Information Technology Reference
In-Depth Information
X
k
x
1
+
X
k
x
2
+···+
X
k
x
n
∂f
2
∂x
1
∂f
2
∂x
2
∂f
2
∂x
n
f
2
(
X
k
+
1
)
=
=
f
2
(
X
k
)
+
0
.
X
k
x
1
+
X
k
x
2
+···+
X
k
x
n
∂f
n
∂x
1
∂f
n
∂x
2
∂f
n
∂x
n
f
n
(
X
k
+
1
)
=
=
f
n
(
X
k
)
+
0
The algorithm for solving a set of nonlinear simultaneous equations by the
Newton - Raphson method is given below. If the equations to be solved are not
analytical, it is necessary to calculate derivatives numerically.
1. Guess
X
k
.
2. Evaluate all
f
i
(
X
k
)
.
3. If all
f
i
(
X
k
)
, the problem is solved.
4. Calculate all partial derivatives at
X
k
.
5. Solve the matrix - vector equations
|
|
<
ε
X
=
P
−
1
(
−
F
)
P
X
=−
F
for
X
where
P
is the matrix of partial derivatives,
X
is the vector of
x
moves, and
F
is the vector of function values.
6. For all
x
i
, calculate
x
k
+
1
i
=
x
i
+
x
i
7. Return to step 2, substituting
X
k
+
1
for
X
k
.
6.4.2 Direct Optimization of an Objective Function
The default regression method in Aspen Plus is maximum likelihood. In this method
the objective function is given by
(T
i
n
data
−
T
i
)
2
σ
(P
i
−
P
i
)
2
σ
(x
i
−
x
i
)
2
σ
(y
i
−
y
i
)
2
σ
ψ =
+
+
+
(6.24)
2
T,i
2
P,i
2
x,i
2
y,i
i
where the superscripts
e
and
m
refer to estimated and measured values. For each point
in the set of data the following constraints apply. For simplicity, Poynting correction
is ignored; then for each component one may write
V
− γ
i
x
i
f
i
y
i
φ
i
P
=
0
(6.25)
γ
i
= γ
(T , x)
(6.26)
i
V
(T,P,y)
φ
= φ
(6.27)
