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S GN NG V SAI SBi ging in t

Nguyn Hng LcTrng i hc Bch Khoa TP HCM

Khoa Khoa hc ng dng, b mn Ton ng dng

TP. HCM 2013.Nguyn Hng Lc (BK TPHCM) S GN NG V SAI S TP. HCM 2013. 1 / 30

S gn ng v sai s Nhng khi nim c bn

Nhng khi nim c bn

nh ngha sai lch gia gi tr gn ng v gi tr chnhxc c gi l sai s.

nh nghaS a c gi l s gn ng ca s chnh xc A,k hiu l a A (c l a xp x A) nu a khc Akhng ng k v c dng thay cho A trongtnh ton.Nguyn Hng Lc (BK TPHCM) S GN NG V SAI S TP. HCM 2013. 2 / 30


nh nghai lng = |a A| c gi l sai s tht sca s gn ng a.

Trong thc t, do khng bits chnh xc A, ta c lng mt i lng dnga cng b cng tt tha iu kin |A a| 6 ac gi l sai s tuyt i ca s gn ng a.

Ch . Trong thc t ta s k hiu A = a a.

Nguyn Hng Lc (BK TPHCM) S GN NG V SAI S TP. HCM 2013. 3 / 30


nh nghai lng = |a A| c gi l sai s tht sca s gn ng a. Trong thc t, do khng bits chnh xc A, ta c lng mt i lng dnga cng b cng tt tha iu kin |A a| 6 ac gi l sai s tuyt i ca s gn ng a.

Ch . Trong thc t ta s k hiu A = a a.



nh nghaSai s tng i ca s gn ng a so vi s chnhxc A l i lng a c tnh theo cng thca =

|A a||A| .

Ch . Trong nhiu trng hp, nu khng bit Ata c th thay th a =

a|a|100%



V d 1. Gi s A = pi; a = 3.14. Do3.13 = 3.14 0.01 < pi < 3.14 + 0.01 = 3.15,nn ta c th chn a = 0.01. Mt khc,3.138 = 3.140.002 < pi < 3.14+0.002 = 3.142,do ta cng c th chn a = 0.002. Nh vy,vi cng mt gi tr gn ng, c th c nhiu sais tuyt i khc nhau. Trong trng hp ny tachn gi tr nh nht ca chng.



V d 2. Vn tc ca mt vt th o c lv = 2.8m/s vi sai s tng i v = 0.5%. Khi sai s tuyt i lv = vv =

0.5

100.2.8m/s = 0.014m/s.

V d 3. o di hai on thng ta ca = 10cm v b = 1cm vi a = b = 0.01cm.Khi a =

0.01

10= 0.1%, b =

0.01

1= 1% hay

b = 10a. T suy ra php o a chnh xc hnphp o b mc d a = b. Nh vy, chnhxc ca mt php o th hin qua sai s tng i.Nguyn Hng Lc (BK TPHCM) S GN NG V SAI S TP. HCM 2013. 6 / 30

S gn ng v sai s Biu din s thp phn

Ch s c ngha

Mi s thc a c th c biu din di dngthp phn hu hn hoc v hna = (mm1 . . . 10.12 . . . n) =

mk=n

k10k ,m, n N,m > 0, n > 1, m 6= 0,

k {0, 1, 2, . . . , 9}.V d 1. 324.59 =3 102 + 2 101 + 4 100 + 5 101 + 9 102.



Mt s vit dng thp phn c th gm nhiuch s. V d 20.25 c 4 ch s, 0.03047 c 6 chs.nh nghaNhng ch s c ngha ca mt s l nhng chs ca s k t ch s khc khng u tintnh t tri sang phi.

V d 2. S 20.25 c 4 ch s c ngha. S0.03047 cng c 4 ch s c ngha.



nh nghaLm trn mt s thp phn a l b mt s cc chs bn phi a sau du chm thp phn cmt s a ngn gn hn v gn ng nht so vi a.

Quy tc. lm trn n ch s th k sau duchm thp phn, ta xt ch s th k + 1 sau duchm thp phn l k+1.

Nu k+1 > 5, ta tngk ln 1 n v; cn nu k+1 < 5 ta gi nguynch s k . Sau b phn ui t ch s k+1tr i.




Quy tc. lm trn n ch s th k sau duchm thp phn, ta xt ch s th k + 1 sau duchm thp phn l k+1. Nu k+1 > 5, ta tngk ln 1 n v;

cn nu k+1 < 5 ta gi nguynch s k . Sau b phn ui t ch s k+1tr i.




Quy tc. lm trn n ch s th k sau duchm thp phn, ta xt ch s th k + 1 sau duchm thp phn l k+1. Nu k+1 > 5, ta tngk ln 1 n v; cn nu k+1 < 5 ta gi nguynch s k . Sau b phn ui t ch s k+1tr i.Nguyn Hng Lc (BK TPHCM) S GN NG V SAI S TP. HCM 2013. 9 / 30


V d 3. Lm trn s pi = 3.1415926535... nch s th 4,3,2 sau du chm thp phn nhnc cc s gn ng ln lt l3.1416; 3.142; 3.14.



nh nghaSai s thc s ca a so vi a c gi l sai s lmtrn. Vy a = |a a|.Sai s tuyt i ca a so vi A c nh gi nhsau: |a A| = |(a a) + (a A)| 6|a a| + |a A| 6 a + a = a. V a > 0 nna > a. Do sau khi lm trn sai s tng ln.V vy, khi tnh ton ta trnh lm trn cc phpton trung gian, ch nn lm trn kt qu cuicng.Nguyn Hng Lc (BK TPHCM) S GN NG V SAI S TP. HCM 2013. 11 / 30


S lm trn s trong bt ng thc

Trng hp lm trn s trong bt ng thc, ta sdng khi nim lm trn ln v lm trn xung.Lm trn ln hay lm trn xung cn lu nchiu bt ng thc.V d 4. a < 13.9236 khi lm trn ln n 2 chs l sau du chm thp phn ta c a < 13.93v b > 78.6789 khi lm trn xung n 2 ch sl sau du chm thp phn ta c b > 78.67.



Ch s ng tin

nh nghaCho a A. Ch s k trong php biu din didng thp phn c gi l ng tin, nua 6

1

2.10k . Trong trng hp ngc li, ch s

k c gi l khng ng tin.

V d 5. Cho s gn ng a = 3.7284 vi sai stuyt i l a = 0.0047 c 3 ch s ng tin l3, 7, 2 v 2 ch s khng ng tin l 8, 4Nguyn Hng Lc (BK TPHCM) S GN NG V SAI S TP. HCM 2013. 13 / 30


Cch vit s gn ng

Chng ta vit s gn ng a ca s chnh xc Avi sai s tuyt i a theo quy tc sau:

1 Vit s gn ng a km theo sai s tuyt ia di dng a a. V d 17.358 0.003.Cch ny thng c dng biu din cckt qu tnh ton hoc php o.

2 Vit s gn ng theo quy c: mi ch s cngha u ng tin. iu ny c ngha l sai stuyt i a khng ln hn mt na n vca ch s cui cng bn phi.



V d a = 23.54 th sai s tuyt ia 6

1

2.102 = 0.005, trong khi nu vit

a = 23.5400 th sai s tuyt ia 6

1

2.104 = 0.00005. Cch ny thng dng

trnh by cc bng s.


Xc nh sai s ca hm s bit sai s ca cc i s Cng thc tng qut ca sai s

Cng thc tng qut ca sai s

Cho hm s kh vi lin tc y = f (x1, x2, . . . , xn)v gi s bit sai s tuyt i xi ca cc i sxi (i = 1..n). Gi Xi ,Y v xi , y (i = 1..n) l ccgi tr chnh xc v cc gi tr gn ng ca i sv hm s. Khi |Y y | = |f (X1,X2, . . . ,Xn)f (x1, x2, . . . , xn)| 6n

i=1

fxi .|Xi xi | 6 n

i=1

fxi .xi . Vy sai s

tuyt i ca hm s y l y =n

i=1

fxi .xi



Sai s tng i ca hm s y l

y =y|y | =

ni=1

fxi .xi|f |

=n

i=1

xi ln f (x1, x2, . . . , xn) .xi



Cng thc tng qut ca sai s

V d 6. Tnh sai s tuyt i v sai s tng ica th tch hnh cu V = 1

6pid3, bit ng knh

d = 3.70cm 0.05cm v pi = 3.14 0.0016.Xem pi v d l nhng i s ca hm s V , ta cV

pi=

1

6d3 =

1

6 (3.70)3 v

V

d=

1

2pid2 =

1

2 (3.14) (3.70)2.



Vy V =Vpi

.pi + Vd .d = 16 (3.70)3

0.0016 +1

2 (3.14) (3.70)2 0.05 = 1.0882.

Do V = 16pi.d3 = 26.5084cm3 1.0882cm3 v

V =1.0882

26.5084= 0.0411.


Xc nh sai s ca hm s bit sai s ca cc i s Sai s ca tng i s

Sai s ca tng i s

Xt hm s y = x1 x2 . . . xn. Khi fxi = 1, (i = 1..n). Do , sai s tuyt i ca

y l y = x1 + x2 + . . . + xn v sai s tngi ca y l y =

y|y | .


Xc nh sai s ca hm s bit sai s ca cc i s Sai s ca tng i s

V d 7. Tnh sai s tuyt i v sai s tng ica y = a + b + c vi a = 47.132 0.003; b =47.111 0.02; c = 45.234 0.5.Sai s tuyt i ca y ly = a+b +c = 0.003+0.02+0.5 = 0.523.

Do Y = y y = 139.477 0.523 nn sai stng i ca y l y =

y|y | = 0.0037.


Xc nh sai s ca hm s bit sai s ca cc i s Sai s ca tch

Sai s ca tch

Xt hm s y = x1.x2 . . . xn. Khi xi ln y = 1|xi |, (i = 1..n). Do , sai s tng

i ca y l y = x1 + x2 + . . . + xn v sai stuyt i ca y l y = y .|y |.


Xc nh sai s ca hm s bit sai s ca cc i s Sai s ca tch

V d 8. Tnh sai s tuyt i v sai s tng ica y = a.b.c vi a = 47.132 0.003; b =47.111 0.02; c = 45.234 0.5.Ta c a =

a|a| , b =

b|b| , c =

c|c| ,

Sai s tng i ca y l y = a + b + c =0.003

47.132+

0.02

47.111+

0.5

45.234= 0.0115.

Do sai s tuyt i ca y ly = y .|y | = ( 0.003

47.132+

0.02

47.111+

0.5

45.234)

(47.132 47.111 45.234) = 1159.2503.Nguyn Hng Lc (BK TPHCM) S GN NG V SAI S TP. HCM 2013. 23 / 30

Bi tp

Bi tp

Bi 1. Bit A c gi tr gn ng l a = 3.5833vi sai s tng i l a = 0.28%. Ta lm trn athnh a = 3.58. Tnh sai s tuyt i ca aGii. Ta ca = a + |a a| = |a|a + |a a| = 0.0134


Bi tp

Bi 2. Lm trn n hai ch s l sau du chmthp phn ca cc s trong cc biu thc sau: a =12.6724; b = 1.5476; c 6 12.8713; d > 1.2354.Gii.

a = 12.6724 a 12.67.b = 1.5476 b 1.55.c 6 12.8713 c 6 12.88.d > 1.2354 d > 1.23.


Bi tp

Bi tp

Bi 3. Cho a = 7.1696 vi sai s tng i la = 0.83%. Tm s ch s ng tin trong cchvit thp phn ca aGii.Ch s ng tin v tr k tha : a 6 1210k k > log(2a) = log(2|a|a) = 0.92kmin = 0. C mt ch s ng tin


Bi tp

Bi tp

Bi 4. Cho biu thc f = x3 + xy + y 3.Bitx = 1.9501 0.0050 v y = 3.4740 0.0083.Tnh sai s tuyt i ca fGii.f = |f x |x + |f y |y = |3x2 +y |x + |x + 3y 2|y= |3 1.95012 + 3.4740| 0.0050 + |1.9501 + 3 3.47402| 0.0083 = 0.3912


Bi tp

Bi tp

Bi 5. Choa = 15.000.02, b = 0.1230.001, c = 1370.5.Hy tnh sai s tuyt i ca

A = a + b + c

B = 20a 100b + cC = a + bc.


Bi tp

Gii.

A = a+b+c A = a + b + c = 0.521.B = 20a 100b + c B =20.a + 100.b + c = 1.

C = a + bc C = a + |c|.b + |b|.c =0.2185.


Bi tp

Bi 6. Cho hm f (x) = 3x5 2x2 + 7 vx = 1.234 0.00015. Tnh f .Gii. Ta c f (x) = 15x4 4x vf = |f (x)|.x nnf = |15 (1.234)4 4 1.234| 0.00015 =0.0045


S gn ng v sai sNhng khi nim c bnBiu din s thp phn

Xc nh sai s ca hm s bit sai s ca cc i sCng thc tng qut ca sai sSai s ca tng i sSai s ca tch

Bi tp

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