计算方法第三章习题

第三章习题

用一次、二次、三次多项式及最小二乘原理拟合这些数据，并写出正规方程组。

2323

X 2=x 1 , x 3=x 1 , x 1x 2=x 1 , x 1x 3=x 1 , x 2x 3=x 1 , x 2=x 1 , x 3=x 1 ,

⎪⎪⎪⎪⎩⎪⎪⎪⎪

⎨⎧

=+++=+++=+++=+++∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑=================-=91

91919191332

323213103919191919

1

2332222

12102919191919

1

1331221121019191919

1

33221109i i i i i i

i i i i i i i i i i i i i

i i i i i i i i i i i i i

i i i i i i i i i i i i

i i i y x a x a x x a x x a x y x a x x a x a x x a x y

x a x x a x x a x a x y a x a x a x a ⎪⎪⎩⎪⎪⎨

⎧

=+=+=+=+2295

.63876.27656.25871.77656.275.34439.87656.275.3121.1875.3931203

120a a a a a a a a 解为：a 0=2.0001,a 1=2.2501 , a 2=0.03131 , a 3=0.002085

f 3(x)=2.0001+2.2501x+0.0313x 2+0.002085x 3 f 2(x)=2.0001+2.2516x+0.0313x 2 f 1(x)=2.0131+2.2516x

2答案：取y=y, x 1=x , 有y=a+bx 1

⎪⎪⎩

⎪⎪⎨⎧

=+=+∑∑∑∑∑===5

112

11515

1

1i i i i i i i i

i y x b x a x y b x na ⎩⎨

⎧

=+=+5.3693217277699

53274.27153275b a b a 解得：a=0.972579 , b=0.050035

y=0.972579+0.050035x 2

Bx

答案：经验公式两边取对数得：lg y=lg A+Bx lg e 令u=lg y , a=lg A ,b=B lg e 则u=a+bx

⎩

⎨

⎧=+=+3398.1330107324

.5104b a b a 解得a=1.9288 , b=-0.19828 所以A=10a

=84.879 ,B=b/lg e=-0.4566 Y=84.879e -0.4566x