Question

已知多组数据X,Y,如何用方程y=a+bx^2+cx^(-2)拟合得到a,b,c的值？

x=[27.270 32.529 32.699 34.495 37.975 38.317 43.219 46.551] y=[3.0741 3.0867 3.0870 3.1061 3.1432 3.1449 3.2098 3.2627] x实为一组频率值，数量级为10的14次方。y 为介电常数。

Answer 1

f(a,b,c)=Sum_{i=1->N}{y(i)-a-b[x(i)]^2
-
c[x(i)]^(-2)}^2
0=df(a,b,c)/da
=
2Sum_{i=1->N}{y(i)-a-b[x(i)]^2
-
c[x(i)]^(-2)}(-1)
...
(1)
0=df(a,b,c)/db
=
2Sum_{i=1->N}{y(i)-a-b[x(i)]^2
-
c[x(i)]^(-2)}{-[x(i)]^2}
...
(2)
0=df(a,b,c)/dc
=
2Sum_{i=1->N}{y(i)-a-b[x(i)]^2
-
c[x(i)]^(-2)}{-[x(i)]^(-2)}
...
(3)
解由
(1),
(2),
(3)构成的关于a,b,c的三元一次方程组就可以求得a,b,c的值.