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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`
199640-24-816-3264-80160-320
199750-11-11-11-5050-50
199862000000000
199958111111585858
200060248163264120240480
---------------------------------
9990270010034013048508168


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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