Etude de la combustion de gaz de synthèse issus d'un processus de ...

Appendix A – Overdetermined linear equations systems
2
r = b −AX
2
(A-4)
in other words, determine X
such that b − AX is close to zero.
Developing the expression (A-4), we obtain:
2
2
T
( ) ( )
r = b − AX = b −AX b −AX
(A-5)
T T T T T
= X A AX − 2X A b + b b
The expression (A-5) is minimized when its gradient with respect to each parameter is
equal to zero. The elements of the gradient vector are the partial derivatives of
respect to each parameter
r
2
with
tel-00623090, version 1 - 13 Sep 2011
∂ r
∂x
i
2
m
∂r
j
= 2 ∑ r
j
.
∂x
j = 1
Substituting r = b −∑ A x and its derivative
Taking
∂ r
∂x
i
j j ji i
i = 1
2
following normal equations
n
2
i
∂r
j
∂x
i
= −A
∂ r
m
n
⎛ ⎞
=−2
Aji ⎜bj − Ajk xk
⎟
∂x i
j= 1 ⎝ k=
1 ⎠
ji
(A-6)
in equation (A-6) follows:
∑ ∑ (A-7)
= 0, for all i=1,…n and upon a rearrangement of (A-7) we obtain the
m m n
∑Ab
+ ∑∑ AA x = 0.
(A-8)
ji j ji jk k
j= 1 j= 1 k=
1
Using matrix notation, the normal equations are given by
T
T
A AX − A b = 0
(A-9)
or equivalently by
T
A AX
=
T
A b
(A-10)
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