ABSTRACT
We wish to represent the approximating curve, yc, as a straight line of the form
y c c xc = +1 2 (9.1)
where c1 and c2 are unknown constants to be determined. Let D be the sum of the square of the errors between the approximating line and the actual points. Then,
D y y x y c c xi c i
= − = − + = =
∑ ∑[ ( )] [ ( )]2 1
1 (9.2)
or
D y c c x y c c x y c c xn n= − + + − + + + − +[ ( )] [ ( )] [ ( )]1 1 2 1 2 2 1 2 2 2 1 2 2 (9.3)
To obtain the best-fit straight-line approximating function, minimize D by taking ∂ ∂ = D c1
0 and ∂∂ = D c2
0. Taking the partial derivative of Equation 9.3 with respect to
c1 gives
∂ ∂ = = − + −
∑Dc y c c x i
1 20 2 1[ ( )][ ]
1= − −
∑ ∑y c x nci i
or
nc x c yi
+
=
∑ ∑ (9.4) Taking the partial derivative of Equation 9.3 with respect to c2 gives
∂ ∂ = = − + −
∑Dc y c c x xi i i i
0 2[ ( )][ ]
= − −
∑ ∑∑x y c x c xi i i
or
x c x c x yi
∑ ∑ ∑ +
=1 1
1 (9.5)
Equations 9.4 and 9.5 describe a system of two algebraic equations in two unknowns, which can be solved by the method of determinants (Cramer’s rule):
c
y x
x y x
n x
x x
y x x x y
n x
2 = =
( )( ) − ( )( ) ∑ ∑ ∑ ∑
(9.6)
c
n y
x x y
n x
x x
n x y x y
n x x x
2 = =
− ( )( ) − ( )(
(9.7)
9.2.2 Best-Fit mth-Degree Polynomial We can generalize the earlier approach for an mth-degree polynomial fit. In this case, take the approximating curve, yc , to be
y c c x cx c x c xc 2 3 m m= + + + + + +1 2 3 4 1 (9.8)
where m ≤ n − 1 and n is the number of data points. The measured values are (xi, yi) for i = 1, 2, … , n. Let yc,i = yc(xi) be the approximated value of yi at the point (xi , yi). Then,
D y y y c c x cx c xi ci
= − = − + + + +( ) =
(9.9)
To minimize D, take
∂ ∂ =
∂ ∂ =
∂ ∂ =+
D c
D c
0 0 0, , ,…
Then,
∂ ∂ = = − + + +( ) −
∂ ∂ = =
D c
y c c x c x
D c
0 2 1
0 2
[ ]
y c c x c x x
D c
y c c x
− + + +( ) −
∂ ∂ = = − + +
1 20 2
[ ]
+( ) −
∂ ∂ = = − + + +
c x x
D c
y c c x c x
im imx( ) − This set of equations reduces to
nc x c x c x c y
x c x c x
+ ( ) + ( ) + + ( ) = ( ) + ( ) +
( ) + + ( ) =
( ) + ( ) + + ( )
c x c x y
x c x c x c
(9.10)
Equation 9.10 can be solved by Gauss elimination (as described in Chapter 4). Alternatively, MATLAB • ’s polyfit function (discussed in Section 9.4) pro-
vides a solution to Equation 9.10, which represents the best-fit polynomial of degree m for the (xi , yi) set of data points.
