Giai on Cuoi Ky 1 In

N TP CUI K MN PHNG PHP TNHBi ging in t

TS. L Xun i

Trng i hc Bch Khoa TP HCMKhoa Khoa hc ng dng, b mn Ton ng dng

TP. HCM 2013.

TS. L Xun i (BK TPHCM) N TP CUI K MN PHNG PHP TNH TP. HCM 2013. 1 / 1

Cho M = 3.3. Shift-STO-MCu 1. Cho phng trnh ex + 2x2 8M = 0 trong khong cch lynghim [1, 2]. Dng phng php Newton, chn x0 ti bin theo iu kinFourier, tnh nghim gn ng x2 v nh gi sai s x2Gii.Ta c f (1) < 0, f (2) > 0, f (x) = ex + 4x > 0,x [1, 2] vf (x) = ex + 4 > 0, x [1, 2] nn chn x0 = 2. Ta xy dng dy (xn)theo cng thc

xn = xn1 f (xn1)f (xn1)

= xn1 exn1 + 2x2n1 8M

ex + 4x

Ta c |f (x)| > min{|f (1)|, |f (2)|} = |f (1)| = m. Shift-STO-A. Do nghim gn ng xn c nh gi sai s so vi nghim chnh xc x nhsau

|x xn| 6 |f (xn)|m

=|exn + 2x2n 8M|

m= xn


n xn xn0 2

1 1.73428805

2 1.70410221 0.000651577

Bm my. Tnh xn

X eX + 2X 2 8M

eX + 4X

CALC x = 2 x1CALC Ans x2Sai s

abs(eX + 2X 2 8M)A

CALC Ans x2Kt qu. x2 1.7041; x2 0.0007


Cu 2. Cho h phng trnh

2.1Mx1 +0.25x2 +0.53x3 = 3.410.32x1 +3.2Mx2 0.13x3 = 4.720.24x1 0.35x2 +4.4Mx3 = 5.12

S dng phng php Jacobi vi x (0) = (0.3, 2.5,1.9)T , tm vc t lpx (3)

Gii.

x1 =1

2.1M (3.41 0.25x2 0.53x3)= 3.412.1M 0.252.1M x2 0.532.1M x3

x2 =1

3.2M (4.72 0.32x1 + 0.13x3)= 4.723.2M 0.323.2M x1 + 0.133.2M x3

x3 =1

4.4M (5.12 0.24x1 + 0.35x2)= 5.124.4M 0.244.4M x1 + 0.354.4M x2


x1x2x3

= 3.412.1M4.72

3.2M5.124.4M

+ 0 0.252.1M 0.532.1M 0.323.2M 0 0.133.2M 0.244.4M 0.354.4M 0

x1x2x3

Khi cng thc lp c dng

X (m) = TjX(m1) + Cj , m = 1, 2, . . .

Chn X (0) =

0.32.51.9

tnh X (1),X (2),X (3)TS. L Xun i (BK TPHCM) N TP CUI K MN PHNG PHP TNH TP. HCM 2013. 5 / 1

MatA =

0 0.25

(2.1M) 0.53(2.1M) 0.32(3.2M) 0 0.13(3.2M) 0.24(4.4M) 0.35(4.4M) 0

, MatB =

3.41(2.1M)4.72

(3.2M)5.12

(4.4M)

,MatC =

0.32.51.9

Bm my. Mode - 6 -Matrix.Dim - MatA - 3 3- ACShift 4 - Dim - MatB - 3 1 - ACShift 4 - Dim - MatC - 3 1 - ACShift 4 - MatB+MatA*MatC= x (1) - ACShift 4 - MatB+MatA*MatAns= x (2) - ACShift 4 - MatB+MatA*MatAns= x (3) - AC


m x(m)1 x

(m)2 x

(m)3

0 0.3 2.5 -1.9

1 0.5472 0.4144 0.4079

2 0.4459 0.4354 0.3535

3 0.4493 0.4378 0.3557

Kt qu. x(3)1 0.4493; x (3)2 0.4378; x (3)3 0.3557

Cu 3. Cho h phng trnh

2.5Mx1 +0.45x2 +0.54x3 = 4.410.32x1 +2.5Mx2 0.33x3 = 1.720.31x1 0.35x2 +4.1Mx3 = 3.12

S dng phng php Gauss-Seidel vi x (0) = [1.3, 0.5, 0.1]T , tm vctlp x (3)

Kt qu. x(3)1 ; x (3)2 ; x (3)3


Bm my. 1.3 Shift-STO-A, 0.5 Shift-STO-B, 0.1 Shift-STO-C, M = 3.3Shift-STO-M.4.41 0.45B 0.54C

2.5MShift-STO-A.

1.72 0.32A + 0.33C2.5M

Shift-STO-B.

3.12 0.31A + 0.35B4.1M

Shift-STO-C.

Thc hin lin tip thm 2 ln na c x (3).

Kt qu. x(3)1 = 0.5091; x

(3)2 = 0.1977; x

(3)3 = 0.2240

Cu 4. Cho bng sx 2.2 2.5 2.7 3.0

y 2.34 M 1.5M 2.1Dng a thc ni suy

Newton, xp x gi tr ca o hm ti x = 2.4Kt qu. y (2.4)


xk f (xk) T sai phn I T sai phn II T sai phn III

2.2 2.34M2.342.52.2

2.5 M 5M+67.86(2.72.2)1.5MM2.72.5

40M31.63(3.02.2)

2.7 1.5M 77.5M3.02.52.11.5M32.7

3.0 2.1

Nh vy cng thc ni suy Newton tin l

N (1)4 (x) = 2.34 +M 2.342.5 2.2 .(x 2.2) +

5M + 6 7.86 (2.7 2.2) (x 2.2)(x 2.5)+

+40M 3 1.63 (3.0 2.2) (x 2.2)(x 2.5)(x 2.7)

y M2.342.52.2 + 5M+67.86(2.72.2) (2x 4.7) + 40M31.63(3.02.2) (3x2 14.8x + 18.19).CALC M=3.3, X=2.4. Kt qu. y (2.4) 7.0600TS. L Xun i (BK TPHCM) N TP CUI K MN PHNG PHP TNH TP. HCM 2013. 9 / 1

Cu 5. Cho bng sx 1.2 1.3 1.4 1.5 1.7

y M 2.5 2M 4.5 5.5S dng phng

php bnh phng b nht, tm hm dng f (x) = A + Bx + Cx2 xp x ttnht bng s trnKt qu. A = , B = , C =Bm my. 3.3 Shift STO M. Bm Mode 3 - STAT. Chn 3- +cx2. Nhpd liu ca 2 ct x , y . AC - Thot ra. Chn Shift 1 - chn 7 - Reg - chn1- A =. Chn Shift 1 - chn 7 - Reg - chn 2- B =. Chn Shift 1 - chn 7- Reg - chn 3- C =.Kt qu. A = 33.9839, B = 48.7636, C = 15.0442Cu 6. Cho tch phn I =

2.51.3

lnx + 2Mdx . Hy xp x tch phn I bng

cng thc hnh thang m rng vi n = 8Kt qu. I Gii.

h =b an

=2.5 1.3

8=

3

20, x0 = 1.3, xk = 1.3 + k .

3

20,

yk = lnxk + 2M = ln

1.3 + k .

3

20+ 2M


Vy I h

2

7k=0

(yk + yk+1) =

3

40

7k=0

(ln

1.3 + k .

3

20+ 2M + ln

1.3 + (k + 1).

3

20+ 2M

)=

Kt qu. I 1.2835Cu 7. Cho bng s

x 1.0 1.2 1.4 1.6 1.8 2.0 2.2

f (x) 2 3.3M 2.4 4.3 5.1 6.2 M.S

dng cng thc Simpson m rng tnh tch phn I =2.21.0

[Mxf 2(x) + x2]dx

Kt qu. I Gii.

h =b an

=2.2 1.0

n= 0.4 n = 3, x0 = 1.0, xk = 1.0 + 0.4k ,

x k =xk1 + xk

2= 1.0 + (2k 1) 0.2, yk = Mxk f 2(xk) + x2k ,

y k = Mxk f

2(x k) + x2k


I h

6

2k=0

(yk + 4yk+1 + yk+1) =

0.4

6(

2k=0

(yk) + 42

k=0

(y k+1) +2

k=0

(yk+1)) =

0.4

6((y0 + y1 + y2) + 4(y

1 + y

2 + y

3) + (y1 + y2 + y3)) =

0.4

6((y0 + 2y1 + 2y2 + y3) + 4(y

1 + y

2 + y

3))

Bm my.M 1.0 22 + 1.02 shift-STO-AM 1.4 2.42 + 1.42 shift-STO-BM 1.8 5.12 + 1.82 shift-STO-CM 2.2 M2 + 2.22 shift-STO-D0.4

6 (A + 2B + 2C + D) shift-STO-X

M 1.2 (3.3M)2 + 1.22 shift-STO-AM 1.6 4.32 + 1.62 shift-STO-BM 2.0 6.22 + 2.02 shift-STO-C4 0.4

6 (A + B + C ) shift-STO-Y

Kt qu: X + YKt qu. I 252.4364TS. L Xun i (BK TPHCM) N TP CUI K MN PHNG PHP TNH TP. HCM 2013. 12 / 1

Cu 8. Xt bi ton Cauchy{y (x) = My2 + x .y(x) + x2, x > 1.1,y(1.1) = 1.6

S dng cng thc Runge-Kutta cp 4, hy xp x gi tr ca hm tix = 1.3 vi bc h = 0.2.Kt qu. y(1.3) Gii.Vi h = 0.2, x1 = x0 + 0.2 = 1.3, y0 = 1.6. Ta c

K 01 = hf (x0, y0) = 0.2(My2 + x .y(x) + x2), K 02 = hf (x0 +

h2 , y0 +

K012 ),

K 03 = hf (x0 +h2 , y0 +

K022 ), K

04 = hf (x0 + h, y0 + K

03 ).

Cng thc tnh nghim gn ng l

y(1.3) y1 = y0 + 16

(K 01 + 2K02 + 2K

03 + K

04 ) =


Bm my. 3.3 Shift-STO-M. 0.2(My2 + x .y + x2).Tnh K 01 . CALC X = 1.1,Y = 1.6. K 01 Shift-STO-ATnh K 02 . CALC X = 1.1 +

0.2

2,Y = 1.6 +

A

2. K 02 Shift-STO-B

Tnh K 03 . CALC X = 1.1 +0.2

2,Y = 1.6 +

B

2. K 03 Shift-STO-C

Tnh K 04 . CALC X = 1.1 + 0.2,Y = 1.6 + C . K 04 Shift-STO-D

y(1.3) y1 = y0 + 16

(K 01 + 2K02 + 2K

03 + K

04 ) =

= 1.6 +1

6(A + 2B + 2C + D)

Kt qu. y(1.2) 40.3086


Cu 9. Cho hm s f (x) = ex . ln(x4 + 1)Mx . S dng sai phn hngtm, xp x gi tr ca f (0.7) v f (0.7) vi bc h = 0.15Kt qu. f (0.7) ; f (0.7) Gii. Cng thc sai phn hng tm

f (x0) f (x0 + h) f (x0 h)

2h f (0.7) f (0.7 + 0.15) f (0.7 0.15)

2 0.15

f (x0) f (x0 + h) 2f (x0) + f (x0 h)

h2

f (0.7) f (0.7 + 0.15) 2f (0.7) + f (0.7 0.15)0.152

Bm my. eX . ln(X 4 + 1)MXCALC X=0.7 + 0.15, M = 3.3 = shift-STO-ACALC X=0.7, M = 3.3 = shift-STO-BCALC X=0.7 0.15, M = 3.3 = shift-STO-Cf (0.7)

A C2 0.15 , f

(0.7) A 2B + C

0.152

Kt qu. f (0.7) 0.5301 ; f (0.7) 11.9020TS. L Xun i (BK TPHCM) N TP CUI K MN PHNG PHP TNH TP. HCM 2013. 15 / 1

Cu 10. Xt bi ton bin{(x + M)y (x) + x3y (x) 15My(x) = x(x + 1), 0 6 x 6 1

y(0) = 0.5M, y(1) = 1.2

S dng phng php sai phn hu hn, xp x gi tr ca hm trong[0, 1] vi bc h = 0.25Kt qu. y(0.25) = , y(0.50) = , y(0.75) =Gii.x0 = 0, x1 = 0.25, x2 = 0.5, x3 = 0.75, x4 = 1.p(x) = x + M, q(x) = x3, r(x) = 15M, f (x) = x(x + 1);p1 = x1 + M, p2 = x2 + M, p3 = x3 + M; q1 = x

31 , q2 = x

32 , q3 = x

33 ;

r1 = r2 = r3 = 15M;f1 = x1(x1 + 1), f2 = x2(x2 + 1), f3 = x3(x3 + 1)

y0 = 0.5M, y4 = 1.2

(p1h2 q12h )y0 + (r1 2p1h2 )y1 + (p1h2 + q12h )y2 = f1




y0 = 0.5M, y4 = 1.2

(r1 2p1h2 )y1 + (p1h2 + q12h )y2 + 0y3 = f1 (p1h2 q12h )y0(p2h2 q22h )y1 + (r2 2p2h2 )y2 + (p2h2 + q22h )y3 = f2

0y1 + (p3h2 q32h )y2 + (r3 2p3h2 )y3 = f3 (p3h2 + q32h )y4

Bm my. Mode-5 - EQN.

r1 2p1h2

= 15M 2.(0.25 + M)(0.25)2

.

p1h2

+q12h

=0.25 + M

0.252+

(0.25)3

2(0.25)0

f1 (p1h2 q1

2h)y0 = 0.25(0.25 + 1) (0.25 + M

0.252 (0.25)

3

2(0.25))0.5M


p2h2 q2

2h=

0.5 + M

0.252 (0.5)

3

2(0.25)

r2 2p2h2

= 15M 2.(0.5 + M)(0.25)2

.

p2h2

+q22h

=0.5 + M

0.252+

(0.5)3

2(0.25)f2 = 16x22 = 0.5(0.5 + 1)

0p3h2 q3

2h=

0.75 + M

0.252 (0.75)

3

2(0.25)

r3 2p3h2

= 15M 2(0.75 + M)0.252

f3 (p3h2

+q32h

)y4 = 0.75(0.75 + 1) (0.75 + M0.252

+(0.75)3

2(0.25))1.2

Kt qu. y(0.25) = 0.7473, y(0.50) = 0.4909, y(0.75) = 0.6225


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Giai on Cuoi Ky 1 In