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Find Fit cubic equation {{1996,1997,1998,1999,2000},{40,50,62,58,60}}

Solution:
Your problem -> Fit cubic equation {{1996,1997,1998,1999,2000},{40,50,62,58,60}}

The equation is y = a + bx + cx^2 + dx^3 and the normal equations are

sum y = an + b sum x + c sum x^2 + d sum x^3

sum xy = a sum x + b sum x^2 + c sum x^3 + d sum x^4

sum x^2y = a sum x^2 + b sum x^3 + c sum x^4 + d sum x^5

sum x^3y = a sum x^3 + b sum x^4 + c sum x^5 + d sum x^6

 X y x = X - 1998 x^2 x^3 x^4 x^5 x^6 x*y x^2*y x^3*y 1996 40 -2 4 -8 16 -32 64 -80 160 -320 1997 50 -1 1 -1 1 -1 1 -50 50 -50 1998 62 0 0 0 0 0 0 0 0 0 1999 58 1 1 1 1 1 1 58 58 58 2000 60 2 4 8 16 32 64 120 240 480 --- --- --- --- --- --- --- --- --- --- --- 9990 270 0 10 0 34 0 130 48 508 168

Substituting these values in the normal equations
270=5a+0b+10c+0d

48=0a+10b+0c+34d

508=10a+0b+34c+0d

168=0a+34b+0c+130d

Solving these 4 equations using inverse matrix method,
Here 5a+10c=270
10b+34d=48
10a+34c=508
34b+130d=168

Now converting given equations into matrix form
[[5,0,10,0],[0,10,0,34],[10,0,34,0],[0,34,0,130]] [[a],[b],[c],[d]]=[[270],[48],[508],[168]]

Now, A = [[5,0,10,0],[0,10,0,34],[10,0,34,0],[0,34,0,130]], X = [[a],[b],[c],[d]] and B = [[270],[48],[508],[168]]

:. AX = B

:. X = A^-1 B

|A| =
 5 0 10 0 0 10 0 34 10 0 34 0 0 34 0 130

=
5 ×
 10 0 34 0 34 0 34 0 130
+0 ×
 0 0 34 10 34 0 0 0 130
+10 ×
 0 10 34 10 0 0 0 34 130
+0 ×
 0 10 0 10 0 34 0 34 0

=5 × (4896)+0 × (0)+10 × (-1440)+0 × (0)

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