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Bo co th c tp tt nghi 2012

Hong Th Thy H nh-L p 08CTT1 Page | 1

ti: ng dng Matlab vo mt sbi ton gii tch s v th Phn 1: Matlap c bn

1.1 Gi i thiu MatlabMatlab l mt phn mm ton h c ca hng Mathworkstnh ton trn cc

sv c tnh tr c quan r t cao, n cng cho php chng ta vcc bi u , th theonhiu cch khc nhau . Matlab c vit tt tMATRIXLABORATORY do m c ch ban u ca Matlab l xy d ng nn m t cng c htr vic tnh ton cc ma tr n mtcch d dng nh t.

Trong mi tr ng Window, sau khi ci Matlab bi u t ng ca n s xut hintrn mn hnh c a my tnh, chng ta c th khi ng Matlab b ng cch double click vo bi u t ng ca n. Trong khi ch y, ty theo yu c u ca ng i sdng, matlab s

to ra m t hoc nhiu ca strn mn hnh. C a squan tr ng nh t l ca s lnh(command window), y l ni chng ta giao tip vi Matlap v cng l ni chngta nhp vo cc l nh v Matlab s cho ra cc k t qu. Chi >> l d u nhc ca chngtrnh Matlab. Khi Matlab ho t ng, con tr chut sxut hin sau d u nhc, lc nyMatlab ang ch ng sdng nh p lnh vo. Sau khi nh p lnh v nh n Enter,Matlab p ng li bng cch in ra cc dng k t qutrong c a slnh hay t o ra m tca shnh nh (Figure Window). Tuy nhin, n u ng i dng ch cn thc thi cu l nhnhng khng cn in k t qura ca sdng l nh, ta thm d u chm phy (;) ngay saucu lnh.Cc cu l nh s c lu trong mt tp tin c ui .m (v d: baitap.m) v c thc thi m t ln khi c hng trnh kh i chy.

to mt tp tin .m, ng i dng vo File ch n New -> M-File hay nh n vo

biu t ng nm trn thanh Toolbar.thc thi m t tp tin .m, ta nh n vo bi u t ng nm trn thanh Editor

Toolbar ho c sdng phm F5.Matlab cng htr cng c debug gip k m tra t ngbc nhm pht hi n li sai trong qu trnh lm vi c. thot kh i chng trnh Matlabchng ta s dng lnh exit ho c quit.

1.2 Php ton , bin, vector, ma trn

1.2.1 BinTrong ngn ng lp trnh Matlab, m t bin (variable) c khai bo v kh i to

thng qua cu l nh gn.Tn bi n bao g m cc k t ch, sv k hi u gch d i ( _ ). Tn bi n phi bt ubng k t ch v c di ty thch. Tuy nhin, Matlab ch ghi nh 31 k t u tin.ng th i, Matlab lun phn bi t ch in v ch thng khi t tn bi n hoc tnchng trnh.


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Cc ki u tn bi n h p l: arg1, no_name, vars, VarsCh :Trong ngn ng lp trnh Matlab, m i bin khi kh i to s c xem nh mt

mng. N u bin c gi tr n th mng c kch th c 11. N u bin l ma tr n hocvector th kch th c ca mng chnh l kch th c ca ma tr n hoc vector . y lmt im khc bi t ca Matlab so v i cc ngn ng lp trnh khc. ly k ch th cca mt bin, ta s dng hm size().

Ngn ng lp trnh Matlab xem chu i k t nh mng mt chiu cha cc k t . Do kch th c ca bin msg l 1 dng 5 c t.1.2.2.Php ton

Php ton D ng i s Matlab

Cng a + b a + b

Tr a - b a - b

Nhn a * b a * b

Chia ph i a / b a / b

Chia tri b \ a a \ b

Ly tha a b a ^ b

Hn na Matlab cn h tr mt shm s hc n gin nh hm lm trn round(), lmtrn ln ceil ( ), lm trn xu ng floor ( ), l y phn d mod( ), tm c chung l n nhtgcd ( ), tm b i chung nh nht lcm( ), v hm l y cn sqrt( ). Ngoi ra cn c cc php ton so snh nh (==), khc (~=), ln hn (>), nh hn (=), v nh hn hoc bng (


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Vector l m t dng c bit ca ma tr n c m t dng ho c mt ct. Trong ccngn ng lp trnh khc, sinh vin c lm quen v i vector thng qua tn g i danhsch (list) ho c mng 1 chi u (1-D array).

kh i to vector dng ch a cc gi tr r i rc, cc ph n t trong vector ph i nmtrong c p ngo c ([]) v c ngn cch nhau b i khong trng ho c du phy (,).kh i to vector dng ch a cc gi tr lin tc (mc nh trong Matlab l 1) ho ccch nhau m t khong gi tr nht nh (cn g i l b c nhy ), Matlab s dng du(:).ng th i gi tr u v cu i ca vector khng c n thit t trong c p du ngo c ([]).Hn na, to mt vector r ng - vector khng ch a gi tr - trong Matlab chng takhai bo nh sau: Ng c li to ra vector c t, chng ta c n ngh ch o vector c t bng cch s dngdu nhy n (') hoc du chm phy (;) ngn cch gia cc ph n t.Gi tr ca mt phn t ti mt v tr bt k trong vector c truy xu t thng qua ch s. Trong Matlab, ch slun b t u t1 v c th l mt gi tr n hoc mt mng.

+ Trch ph n t th i: X(i)

+Trch nhi u phn t: X([danh sch v tr])

xa m t phn t trong vector, chng ta s gn ph n t v i vector r ng.

1.2.4.Ma trn

Trong Matlab, ma tr n i din cho m ng nhi u chiu c nhi u dng v nhi uct. Phng thc khai bo v kh i to ma tr n tng t nh vector. Tuy nhin k tthc m t dng trong ma tr n, chng ta s dng du chm phy (;).

ng thi, Matlab cng htr mt shm c th kh i to cc ma tr n c bit nhsau:

Ma tr n khng: zeros(s dng, s ct) Ma tr n vung khng c p n: zeros(n)

Ma tr n n v: eye(n) Ma tr n ng cho: diag([cc ph n t trn ng cho chnh]) Ma tr n thc ngu nhin trong kho ng [0,1]: rand(s dng, s ct) hoc rand (n)

(ma tr n vung c p n) Ma tr n ton s mt: ones(s dng, s ct)


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Ma tr n vung m t cp n: ones(n)Tng t nh vector, gi trca mt phn t ti mt v tr bt k trong ma tr n c

truy xu t thng qua ch sdng v ch sct. Trch ph n t ti dng i c t j: A(i, j) Trch nhi u phn t: A([danh sch cc dng, danh sch cc c t]) Trch ng cho chnh c a ma tr n: diag(A) Trch t t cphn tca ma tr n: A(:) Trch t t ccc ph n t ti ct i: A(: , i) Trch t t cphn t ti dng j: A(j, :)

Ch : Trong Matlab, ch scui cng c a dng hay c t ca ma tr n hoc vector c th thay th b i chend.1.3.Biu th c Logic1.3.1.Cc ton t Logic

Mt biu thc logic trong Matlab c xy d ng t6 ton t quan h l >, =, < >=


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8. & (AND)

9. | (OR)

1.3.2.Vector v biu th c logicBiu thc logic cho php truy xu t mt cch linh ho t n cc thnh ph n ca

mt vector hay ma tr n.

- x(x>0): xu t nhng gi tr dng ca vector x.- x(x>2 & x 2 cho k t qul 0 0 0 1 1 1 1 1 1 l vector ch a k t quso snh t ng

phn t tng ng ca x v i 2 v x(x>2) s xut ra cc gi tr ln hn 2.

1.3.3.Cc hm logic: ALL, ANY, FINDMt shm logic thng d ng l: all, any, find

any : Kim tra xem c t n ti mt phn tno ca vector th a iu kin khng? N u cth cho k t qu l 1, ng c li l 0.V d: any(x>0) : Ki m tra xem c t n ti phn tno ca vector x dng hay khng?

all : Kim tra t t ccc ph n tca vector c th a iu kin hay khng? K t qul 1nu ng.

V d: all(x


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If biu thc logicLnh th1Lnh th2.

End

If (a>0)b=a;disp( a is positive);

End

If biu thc logicCc lnh thc thi khi bi uthc logic ng

ElseCc lnh thc thi khi bi uthc logic sai

End

If (temperature>100)Disp( above boiling.);

elsedisp(temperature is ok); toohigh=0

End

If biu thc logic1Cc l nh thc thi khi bi uthc logic 1 ng

Elseif bi u thc logic 2

Cc l nh thc thi khi bi uthc logic 2 ng Else

Cc l nh c thc thi khikho c bi u thc no ng

End

If (height>190)Disp( very tall);

Elseif (height >170)Disp ( tall)

Elseif (height 0, x=sqrt(x); end.

Cu trc ifelseifelseend

C php V d

C php V d

C php V d


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For index=first:step:endLnh thc thi

End

Sumx=0;For i=1:length(x)

Sumx=sumx+x(i);End

while bi u thc iu kinpht bi u 1;pht bi u 2;.

End

N=100;Iter=1; msum=0;while inter


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switch Bi u thc iu kin

case Gi tr th1

Khi lnh 1

case Gi tr th2

Khi lnh 2

...

otherwise

Khi lnh n

end

Method=2;

switch method

case {1,2}

disp(method is linear);

case 3

disp(method cubic); otherwise

disp(unknown method);

end

Lnh switch s ln l t thc hin cc kh i lnh tng ng v i tng gi tr th trongbiu thc iu kin. Biu thc iu kin phi c d ng shoc chu i.1.4.5.Script v HmScript : L cc dng l nh Matlab c cha trong m t file, c ph n m rng .m; filescript c th c son tho bng Matlab Editor ho c cc chng trnh son tho khc.thc thi script ch cn gi tn file trong c a sdng l nh ca Matlab.Hm: Cng l cc on lnh Matlab c son tho trong file .m, hm nh n cc thamstruyn vo, x l v xu t ra gi tr . Tn c a hm ph i ging nh tn ca file .m,trnh t tn c a hm trng v i cc hm c s n ca Matlab. Dng u tin c a hm(trphn ch thch) ph i c d ng nh sau:

function [Cc gi tr xut] = Tn_hm(Cc gi tr nhp)Sau dng ny, cc dng ch thch b t u bng du (%) s xut hin khi g i lnh helptn_hm.

1.5.V th 1.5.1.V th trong 2-D

C php V d


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Lnh c bn: plot(x, f(x))

V i x: vector ch a min gi tr ca hm f, f(x): cc gi tr ca f ng v i x.

Ch thch trn th:text(x, y, '...') : t mt ch thch ( trong d u ' ') ln th ti ta (x, y).gtext('...'): t ch thch ln th, v tr c xc nh b i click chu t.

title('...'): t a ca th

xlabel('...'): ghi nhn cho tr c Ox

ylabel('...'): ghi nhn cho tr c Oy

hold on/off : b t/tt ch cho php v nhiu th trong cng m t htrc ta .Cc ty chnh vnt v, du v mu sc:

Lnh: plot(x, y, 'Nt v _Du_Mu s c')

Nt v :

D u (marker):


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Mu s c: gm 8 ty ch n l ' r ' - , ' k ' - en, ' w '- trng, ' y ' - vng, ' c ' - cam, ' b ' -xanh n c bin, ' g ' - xanh l cy, ' m ' - tm.

V d:

Ty chnh mu sc v l n ca nt v:LineWidth: rng ca nt v , tnh b ng pt.

MarkerEdgecolor: mu c a ng vin du (marker).MarkerFacecolor: mu bn trong d u.

Markersize: l n ca du, tnh b ng pt.Xc nh ta :axis([xmin xmax ymin ymax])

xlim([xmin xmax])

ylim([ymin ymax])

Ty chnh cc kiu trc ta :

axis on/off/auto


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axis normal/square/equal/tight

axis ij/xy

grid on/off subplot - Vnhiu th trong cng mt c a s subplot(m, n, p): t o ra m t ma tr n m hng, n c t cha mn th, p l v tr ca tngth, th t t trn xu ng d i theo hng.


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1.5.2.V th trong 3-D:

Lnh c bn: plot3(x, y, z). Trong plot3, ta c n xc nh cc vector (x, y, z).

vmt (x, y, z = f(x,y)), s dng lnh meshgrid(x, y).V d:

Mt slnh v th trong 3-D khc contour / contourf / contour3 mesh / meshc / meshz surf / surfc waterfall bar3 / bar3h pie3 / fill3 comet3 / scatter3 / stem3


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Phn 2: p dng vo bi ton c th

2.1 Phng php ni suy Lagrange

2.1.1 t ng bi ton

Cho cc m c ni suy a= x 0


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Tm 1 ( ) L x v i

1 0

1 1

1

( ) 0

( ) 1

........

( ) 0n

L x

L x

L x

1 21

1 0 1 2 1

( )( )...( )( )

( )( )...( )n

n

x x x x x x L x

x x x x x x

Khi 1 2 1 10 1 1 1

( )( )...( )( )...( )( )

( )( )...( )( )...( )i i n

ii i i i i i i n

x x x x x x x x x x L x

x x x x x x x x x x

0

1

1

1

( ) 0

( ) 0...

( ) 0

( )

( ) 0

......

( ) 0

i i

i i

i i i

i i i i

i i i

i i n

y L x

y L x

y L x

y L x y

y L x

y L x

t 0 0 1 1( ) ( ) ( ) ... ( )n n L x y L x y L x y L x . a thc L(x) l a thc ni suyLagrange

2.1.1 Thu t ton:

u vo : x = [x0 x1 ... xn], y = [y0 y1 ... yn]

u ra: l = h sca a thc Lagrange b c n

L= a thc Lagrange

Code Matlab

function [l,L] = lagrange(x,y)n = length(x) - 1; %bac cua da thucl l = 0;for m = 1:n + 1p = 1;


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for k = 1:n + 1if k ~= mp = conv(p, [1 -x(k)])/(x(m) - x(k));

end end L(m, : ) = p; %da thuc Lagrange l = l + y(m)*p;a = x(1,1):0.2:x(1,n);hamso=polyval(l,a);plot(x,y, 'o' ,a,hamso);end

V d :f(-2)=3, f(0)=6 ,f(1)=2 ,f(4)= -3 .Tnh x p x ti f(2.5) ta g i hm trn

command window:clear all,clcx=[-2 0 1 4]; y=[3 6 2 -3]l=lagrange(x,y);yx=polyval(l, 2.5)Ta thu c k t qusau:

l = 0.4028 -1.4306 -2.9722 6.0000yx = -4.0781

th


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2.2.Phng php lp n

2.2.1. t ng bi ton

Gii ( ) 0 , f x tr n a b (1)

Chuyn phng trnh (1) vdng tng ng ( ) x g x (2)

Sao cho c o hm lin t c trn [a,b] v ( ) 1 [ , ]g x q x a b

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

Khi (2) c duy nht nghi m trn [a,b] v dy {x n} v i 1( )n n x g x , [ , ]n x a b hi t

t i nghi m duy nh t x* c a (1)[a, b] l kho ng cha nghi m ca phng trnh (1) tc l c a (2),m i xn

tnh theo (3) u thu c [a, b].

2.2.2 Thut ton

Gii phng trnh x = g(x) tx0 bng cch l p

u vo : g(x), x 0 = hm v gi tr ban utolx = sai s mong mu nmaxiter = S ln lp maxu ra: x = nghiemerr = sai so |x(k) - x(k - 1)|xx = cc gi tr trung gian

Code Matlab

function [x, err, xx] = simpiter(g, x0, tolx, maxiter)if nargin < 4

maxiter = 100;

endif nargin < 3

tolx = 1e-6;endxx(1) = x0;for k = 2:maxiter

xx(k) = feval(g, xx(k - 1));


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err = abs(xx(k) - xx(k - 1));if err < tolx

break;

endendx = xx(k);if k == maxiter

fprintf('Khong hoi tu sau %d lan lap\n', maxiter)else

fprintf('Hoi tu sau %d lan lap\n',k)endtnh li v d trn ta dng chng trnh ctsimpiter4_2.mclear all, clc

f = inline('-0.5*((x - 1).^2 - 3)');[x, ss, xx] = simpiter(f, 0.5,.00001,200)Kt qu thu c:Hoi tu sau 13 lan lapx = 1.4142ss = 7.9566e-006xx =

Columns 1 through 9

0.5000 1.3750 1.4297 1.4077 1.4169 1.4131 1.4147 1.41401.4143

Columns 10 through 13

1.4142 1.4142 1.4142 1.41422.3 Tm nghim gn ng bng phng php chia i

2.3.1 t ng bi ton

Xt phng trnh ( ) 0 f x c nghim chnh xc x trong khong cch ly nghim[ , ]a b v ( ). ( ) 0 f a f b . t 0 0 0 0 0, ,a a b b d b a b a v 0 x l im gia ca on [ , ]a b . Tnh gi tr 0( ) f x . Nu th 0( ) f x =0 , 0 x chnh l nghim v qu trnh dng li.

Ngc li ta xt du ca 0( ) f x . Nu 0( ). ( ) 0 f x f b , t 1 0 1 0,a a b x . Nu0 0( ). ( ) 0 f x f b , t 1 0 1 0,a x b b . Nh vy, ta thu c1 1 0 0[ , ] [ , ]a b a b v di


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01 1 1 2 2

d b ad b a .Tip tc qu trnh chia i nh vy n ln, ta c kt qu

sau:

(1) ,

2

( ) ( ) 0,2

n nn n n n n

n n n n n n

a ba x b a x b

b a f a f b d b a

vi 0,1,2...n

Tm nghim phng trnh (mu ) bng phng php chia i

Nh vy, ta c 0na l dy tng v b chn trn, cn 0nb l dy gim v b chndi. Do chng cng hi t. T (1), ta c: lim lim limn n n x x x

a b x x

Thng thng, ta s dng cng thc nh gi sai s sau: 12nb a

x x

2.3.2. Gii thut
http://web.srasgroup.com/wp-content/uploads/2010/11/Bisection_method.jpg


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2.3.2.1. Trnh t gii

u vo:hm dng gii phng trnhphi tuy n bng phng php chia i

u ra:K t quca chng trnh l bng sk t qutnh, bao g m:

+ n: S ln lp thc t + a: Gi tr bin bn tri+ b: Gi tr bin bn ph i+ x: nghi m ca phng trnh + f: gi tr ca hm f(x)+ e: sai s

function c2bisectformat long ;%Thiet lap cac cac gia tri ban dau

N = 100; %So lan lap toi da eps = 1.0E-6; %Sai so cho phep a = 1; b = 2; %Gia tri nghiem trong khoang [a,b] err = eps+1; %Sai so n = 0; %Thu tu lan lap

%Trang tri cho viec the hien ket qua fprintf( '%3c %12c %12c %12c %12c %12c\n' ,['n' 'a' 'b' 'x' 'f' 'e']);fprintf( '----------------------------------------------------------------------\n' );q = [];w = [];while (neps)

x = (a + b)/2;a2 = a; b2 = b; %Ket qua cu cua a, b (dung cho the hien ket qua) if f(a)*f(x)>0 %Nghiem khong nam trong khoang [a,c]

a = x;else %Nghiem thuoc [a,c]

b = x;end err = b - a;

n = n + 1;%Xuat ket qua ra man hinh fprintf( '%3d %12.8f %12.8f %12.8f %12.8f %12.8f\n' ,[n a2 b2 x f(x) err])q = [q x];


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w = [w f(x)];end plot(q, w, 'b-' );

hold on plot(q, w, 'r*' );end %Thay doi ham fx o dong 46 function fx = f(x)fx = 2*x-cos(x);end

V d:Tm nghi m gn ng ca phng trnh 2.x-cos(x)=0.Khi f(x)=2.x-cosx

Ta c k t qusau:n a b x f e-------------------------------------------------------------------------------------

1 1.00000000 2.00000000 1.50000000 0.48168907 0.500000002 1.00000000 1.50000000 1.25000000 -0.00965704 0.250000003 1.25000000 1.50000000 1.37500000 0.20507672 0.125000004 1.25000000 1.37500000 1.31250000 0.09045074 0.062500005 1.25000000 1.31250000 1.28125000 0.03863834 0.031250006 1.25000000 1.28125000 1.26562500 0.01405786 0.015625007 1.25000000 1.26562500 1.25781250 0.00209306 0.007812508 1.25000000 1.25781250 1.25390625 -0.00380873 0.003906259 1.25390625 1.25781250 1.25585938 -0.00086453 0.00195313

10 1.25585938 1.25781250 1.25683594 0.00061259 0.0009765611 1.25585938 1.25683594 1.25634766 -0.00012639 0.0004882812 1.25634766 1.25683594 1.25659180 0.00024299 0.0002441413 1.25634766 1.25659180 1.25646973 0.00005827 0.0001220714 1.25634766 1.25646973 1.25640869 -0.00003406 0.0000610415 1.25640869 1.25646973 1.25643921 0.00001210 0.0000305216 1.25640869 1.25643921 1.25642395 -0.00001098 0.0000152617 1.25642395 1.25643921 1.25643158 0.00000056 0.0000076318 1.25642395 1.25643158 1.25642776 -0.00000521 0.0000038119 1.25642776 1.25643158 1.25642967 -0.00000232 0.0000019120 1.25642967 1.25643158 1.25643063 -0.00000088 0.00000095

th:


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2.4. Phng php lp jacobi

2.4.1 t ng bi ton

Cho h AX=B (1) , ma tr n A=(a ij)n, X=(x 1,x2,..x n), B=(b 1,b2,bn)T

Nu A l ma tr n cho tr i th ta lun a h(1) vdng X=CX+D v i ||C||

Ma tr n A=(a ij)n gi l ma trn cho tr i nu tha mn 1 trong 2 iu kin sau

1) ii ij1

1,n

j

a a j n

2) ii ij1

1,n

ii j

a a i n


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Khi xt dy tng {xn} v i1n n X CX D

th hi tvnghim duy nh t ca

(1) v c sai s * 1 01

k

k C X X X X

C hoc * 11k k k

C X X X X C

2.4.2 Thut ton

function X = jacobi(A,B,X0,kmax)if nargin < 4, tol = 1e-6; kmax = 100;

elseif kmax < 1, tol = max(kmax,1e-16); kmax = 100;else tol = 1e-6;

end

if nargin < 3, X0 = zeros(size(B));end NA = size(A,1);X = X0; At = zeros(NA,NA);

for m = 1:NAfor n = 1:NA

if n ~= m, C(m,n) = -A(m,n)/A(m,m);end

end B(m,:) = B(m,:)/A(m,m);

end for k = 1: kmaxX = C*X + D;

if nargout == 0, X,end if norm(X - X0)/(norm(X0) + eps) < tol, break ;end

X0 = X;End

End

V d : Gii h phng trnh1 2

1 2

3 2 1

1 2 1

x x

x x

, v i 0

0

0 X

Khi ta c 3 21 2

A

v1

1 B

Ta nh p vo c a scommand window cc dng l nh sau

X 0 chn ty


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>>A = [3 2;1 2]

>>B = [1 - 1]

>>X0=[0 0]

>>x = jacobi(A,B,X0,20) % xem k t qucui cng

Ta c k t qu nh sau

x = 1.0000-1.0000

>>jacobi(A,b,x0,20) % xem k t qun ln lp

Ta c k t qusau;

X = 0.3333 0.6667 0.7778 0.8889 0.9259 ......-0.5000 -0.6667 -0.8333 -0.8889 -0.9444 ......


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MC LC

Chng 1: Matlab c bn

1.1 Gi i thiu Matlab ........................................................................................

1.2 Php ton , bi n, vector, ma tr n ................................................................

1.2.1 Bi n ...............................................................................................

1.2.2 Php ton .......................................................................................

1.2.3 Vector ............................................................................................

1.2.4 Ma tr n ..........................................................................................

1.3 Bi u thc Logic ...........................................................................................

1.3.1 Cc ton t logic ...........................................................................

1.3.2 Vector v bi u thc Logic .............................................................

1.3.3 Cc hm logic: ALL, Any v Find ...............................................

1.4 Lnh iu kin v vng l p ..........................................................................

1.4.1 L nh IF ..........................................................................................

1.4.2 L nh FOR ......................................................................................

1.4.3 L nh WHILE .................................................................................

1.4.4 L nh SWITCHCASE................................................................


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1.4.5 Script v hm .................................................................................

1.5 V th ......................................................................................................

1.5.1 V th trong 2-D .......................................................................

1.5.2 V th trong 3-D .......................................................................

Chng 2: p dng vo m t sbi ton c th

2.1 Phng php ni suy Lagrange ...................................................................

2.2 Phng php lp n...................................................................................

2.3 Tm nghi m gn ng bng phng php chia i.....................................

2.4 Phng php lp Jacobi ...............................................................................

Chng 3: Thc hnh ......................................................................................................


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