = b vi7 3 5 1

3 10 2 2

5 2 4 1

1 2 1 4

A

4

18

7

2

b


133/480

133

Chun vct v chun ma trn

Khng gian tuyn tnh thc Rn. Chun ca vctx Rn l mt s thc x tha

x Rn, x 0; x = 0 x = 0 x Rn, R, x= x x, y Rn, x + y x + y (bt ng thc

tam gic) Xt ch yu chun thng dng sau

x =[x1,x2,,xn]T

Rn 1 21 1...

n

n k

kx x x x x

1 2

1,max , ,..., max

n kk n

x x x x x


134/480

134

Chun vct v chun ma trn (tt)

Chun ma trn tng ng vi chun vect

1 0max max

x x

AxA Ax

x

1 11

maxn

ijj n

i

A a

1

1

maxn

iji n

j

A a

11

n

k

k

x x

1,max kk nx x

theo hngtheo ct


135/480

135

V d chun vct v chun ma trn

0

3

4

x

17

4

x

x

3 7 2

3 5 1

6 4 0

A

1

max 12,16,3 16

max 12,9,10 12

A

A

V d h t h t


136/480

136

V d chun vct v chun ma trn(tt)

x = [1, 1, 2, 2]T x1 = 6

x = 2

A1 = 6

A = 7

43

21A


137/480

137

Vector Norm

Given an n -dimensional vectorx =[x1,x2,,xn]T

a general vector norm x, is a nonnegative norm

defined such that x > 0 whenx 0 ; x = 0 iffx = 0 x= xfor any scalar x + y x + y

The vector norm xp forp = 1, 2, ... is defined aspn

i

p

ipxx

1

1


138/480

138

Vector Norm (cont)

pn

i

p

ipxx

1

1

1 211

...n

n k

k

x x x x x

1 2 1,max , ,..., maxn kk nx x x x x 22

2

2

1

21

1

2

2...

n

n

ii

xxxxx


139/480

139

Matrix Norm

Given a square complex or real matrixA, a matrixnormAis a nonnegative number associatedwithA having the properties

A > 0 whenA 0 and A = 0 iffA = 0 k A = |k| A for any scalark A + B A + B

A B AB


140/480

140

S hi t ca dy cc vct

Dy cc vct vi x(k) Rn hi t v vctkhi k +

( ) 0kx x khi k+ (hi t theo chun)( )lim k

kx x

iu kin cn v : dy hi t v(hi t theo ta )

, 1,m m n

0kkx

km

x


141/480

141

Tng quan phng php lp

Chng 1: Phng php lp n vi phng trnhf(x) = 0

1( )

( ) 0

( ) ( ) : ' 1

n n

x g xf x x g x

g x g y q x y g x q

HAx = b x = T x + c =g(x), T: ma trn, c: vect.

iu kin: g(x) g(y) qx y

Dy lp:x(k+1) = T x(k) + c

Nu T


142/480

142

Tng quan phng php lp (tt)

Nu T


143/480

Ma trnA l ma trn ng cho tri nghim ngt:1,

ij ii

j j i

a a

11 12 1

21 22 2

1 2

...

...

... ... ... ...

...

n

n

n n nn

a a a

a a aA

a a a

11

22

0 ... 0

0 ... 0

... ... ... ...

0 0 ... nn

a

a

a

12 1

21 2

1 2

0 0 ... 0 0 ...

0 ... 0 0 0 ...

... ... ... ... ... ... ... ...

... 0 0 0 ... 0

n

n

n n

a a

a a

a a

D L U

Ax = b (D L U)x = b

theo hng

Phng php lp Jacobi


144/480

Ax = b (D L U)x = b

Dx = (L + U)x + bx = D (L + U)x + DbK hiu Tj = D (L + U) cj = Db

x(k+1) = Tjx(k) + cj , k =1,2,31

( 1) ( ) ( )

1 1

1 i nk k ki ij j ij j i

j j iii

x a x a x ba

i = 1, 2,, n

1,1

1, 1,1,max max 1

n

ijnij j j i

j i n i nj j i ii ii

aa

T D L U a a

V d phng php lp Jacobi (tt)1 2 310 7x x x

Vi vect x(0) = (2, 2, 2)T, tm nghim


145/480

1 2 3

1 2 3

1 2 3

2 10 10

4 10 11

x x x

x x x

1 2 3

2 1 3

3 1 2

0.1 0.1 0.7

0.2 0.1 1.0

0.4 0.1 1.1

x x x

x x x

x x x

1 2 3

2 1 3

1 23

0.1 0.1 0.7

0.2 0.1 1.0

0.4 0.1 1.1

x x x

x x x

x xx

x = T x + c

Vi vectx (2, 2, 2) , tm nghimxp x x(k) bng php lp Jacobi ca hphng trnh sau v nh gi sai s

Nghim chnh xcx = (0.5, 1, 1)T

1.1

0.1

7.0

01.04.0

1.002.0

1.01.00

xx

V d phng php lp Jacobi0.7 0 0.1 0.1 x(k+1) = T x(k) + c


146/480

0.7

1.0

1.1

c

0 0.1 0.1

0.2 0 0.1

0.4 0.1 0

T

x( ) T x( ) + c max 0.2,0.3,0.5

0.5 1

T

k x1(k) x2(k) x3(k) x(k) -x(k-1)0 2 2 2

1 0.3 0.8 0.5 1.7

3 0.495 0.992 0.971 0.0892 0.57 0.99 1.06 0.56

2

2

20x

5.0

8.0

3.01x

06.1

99.0

57.02x

971.0

992.0

495.03x

V d phng php lp Jacobi1 2 310 7x x x

1 2 310 7x x x


147/480

1 2 3

1 2 3

1 2 3

2 10 10

4 10 11

x x x

x x x

1 2 3

2 1 3

3 1 2

10 2 10

10 4 11

x x x

x x x

11 2 3

12 1 3

13 1 2

10 7

10 2 10

10 4 11

k k k

k k k

k k k

x x x

x x x

x x x

k x1(k) x2(k) x3(k) x(k) -x(k-1)0 2 2 2

1 0.3 0.8 0.5 1.7

3 0.495 0.992 0.971 0.0892 0.57 0.99 1.06 0.56

V d phng php lp Jacobi (tt)1 2 310 7x x x

Vi vectx(0) = (0, 0, 0)T, tm nghim


148/480

1 2 3

1 2 3

1 2 3

2 10 10

4 10 11

x x x

x x x

1 2 3

2 1 3

3 1 2

0.1 0.1 0.7

0.2 0.1 1.0

0.4 0.1 1.1

x x x

x x x

x x x

1 2 3

2 1 3

1 23

0.1 0.1 0.7

0.2 0.1 1.0

0.4 0.1 1.1

x x x

x x x

x xx

x = T x + c

( , , ) , g xp x x(k) bng php lp Jacobi ca hphng trnh sau v nh gi sai s


1.1

0.1

7.0

01.04.0

1.002.0

1.01.00

xx

V d phng php lp Jacobi0.7

0 0.1 0.1

x(k+1) = T x(k) + c


149/480

1.0

1.1

c

0.2 0 0.1

0.4 0.1 0

T

x T x + c max 0.2,0.3,0.5

0.5 1

T

(0)

0

0

0

x

(1)

0.7

1.0

1.1

x

(2 )

0.49

0.97

0.92

x

k x1(k) x2(k) x3(k) x(k) -x(k-1)0 0 0 0

1 0.7 1.0 1.1 1.1

3 0.511 0.994 1.001 0.0812 0.49 0.97 0.92 0.21

V d phng php lp Jacobi1 2 310 7x x x

1 2 310 7x x x


150/480

k x1(k) x2(k) x3(k) x(k) -x(k-1)0 0 0 0

1 0.7 1.0 1.1 1.1

3 0.511 0.994 1.001 0.0812 0.49 0.97 0.92 0.21

1 2 3

1 2 3

1 2 3

2 10 10

4 10 11

x x x

x x x

1 2 3

2 1 3

3 1 2

10 2 10

10 4 11

x x x

x x x

11 2 3

12 1 3

13 1 2

10 7

10 2 10

10 4 11

k k k

k k k

k k k

x x x

x x x

x x x

V d phng php lp Jacobi (tt)10 2 2 4x x x x ( 1) ( ) ( ) ( )10 2 2 4k k k k x x x x


151/480

1 2 3 4

1 2 3

1 2 3 4

1 4

10 2 2 4

5 1

2 5 2

3 9 2

x x x x

x x x

x x x x

x x

Nghim ngx =(0.3333, 0.2222, 0.4444, 0.1111)T

1 2 3 4

2 1 3

3 1 2 4

4 1

( ) ( ) ( ) ( )

( 1) ( ) ( )

( 1) ( ) ( ) ( )

( 1) ( )

10 2 2 4

5 1

5 2 2

9 3 2

k k k

k k k k

k k

x x x x

x x x

x x x x

x x

k x1(k)

x2(k)

x3(k)

x4(k)

x(k)

-x(k-1)

0 0 0 0 01 0.4000 0.2000 0.4000 0.2222 0.4000

2 0.3644 0.2000 0.4444 0.0889 0.1333

3 0.3333 0.2160 0.4249 0.1007 0.0311

x(0) = (0, 0, 0, 0) T

V d phng php lp Jacobi (tt)1 2 3 47 2 2 3x x x x

1 2 3 4( 1) ( ) ( ) ( )7 2 2 3k k k k x x x x


152/480

1 2 3 4

1 3 4

2 3 4

2 8 3 2

5 2 52 4 4

x x x x

x x xx x x

k x1(k) x2(k) x3(k) x4(k) x(k) -x(k-1)0 0 1 1 1

1 2 3 4

2 1 3 4

3 1 4

4 2 3

( 1) ( ) ( ) ( )

( 1) ( ) ( )

( 1) ( ) ( )

8 2 3 2

5 2 5

4 2 4

k k k k

k k k

k k k

x x x x

x x x

x x x

Nghim ngx =( 0.1752, -0.5338, 0.4166, 1.3710)T

1 0.2857 0.7500 0.6000 1.7500 0.7500

4 0.1339 0.4850 0.4073 1.2983 0.0916

2 0.3714 0.6223 0.2429 1.5250 0.3571

3

0.2196

0.4388 0.3157 1.3719 0.1835

Phng php lp Gauss-SeidelTng t lp Jacobi nhng vi thng tin cp nht ho


153/480

Tng t lp Jacobi nhng vi thng tin cp nht ho

1 2 3 4

1 2 3

1 2 3 4

1 4

10 2 2 4

5 1

2 5 2

3 9 2

x x x x

x x x

x x x x

x x

( 1) ( ) ( ) ( )

1 2 3 4

( 1) ( ) ( )

2 1 3

( 1) ( ) ( ) ( )

3 1 2 4

( 1) ( )

4 1

0.2 0.1 0.2 0.4

0.2 0.2 0.2

0.2 0.4 0.2 0.4

0.3333 0.2222

k k k k

k k k

k k k k

k k

x x x x

x x x

x x x x

x x

LpJacobi

( 1) ( ) ( ) ( )

1 2 3 4

( 1) ( 1) ( )

2 1 3

( 1) ( 1) ( ) ( )

3 1 2 4

( 1) ( 1)

4 1

0.2 0.1 0.2 0.4

0.2 0.2 0.2

0.2 0.4 0.2 0.4

0.3333 0.2222

k k k k

k k k

k k k k

k k

x x x x

x x x

x x x x

x x

LpGauss-Seidel


154/480

154

1 2

1 2

23 2 3

2 17 4

x x

x x

11 2

12 1

23 3 2

17 4 2

k k

k k

x x

x x

LpJacobi

Vix= (0,0); tmx(3) bng lp Jacobi


155/480

155

Phng php lp Gauss-Seidel

1 2

1 2

23 2 3

2 17 4

x x

x x

11 2

12 1

23 3 2

17 4 2

k k

k k

x x

x x

11 2

1 12 1

23 3 2

17 4 2

k k

k k

x x

x x

LpJacobi

Lp Gauss-Seidel

Tng t lp Jacobi nhng vi thng tin cp nht ho


156/480

Phng php lp Jacobi v Gauss-Seidel

710 701010


157/480

11104

10102

710

321

321

321

xxx

xxx

xxx

1.11.04.0

0.11.02.0

7.01.01.0

213

312

321

xxx

xxx

xxx

1.11.04.00.11.02.0

7.01.01.0

21

1

3

31

1

2

32

1

1

kkk

kkk

kkk

xxxxxx

xxx

Jacobi

Gauss-Seidel

1.11.04.00.11.02.0

7.01.01.0

1

2

1

1

1

3

3

1

1

1

2

32

1

1

kkk

kkk

kkk

xxxxxx

xxx

Phng php lp Gauss-Seidel1 2 310 7x x x



158/480

1 2 3

1 2 3

2 10 10

4 10 11

x x x

x x x

1 2 3

2 1 3

3 1 2

( 1) ( ) ( )

( 1) ( 1) ( )

( 1) ( 1) ( 1)

0.1 0.1 0.7

0.2 0.1 1.0

0.4 0.1 1.1

k k k

k k k

k k k

x x x

x x x

x x x

xp x x(k) bng php lp Gauss-Seidelca h phng trnh v nh gi sai s


k x1(k) x2(k) x3(k) x(k) -x(k-1)

0 2 2 2

1 0.3 1.14 1.094 1.72 0.4766 1.1014 1.0108 0.1766

3 0.4975 1.0016 1.0012 0.0210

V d phng php lp Gauss- Seidel1 2 310 7x x x

1 2 310 7x x x


159/480

1 2 3

1 2 3

2 10 10

4 10 11

x x x

x x x

2 1 3

3 1 2

10 2 10

10 4 11

x x x

x x x

11410

10210

710

1

2

1

1

1

3

3

1

1

1

2

32

1

1

kkk

kkk

kkk

xxx

xxx

xxx

k x1(k) x2(k) x3(k) x(k) -x(k-1)

0 2 2 2

1 0.3 1.14 1.094 1.72 0.4766 1.1014 1.0108 0.1766

3 0.4975 1.0016 1.0012 0.0210

Phng php lp Gauss-Seidel1 2 310 7x x x



160/480

1 2 3

1 2 3

2 10 10

4 10 11

x x x

x x x

1 2 3

2 1 3

3 1 2

( 1) ( ) ( )

( 1) ( 1) ( )

( 1) ( 1) ( 1)

0.1 0.1 0.7

0.2 0.1 1.0

0.4 0.1 1.1

k k k

k k k

k k k

x x x

x x x

x x x

xp x x(k) bng php lp Gauss-Seidelca h phng trnh v nh gi sai s


k x1(k) x2(k) x3(k) x(k) -x(k-1)

0 0 0 0

12

0.7 0.86 0.906 0.9060.5234 0.9859 0.9892 0.1766

V d phng php lp Gauss- Seidel1 2 310 7x x x

1 2 310 7x x x


161/480

k x1(k) x2(k) x3(k) x(k) -x(k-1)0 0 0 0

1 0.7 0.86 0.906 0.906

3 0.50248 0.99843 0.99885 0.0212 0.52340 0.98592 0.98923 0.1766

1 2 3

1 2 3

2 10 10

4 10 11

x x x

x x x

2 1 3

3 1 2

10 2 10

10 4 11

x x x

x x x

11410

10210

710

1

2

1

1

1

3

3

1

1

1

2

32

1

1

kkk

kkk

kkk

xxx

xxx

xxx

V d phng php lp Gauss-Seidel


162/480

1 2 3 4

1 2 3

1 2 3 4

1 4

10 2 2 4

5 1

2 5 2

3 9 2

x x x x

x x x

x x x x

x x

Nghim ngx =(0.3333, 0.2222, 0.4444, 0.1111)T

k x1(k) x2(k) x3(k) x4(k) x(k) -x(k-1)0 0 0 0 01 0.4000 0.1200 0.3680 0.0889 0.4000

2 0.3570 0.2022 0.4273 0.1032 0.0822

3 0.3375 0.2180 0.4403 0.1097 0.0195

239

225

15

42210

14

4213

312

432

1

1

kk

kkkk

kkk

kkkk

xx

xxxx

xxx

xxxx

V d phng php lp Gauss-Seidel (tt)Dng phng php lp Gauss-Seidel tm


163/480

1 2 3 4

1 2 3 4

1 3 4

2 3 4

7 2 2 3

2 8 3 2

5 2 52 4 4

x x x x

x x x x

x x xx x x

g p g p p px(2) lm trn n 4 ch s

x(0) = (0, 1, 1, 1) T

V d phng php lp Gauss-Seidel (tt)

7 2 2 3 ( 1) ( ) ( ) ( )

7 2 2 3k k k k

x x x x


164/480

1 2 3 4

1 2 3 4

1 3 4

2 3 4

7 2 2 3

2 8 3 2

5 2 5

2 4 4

x x x x

x x x x

x x x

x x x

1 2 3 4

2 1 3 4

3 1 4

4 2 3

( 1) ( 1) ( ) ( )

( 1) ( 1) ( )

( 1) ( 1) ( 1)

7 2 2 3

8 2 3 2

5 2 5

4 2 4

k k k k

k k k

k k k

x x x x

x x x x

x x x

x x x

k x1(k) x2(k) x3(k) x4(k) x(k) -x(k-1)0 0 -1 1 1

Nghim ngx =(-0.1752, -0.5338, 0.4166, 1.3710)T

1 -0.2857 -0.6786 0.5429 1.4750 0.47502 -0.2643 -0.5719 0.3571 1.3752 0.18573 -0.1788 -0.5111 0.4142 1.3591 0.08554 -0.1650 -0.5340 0.4234 1.3728 0.0228


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165

H phng trnh b nhiu

V d: Gii cc h phng trnh

2 5

2.01 5.02

x y

x y

2 5

2.01 5.1

x y

x y

1

2

x

y

15

10

x

y

Nhiu v phi

H gn nhau, nghim xa nhau!

2 5

1.99 5.02

x y

x y

9

2

x

y

Nhiu v tri


166/480

166

S iu kin ca ma trn

Xt h phng trnhAx = b c nghimx =A1 bNu b c s gia b thx c s gia tng ng xA x = b x =A1 b

1 1x A b A b

b Ax A x

1x b

A Ax b

1Cond( )k A A A A S iu kin ca ma trn A

k(A) cng gn 1 h cng n nh

k(A) cng ln h cng mt n nh h iu kin xu


167/480

167

S iu kin ca ma trn (tt)

Xt h phng trnh

2 5

2.01 5.02

x y

x y

1 3.01 401 1207.01 1k A A A

1 2

1 2.01A

1201 200

100 100A


168/480

168

1

25 41 10 6

41 68 17 10

10 17 5 36 10 3 2

A

133, 136, cond 4488A A A

10 7 8 7

7 5 6 5

8 6 10 97 5 9 10

A

1.003 58.095.550 321.8

A

35.327A

V d

1 879.5 158.815.17 2.741

A


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169

n chng 2

A. Cc phng php gii ng 1/ Phng php kh Gauss (phn t tr). 2/ Phn tch nhn tLU 3/ Phn tch Cholesky

B. Cc phng php lp 1/ Lp Jacobi 2/ Lp Gauss Seidel

C. S iu kin v h iu kin xu


170/480

170

Bi tp n chng 2

Tm cc gi tr ca ma trn l xc nh dng

1 11 2 1

1 1 4

1 24 1

2 1 8

1 11 3 1

1 5

> 8/7 1 31 1 314 4

1 7 1 7

3 3


171/480

171

Bi tp n chng 2 (tt)

S dng phng php Cholesky gii h

1 2

1 2 3

2 3

2 2

2 12 2

x x

x x xx x

1 2 3

1 2 3

1 2 3

3 2 1

3 4 2 42 2 3

x x x

x x xx x x

1 2 3 4

1 2 3 4

1 2 3

1 2 4

4 1

3 0

2 1

2 2

x x x x

x x x x

x x x

x x x


172/480

172

1474

7102

424

A

132

031

002

B


173/480

173

Bi tp n chng 2 (tt)

1.4142 0 0

0.7071 1.2247 0

0 0.8165 1.1547

B

1 0 0

3 5 0

4 12

5 5

B i

i

2 0 0 0

0.5 1.6583 0 0

0.5 0.7538 1.0871 0

0.5 0.4523 0.0836 1.2403

B


174/480

174

Bi tp n chng 2 (tt)

Tm cc gi tr ca > 0 v > 0 ma trn lng cho tri nghim ngt

4 12 5 4

2

3 25

2 1

2 42

4 1

3

0.5

2

1

5

3

6

2

5


175/480

175

Bi tp n chng 2 (tt)

S dng phng php Jacobi tm nghim gn ngvi sai s 10-. Chn chun v cng

1 2 3

1 2 3

1 2 3

4 13 8 2 0

3 3 10 4

x x xx x x

x x x

1 2

1 2 3

2 3

10 910 2 7

2 10 6

x xx x x

x x

1 2

1 2 3

2 3 4

3 4

10 5 6

5 10 4 254 8 11

5 11

x x

x x xx x x

x x

1 2 3

1 2 4

1 3 4

2 3 4

4 2

4 14 0

4 1

x x x

x x xx x x

x x x


176/480

176

Cho h phng trnh

Vi x(0) = [1, 2, 3]Ttnh vect x(1) theo

phng php Jacobi

1 2 3

1 2 3

1 2 3

25 2 16

2 18 35

2 2 20 58

x x x

x x x

x x x

V d phng php lp Jacobi (tt)


177/480

177

V d phng php lp Jacobi (tt)

1 2 3 4

1 2 3 4

1 3 4

2 3 4

7 2 2 3

2 8 3 2

5 2 5

2 4 4

x x x x

x x x x

x x x

x x x

Dng phng php lp Jacobi tm x(1) lmtrn n 4 ch s

x(0) = (0, 1, 1, 1) T

Ma trn lp trong cc phng phpMa trnA l ma trn ng cho tri nghim ngt:

1

n

ij ii

j j i

a a


178/480

1,j j i

11 12 1

21 22 2

1 2

...

...

... ... ... ...

...

n

n

n n nn

a a a

a a aA

a a a

11

22

0 ... 0

0 ... 0

... ... ... ...

0 0 ... nn

a

a

a

12 1

21 2

1 2

0 0 ... 0 0 ...

0 ... 0 0 0 ...

... ... ... ... ... ... ... ...

... 0 0 0 ... 0

n

n

n n

a a

a a

a a

D L U

Ax = b (D L U)x = b

theo hng

Tm ma trn lp trong phng phpJ bi


179/480

179

Jacobi

Ax = b (D L U)x = bDx = (L + U)x + b

x = D

(L + U)x + D

b=> Tj = D (L + U)

nna

a

a

D

...00............

0...0

0...0

22

11

0...............

...0

...0

21

221

112

nn

n

n

aa

aa

aa

UL

Ma trn lp trong phng php Jacobi (tt)Tj = D (L + U)


180/480

j ( )

nna

aa

D

1...00

............

0...100...01

22

11

1

0...

............

...0

...0

21

2222221

1111112

1

nnnnnn

n

n

j

aaaa

aaaa

aaaa

ULDT

Ma trn Tj c ng cho chnh bng 0

Tm ma trn lp trong phng phpGauss Seidel


181/480

181

Gauss-Seidel

Ax = b (D L U)x = b(D L) x = U x + b

x = (D L)

U x + (D L)

b=> Tg= ( D L ) U

Tm ma trn lp trong phng phpGauss-Seidel (tt)


182/480

Tg

= Tg

= ( D L ) U

nna

a

a

D

...00

............

0...0

0...0

22

11

nnnn aaa

aa

a

LD

...

............

0...

0...0

21

2221

11

0...00

............

...00

...0

2

112

n

n

a

aa

U

Ma trn Tgc ct th nht bng 0

Ma trn lp phng php Gauss-S i i i


183/480

183

Seidel cho h hai phng trnh hai n

Tg= Tg= ( D L ) U

2221

110

aa

aLD

00

012

aU

22

2211

21

111

1

01

aaa

a

a

LD

2211

2112

11121

0

0

aa

aa

aa

ULDTg


184/480

Chng 3. Ni suy v phng phpbnh phng cc tiu

Mn hc:Phng php tnh

Ni d


185/480

18

5

Ni dung

1. Ni suy a thc Lagrange2. Sai s ni suy Lagrange

3. Ni suy Newton, mc ni suy cch u4. Ni suy spline bc ba5. Phng php bnh phng cc tiu

T i


186/480

18

6

Tng quan v ni suy

Bi ton dn n ni suy Ni suy: Bng cha (n+1) cp d liu

{ (xk,yk) }, k= 0 nMc ni suy x0 x1 x = xk xn-1 xn

Gi tr ni suy y0 y1 y = ? yn-1 yn

xk: mc ni suy,yk: gi tr (hm) ni suyT bng ny, ni suy gi try ti imx = ?

i ki i


187/480

18

7

iu kin ni suy

Hmf(x) lin tc trn on [x0,xn]f(xk) =yk

C v s hm tha iu kin trn! Minh ha hnh hc

1

1.2

-1 -0.5 0 0.5 1


188/480

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Ni th


189/480

18

9

Ni suy a thc

Ni suy a thc: Xc nh a thc y = P(x) thoiu kin ni suy

P(xk) = yk, k= 0 n y () P()

a thc

L do chn a thc

C th xp x hm lin tc vi sai s ty C th ly o hm v tch phn bao nhiu ln ty Tnh gi tr a thc v o hm d dng

11 1 0n n

n n nx a x a x a x a

Mi h h i th


190/480

19

0

Minh ha ni suy a thc

Mc ni suy 1 0 1Gi tr ni suy 0.5 1 2

3 mc n = 2 P(x) = ax+ b x + c (3 h s cn tm)P(1) = 0.5 a b + c = 0.5P(0) = 1 c = 1P(1) = 2 a + b + c = 2

a = 0.25, b = 0.75, c = 1P(x) = 0.25x + 0.75x + 1x = 0.8 P(x) = 0.25 (0.8) + 0.75 0.8 + 1 = 1.76So snh vi gi tr chnh xc: f(x) = 2x ; f(0.8) 1.7411

x = 0.8,y ?

Khi n tng th s iu kin ca ma trn h s cng tng ln!

Cond(A)


191/480

1.0E+00

1.0E+02

1.0E+04

1.0E+06

1.0E+08

1.0E+10

1 2 3 4 5 6 7 8 9 10

a thc ni suy LagrangeXy dng cc a thcph c bc n v tha

( )1 khi

0 khi

k

n j

j kp x

j k


192/480

ph c bc n v tha 0 khij j k

Do cc a thc ph c bc n v c n nghimx0,x1, xk-1,xk+1,xn, nn

( ) 1kn kp x

11110 nkkkkkkkkk

n xxxxxxxxxxCxp

nkkkkkkk

xxxxxxxxxxC

1110

1

nkkkkkkk

nkkk

nxxxxxxxxxx

xxxxxxxxxxxp

1110

1110

nkk

k

n xxxxxxxxxxCxp 1110

a thc ni s Lagrange (tt)


193/480

19

3

a thc ni suy Lagrange (tt)

a thc ni suy Lagrange c dng ( )0

( )n

k

n k

k

L x p x y

x 1 0 1y 0.5 1 2

(0 )

2

1

2

x xp x

(1)

21 1p x x x

(2)

2

1

2

x xp x

(0) (1) (2)2 2 2( ) 0.5 1 2L x p p p

2

2

1 3( ) 1

4 4

0.25 0.75 1

L x x x

x

L(0.8) = 1.76



194/480

19

4


0 1 nw x x x x x x x

( )k

n

k k

w xp x

w x x x

0

n

k

k k k

w xL x y

w x x x



195/480

19

5


x x0 x1 xnx0 x x0 x0 x1 x0 xn D0x1 x1 x0 x x1 x1 xn D1

xn xn x0 xn x1 x xn Dnw(x)

n

k k

kn

D

yxwxL

0

0

n

k

k k k

w xL x y

w x x x



196/480

19

6


Trng hp cc mc cch u xk+1 xk= h = const xk=x0 + kh. t q = ( x x0)/h

kx x q k h

i jx x i j h

0

1 1

! !

n kn

k

k

q q q nx y

k n k q k

0 1 1 1( )

0 1 1 1

k k nk

n

k k k k k k k n

x x x x x x x x x xp x

x x x x x x x x x x

( )1 1 1 1

! !

n k

k

n

q q q k q k q np x

k n k

Sai s ni suy Lagrange


197/480

19

7

Sai s ni suy Lagrange

f x L x Kw x

Gi sf(x) c o hm n cp n+1 lin tc trn [a, b]

ChnKsao cho (x*) = 0 (x* xi)

Xt hm ph

f x L x Kw x

Tn ti [a, b] sao cho (n+1) () = 0

( 1) ( 1) 1 ! 0n nf K n

( 1)

1 !

nfK

n

( 1)

*

1 !

nMf x L x w x

n

a thc ni suy Lagrange: v d


198/480

19

8

a thc ni suy Lagrange: v d

x 1 0 1y 0.5 1 2

x = 0.8,y ?

3(3) ln 2 2xf x

f(x) = 2x, n = 2

3 3(3)

1,1max ln 2 2 2 ln 2x

xM

3

2 ln 20.8 1 0.8 0.8 1 0.032

3!f x L x

0.82 1.76 0.0189f x L x

a thc ni suy Lagrange: bi tp


199/480

19

9


x 0 0.25 0.75 1y 1.3 1.5 1.6 1.8

Xy dng a thc ni suy Lagrange v s dng xp x gi trca hm tix*

x* = 0.5



200/480

20

0


x 0 0.25 0.75 1y 1.3 1.5 1.6 1.8

x* = 0.5

( )0

( )n

k

n k

k

L x p x y

(0 )3 0.25 0.75 1

0.25 0.75 1x x xp x

(1)

3

0 0.75 1

0.25 0.5 0.75

x x xp x

(2)3

0 0.25 1

0.75 0.5 0.25

x x xp x

(3)

3

0 0.25 0.75

1 0.75 0.25

x x xp x

(0 ) 0.5 0.1667p

(1) 0.5 0.6667p

(3) 0.5 0.1667p

(3) 0.5 0.6667p

(0.5) 1.5500L



201/480

20

1


Xy dng a thc ni suy Lagrange v s dng xpx gi tr ca hm tix*

x 1 0 1 2y 4 2 0 1 x* = 1.25

Bi tp


202/480

20

2

Bi tp

Xy dng a thc ni suy Lagrange v s dng xpx gi tr ca hm tix*

x 0 0.5 1 1.5y 2 0 8 1 x* = 0.75

x 0 0.5 1 1.5

y 0.5 1 1 4

x* = 0.75

Bi tp


203/480

20

3

Bi tp

Xy dng a thc ni suy Lagrange v s dng xpx gi tr ca hm tix* = 345; 355; 365. nh gi sais xp x viy =f(x) = Lg(x)

x 340 350 360 370y 2,531 2,544 2,556 2,568


204/480


205/480

Xy dng a thc ni suy Newton


206/480

20

6

Xy dng a thc ni suy Newton

nh ngha t sai phn cpmt ca hmf(x)

0

00,

xx

yxfxxf

000 , xxxxfyxf

nh ngha t sai phn cp hai ca hmf(x)

1

10010

,,,,

xx

xxfxxfxxxf

10100100 ,,, xxxxxxxfxxxxfyxf

110100 ,,,, xxxxxfxxfxxf

Cng thc Newton tin


207/480

20

7

Cng thc Newton tin

Lp li n bc th n

nn

nn

xxxxxxxxxxf

xxxxxxxxxf

xxxxxxxfxxxxfyxf

...,...,,,

...,...,,

...,,,

1010

11010

102100100

11010

102100100

)1(

...,...,,

...,,,

nn xxxxxxxxxf

xxxxxxxfxxxxfyxN

nnn xxxxxxxxxxfxR ...,...,,, 1010

xRxNxf n)1(

Cng thc Newton tin xut pht tntx0

Cng thc Newton li


208/480

20

8

Cng thc Newton li

nn

nnnnnnnnn

xxxxxxxxxf

xxxxxxxfxxxxfyxN

...,...,,

...,,,

2110

1121

)2(

xRxNxf n )2(Cng thc Newton li xut pht tntxn

( 1)

*

1 !

nMf x L x w x

n

Sai s ca cng thc ni suy Newton: tng t Lagrange

V d cng thc ni suy Newton tin


209/480

20

9

V d cng thc ni suy Newton tin

x 1 0 1y 0.5 1 2

xk

f(xk

) f[xk

, xk+1

] f[xk

, xk+1

, xk+2

]-1 0.5

0.50 1 0.25

11 2

x = 0.8,y ?

N(x) = 0.5 + 0.5 (0.8 + 1) + 0.25 (0.8 + 1) (0.8 0) = 1.76

Cng thc ni suy Newton trng hpmc cch u


210/480

21

0

Sai phn hu hn cp 1 ca hm ti imxkyk=yk+1 yk

Sai phn hu hn cpp ca hm ti imxkpyk= (p1yk)

Quan h gia t sai phn v sai phn hu hn

1, ,...,

!

p

kk k k p p

yf x x x

p h

q = (xx0)/h p = (xxn)/h

Cng thc ni suy Newton trng hpmc cch u (tt)

)1( xxxxxxxfxxxxfyxN


211/480

11010102100100

...,...,,

...,,,

nn xxxxxxxxxf

xxxxxxxfxxxxfyxN

2(1) 0 0

0

0

1 ...1! 2!

1 ... 1

!

n

y yN x y q q q

yq q q n

n

nn

nnnnnnnnn

xxxxxxxxxf

xxxxxxxfxxxxfyxN

...,...,,

...,,,

2110

1121

)2(

2(2) 1 2

0

1 ...1! 2!

1 ... 1!

n n

n

n

y yN x y p p p

yp p p n

n

a thc ni suy Lagrange: bi tpXy dng a thc ni suy Lagrange v s dng xp x gi trca hm tix*


212/480

x 1 0 1 2y 4 2 0 1

x* = 1.75

(0 )

1 2

1 2 3

x x xp

(1)

1 1 2

1 1 2

x x xp

(3) 1 1

3 2 1

x x xp

3

1 2 1 1 2 1 14 2

1 2 3 1 1 2 3 2 1

1 52

2 2

x x x x x x x x xL x

x x

3047.0275.12575.1

2175.1

3 L

Cng thc ni suy NewtonXy dng a thc ni suy Newton v xp x gi tr tix*

x 1 0 1 2 x* = 1 75


213/480

x 1 0 1 2

y 4 2 0 1

x* = 1.75

(1) 31 1 5

4 2 1 1 1 22 2 2

x x x x x x x

xk f(xk) f[xk, xk+1 ] f[xk, xk+1, xk+2 ] f[xk,xk+1,xk+2,xk+3]

1 4

0 21 0

2 1

2

2

1

01.5

0.5

3047.0275.12

575.1

2

175.1

3 L

Cng thc ni suy Newton mc cch uXy dng a thc ni suy Newton v xp x gi tr tix*

x 1 0 1 2 x* = 1 75


214/480

x 1 0 1 2

y 4 2 0 1

x* = 1.75

xk f(xk) yk yk yk1 4

0 21 0

2 1

2

21

0

3 3

21!3

1!2

75.1 03

0

2

001

qqq

yqq

yqyyN

75.20*

h

xxq

25.0

*

hxx

pn

21!3

1!2

75.1 03

1

2

232

ppp

ypp

ypyyN

a thc ni suy Lagrange: bi tpXy dng a thc ni suy Lagrange v s dng xp x gi trca hm tix*


215/480

x 1 0 1 2y 4 2 0 1

x* = 1.25

(0 )

1 2

1 2 3

x x xp

(1)

1 1 2

1 1 2

x x xp

(3) 1 1

3 2 1

x x xp

3

1 2 1 1 2 1 14 2

1 2 3 1 1 2 3 2 1

1 52

2 2

x x x x x x x x xL x

x x

31 51.25 1.25 1.25 2 0.14842 2

L

Cng thc ni suy NewtonXy dng a thc ni suy Newton v xp x gi tr tix*

x 1 0 1 2 x* = 1 25


216/480

x 1 0 1 2

y 4 2 0 1

x* = 1.25

(1) 31 1 5

4 2 1 1 1 22 2 2

x x x x x x x

31 5

1.25 1.25 1.25 2 0.14842 2

L

xk f(xk) f[xk, xk+1 ] f[xk, xk+1, xk+2 ] f[xk,xk+1,xk+2,xk+3]

1 4

0 21 0

2 1

2

2

1

01.5

0.5

Trng hp mc cch u v d

Cho bng gi tr sin x t 15 55. Xy dng a


217/480

thc ni suy tin (li) cp 3 & tnh sin16 (sin54)x y y 2y 3y15 0.2588

20 0.3420

25 0.4226

30 0.535 0.5736

40 0.6428

45 0.7071

50 0.7660

55 0.8192

0.0832

0.0806

0.0774

0.0736

0.0692

0.0643

0.0589

0.0532

0.0026

0.0032

0.00380.0044

0.0049

0.0054

0.0057

0.0006

0.00060.0006

0.0005

0.0005

0.0003

Trng hp mc cch u v d (tt)


218/480

21

8

g p ( )

a thc ni suy tin:x

giáo trình phương pháp tính

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