Graphics Reference
In-Depth Information
F.1.2. Theorem.
Let A and B be m ¥ n matrices and let a Œ
R
.
(1) ||A|| ≥ 0 and ||A|| = 0 if and only if A is the zero matrix.
(2) ||aA|| = |a| ||A||.
(3) ||A + B|| £ ||A|| + ||B||.
(4) If m = n, then ||AB|| £ ||A|| ||B||.
Proof.
Straightforward.
Definition.
Let A be a nonsingular n ¥ n matrix. The product ||A|| ||A
-1
|| is called the
condition number
of A and is denoted by cond(A).
The condition number of a matrix plays an important role in numerical analysis
because it has a direct bearing on the accuracy of numerical solutions to linear
systems of the form
x
A =
b
. In this context, one usually says that the linear system or
matrix is
ill-conditioned
if cond(A) is “large.” Numerical solutions to ill-conditioned
systems are typically very inaccurate.
F.2
Approximation and Numerical Integration
The problem addressed in this section is how, given a function f : [a,b] Æ
R
, one can
best approximate the integral
b
=
Ú
I
f
(F.1)
a
numerically if no antiderivative of f is available. In no way does this section intend to
cover the subject of numerical integration. We simply want to explain the gist of one
important technique called Gaussian quadrature because it does come up in geomet-
ric modeling. For example, see Chapter 14 of [AgoM05]. For more details see
[ConD72] or [Hild87].
We assume that the reader is familiar with the trapezoidal and Simpson's rule
from calculus. These methods approximate the integral I in (F.1) by approximating
the function f by a straight line and parabola, respectively.
1
2
(
)
(
( )
+
()
)
Trapezoidal rule:
I
ª-
ba f a
f b
1
12
3
¢¢
()
(
)
with the error
E
=-
f
x
b
-
a
for some xŒ(a,b)
1
6
ab
+
Ê
Ë
Ê
Ë
ˆ
¯
ˆ
¯
(
)
( )
+
+
()
Simpson's rule:
I
ª-
ba
f a
4
f
fb
2
5
1
90
ba
-
Ê
Ë
ˆ
¯
()
4
()
with the error
E
=-
f
x
for some xŒ(a,b)
2
