AIMS Lecture Notes 2006
Peter J. Olver
9. Numerical Solution of Algebraic Systems
In this part, we discuss basic iterative methods for solving systems of algebraic equa
tions. By far the most common is a vectorvalued version of Newton’s Method, which will
form our primary object of study.
9.1.
Vector–Valued Iteration.
Extending the scalar analysis to vectorvalued iterative systems is not especially dif
ficult. We will build on our experience with linear iterative systems.
We begin by fixing a norm
k · k
on
R
n
. Since we will also be computing the associated
matrix norm
k
A
k
, as defined in Theorem 7.13, it may be more convenient for computations
to adopt either the 1 or the
∞
norms rather than the standard Euclidean norm.
We begin by defining the vectorvalued counterpart of the basic linear convergence
condition (2.21).
Definition 9.1.
A function
g
:
R
n
→
R
n
is a
contraction
at a point
u
?
∈
R
n
if there
exists a constant 0
≤
σ <
1 such that
k
g
(
u
)

g
(
u
?
)
k ≤
σ
k
u

u
?
k
(9
.
1)
for all
u
sufficiently close to
u
?
, i.e.,
k
u

u
?
k
< δ
for some fixed
δ >
0.
Remark
: The notion of a contraction depends on the underlying choice of matrix
norm.
Indeed, the linear function
g
(
u
) =
A
u
if and only if
k
A
k
<
1, which implies
that
A
is a convergent matrix. While every convergent matrix satisfies
k
A
k
<
1 in
some
matrix norm, and hence defines a contraction relative to that norm, it may very well have
k
A
k
>
1 in a particular norm, violating the contaction condition; see (7.31) for an explicit
example.
Theorem 9.2.
If
u
?
=
g
(
u
?
)
is a fixed point for the discrete dynamical system
(2.1)
and
g
is a contraction at
u
?
, then
u
?
is an asymptotically stable fixed point.
Proof
: The proof is a copy of the last part of the proof of Theorem 2.6. We write
k
u
(
k
+1)

u
?
k
=
k
g
(
u
(
k
)
)

g
(
u
?
)
k ≤
σ
k
u
(
k
)

u
?
k
,
3/15/06
141
c
2006
Peter J. Olver
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
using the assumed estimate (9.1). Iterating this basic inequality immediately demonstrates
that
k
u
(
k
)

u
?
k ≤
σ
k
k
u
(0)

u
?
k
for
k
= 0
,
1
,
2
,
3
,
. . . .
(9
.
2)
Since
σ <
1, the right hand side tends to 0 as
k
→ ∞
, and hence
u
(
k
)
→
u
?
.
Q.E.D.
In most interesting situations, the function
g
is differentiable, and so can be approxi
mated by its first order Taylor polynomial
g
(
u
)
≈
g
(
u
?
) +
g
0
(
u
?
) (
u

u
?
) =
u
?
+
g
0
(
u
?
) (
u

u
?
)
.
(9
.
3)
Here
g
0
(
u
) =
∂g
1
∂u
1
∂g
1
∂u
2
. . .
∂g
1
∂u
n
∂g
2
∂u
1
∂g
2
∂u
2
. . .
∂g
2
∂u
n
.
.
.
.
.
.
.
.
.
.
.
.
∂g
n
∂u
1
∂g
n
∂u
2
. . .
∂g
n
∂u
n
,
(9
.
4)
denotes the
n
×
n
Jacobian matrix
of the vectorvalued function
g
, whose entries are the
partial derivatives of its individual components.
Since
u
?
is fixed, the the right hand
side of (9.3) is an affine function of
u
. Moreover,
u
?
remains a fixed point of the affine
approximation. Proposition 7.25 tells us that iteration of the affine function will converge
to the fixed point if and only if its coefficient matrix, namely
g
0
(
u
?
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 Olver
 Algebra, Numerical Analysis, Equations, Fixed points, Jacobian matrix, Peter J. Olver

Click to edit the document details