# Lets take an example with four unknowns n 4 which we

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also more normal equations. Let’s take an example with four unknowns ( N = 4), which we will denote by A, B, C, D ; this would be like fitting a cubic. With N = 4 we need at least five datapoints ( M = 5), so there must be at least five equations of condition. The generalization of equation 0.4 is the M equations As m + Bt m + Cu m + Dv m = y m , (1.1) with m = 0 ( M 1). Again, the least-squares-fitting process assumes that the s m , t m , u m , v m are known with zero uncertainty; all of the uncertainties are in the measurements of y m . We then form the four normal equations; the generalization of equation 0.5 written in matrix format is:

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– 7 – [ ss ] [ st ] [ su ] [ sv ] [ ts ] [ tt ] [ tu ] [ tv ] [ us ] [ ut ] [ uu ] [ uv ] [ vs ] [ vt ] [ vu ] [ vv ] A B C D = [ sy ] [ ty ] [ uy ] [ vy ] (1.2) The N × N matrix on the left is symmetric. With N equations and N unknowns, you can actually solve for A, B, C, D ! 2. FAR, FAR BEST AND EASIEST: MATRIX ALGEBRA The above equations are terribly cumbersome to solve in a computer code because they require lots of loops. However, it becomes trivial if we use matrices. Here we designate a matrix by boldface type. We illustrate the matrix method by carrying through the above N = 4 example, and we assume that there are 5 independent measurements ( M = 5). We first define the matrices X = s 0 t 0 u 0 v 0 s 1 t 1 u 1 v 1 s 2 t 2 u 2 v 2 s 3 t 3 u 3 v 3 s 4 t 4 u 4 v 4 (2.1a) a = A B C D (2.1b) Y = y 0 y 1 y 2 y 3 y 4 (2.1c) so, in matrix form, the equations of condition (equation 1.1) reduce to the single matrix equation
– 8 – X · a = Y . (2.2) The notation for these equations corresponds to NR’s. We write them with subscripts σ to empha- size that they are calculated without dividing by σ meas , i.e. that we are doing least squares instead of chi-square fitting. For chi-square fitting, see § 8 and 9. Our matrix X corresponds to NR’s “design matrix” A of Figure 15.4.1, except that our elements are not divided by σ meas,m , and the matrix equation of condition (equation 2.2) is identical to the expression inside the square brackets of NR’s equation 15.4.6. The differences arise because here we are discussing least-squares fitting instead of chi-square fitting, i.e. we have omitted the factors involving σ meas,m , the intrinsic measurement uncertainties ( § 8). Again, there are more equations than unknowns so we can’t solve this matrix equation directly. So next we form the normal equations from these matrices. In matrix form, the normal equations (equation 1.2) reduce to the single equation [ α ] · a = [ β ] , (2.3) (NR equation 15.4.10), where [ α ] = X T · X (2.4a) [ β ] = X T · Y . (2.4b) The matrix [ α ] is known as the curvature matrix because each element is twice the curvature of σ 2 (or χ 2 ) plotted against the corresponding product of variables.

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