Fitting an equation to a set of multiple equations.

Hi,

I need to know how can I get a single equation for a given set of equations.

For e.g.:- I have,

Y1=0.106X1+0.103, for a1=1mm
Y2=0.092X2-0.03, for a2=0.5mm
Y3=0.028X3+0.078, for a3=2mm

Now I need a single equation for Y which can include 'a', say;

Y=pX+qa+r or something like that. Is it possible? Is there any software in order to achieve this?

Please help me!

OK, in general you have a linear equation whose coefficient are function of a variable "a", that is:

Y=m(a)*x+n(a)

so you want to determine the funtions m(a) and n(a)

from your data we have (ordered by increasing value of "a"):

m(a): m(0.5)=0.092, m(1)=0.106, m(2)=0.028
n(a): n(0.5)=-0.03, n(1)=0.103, n(2)=0.078

we can see that "m" and "n" arent' linear with respect to the variable "a", so you can't have an expression like that you showed (I mean: Y=pX+qa+r) unless you can accept very large errors.

You can instead use a quadratic model; that is m(a)=hm*a^2+km*a+qm and n(a)=hn*a^2+kn*a+qn. This because it's always possible to fit a parabola given 3 points.

numerically you have to solve (with respect to h,k and q) the two linear systems of equations:

0.092 =0.25*hm+0.5*km+qm
0.106=hm+km+qm
0.028=4*hm+2*km+qm

-0.03 =0.25*hn+0.5*kn+qn
0.103=hn+kn+qn
0.078=4*hn+2*kn+qn

Then Y=m(a)*x+n(a) will be:

Y=(hm*a^2+km*a+qm)*x+hn*a^2+kn*a+qn

NOTE: this last will be exact only in the three point (a=0.5, 1 and 2 mm)