Page 84 - Numerical Methods for Chemical Engineering

Newton’s method for multiple nonlinear equations 73

Assuming that x [k] is sufficiently close to the true solution x s that we introduce little error
by dropping the terms of quadratic and higher order,

N
[k] ∂ f i [k]
f i (x s ) = 0 ≈ f i x + x s,m − x (2.52)
m
∂x m
m=1 x [k]
For convenience, we collect the first partial derivatives into the N × N Jacobian matrix
[k]
J [k] = J(x ), with elements

[k] ∂ f i
J im = (2.53)

∂x m x [k]
The truncated Taylor series expansion then becomes
N
[k] [k] [k]
0 ≈ f i x + J x s,m − x (2.54)
im m
m=1
We thus generate the new estimate of the solution, x [k+1] ≈ x s by solving the set of linear
algebraic equations
N
[k] [k] [k+1] [k]
0 = f i x + J x − x (2.55)
im m m
m=1
Defining the update vector
[k] [k+1] [k]
x ≡ x − x (2.56)
the set of Newton update linear equations is
N
[k] [k] [k]
J x =− f i x (2.57)
im m
m=1
We recognize the term on the left-hand side as the ith component of a matrix–vector product
and write the linear system as
[k]

J x [k] =− f x [k] (2.58)
At each Newton iteration, we must solve a linear system of N equations, e.g. through
Gaussian elimination. This procedure is repeated until the norm of the function vector
becomes smaller than some tolerance value,
& & & & & [0]
f x [k] & and/or f x [k] & ≤ δ rel f x (2.59)
& ≤ δ abs & &
The choice of norm is somewhat subjective, but a common selection is

&
&
& f x [k] & = max{| f 1 | , | f 2 | ,..., | f N |} (2.60)
∞
As was the case for a single equation, convergence in the close vicinity of the solution is
quadratic. The proof is somewhat more involved with multiple equations, and is presented
in the supplemental material in the accompanying website.