Download 2.13: How to compute matrix norms

2.13: How to compute matrix norms
Matrix norms are computed by applying the following formulas:
1-norm (Th. 2.8): kAk1 = maxj=1:n
Pn
∞-norm (Th. 2.7): kAk1 = maxi=1:n
i=1 |aij |
Pn
j=1 |aij |
maximal column sum
maximal row sum
p
2-norm (Th. 2.9): kAk2 = maxi=1:n λi(ATA)
where λi(ATA) is the ith eigenvalue of ATA.
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2.14: Condition of a Problem
A mathematical problem can be viewed as a function mapping in-data to
out-data (solution):
f :D⊂V→W
Condition number is a measure for the
sensitivity of the out-data with respect to
perturbations in in-data.
Scalar case: y = f (x) and ŷ = f (x + δx):
1 00
ŷ − y = f (x + δx) − f (x) = f (x)δx + f (x + θδx)(δx)2
2!
0
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2.15: Condition of a Problem (Cont.)
Scalar case, relative:
ŷ − y
=
y
absolut (local) condition number
Condx = |f 0(x)|
0
xf (x)
f (x)
δx
+ O(δx2)
x
relative (local) condition number
0 xf (x) condx(f ) = f (x) C. Führer: FMN081-2005
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2.16: Condition of a Problem (Cont.)
In general
absolut (local) condition number
relative (local) condition number
Condx = kf 0(x)k
kxk kf 0(x)k
condx(f ) =
kf (x)k
and we get
krel.output errork ≤ condx(f )krel.input errork
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2.17: Examples
Example 1: (Summation)
x1
Problem: f : R2 −→ R1 with
7→ x1 + x2
x2
0
T
Jacobian:
f (x1, x2) = 1, 1
In 1-norm:
Condx1,x2 (f ) = 1
|x1| + |x2|
|x1 + x2|
Problem if two nearly identical numbers are subtracted.
condx1,x2 (f ) =
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2.18: Examples (Cont.)
Example 2: (Linear Systems)
Problem: f : R −→ R with b 7→ x = A−1b
0
Jacobian:
f b = A−1
n
In 1-norm:
n
Condb(f ) = kA−1k
kbkkA−1k kAkkxkkA−1k
−1
condb(f ) =
≤
=
kAkkA
k =: κ(A)
−1
kA bk
kxk
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2.19: Examples (Cont.)
The estimate is sharp, i.e. there is a worst case perturbation
Example:
A=
3
10
0
0
10−3
1
0
b=
δb =
0
10−5
(see the exercises of the current week and Exercise 2.14 in the book.)
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Unit 3: Systems of nonlinear functions
Example (p. 107)
F (x) =
1
2
2
f1(x1, x2)
x1 + x2 − 1
=
=0
f2(x1, x2)
5x21 + 21x22 − 9
f1
0.5
f2
p
p
T
ξ1 = (− 3/2, 1/2) ξ2 = ( 3/2, 1/2)T
p
p
T
ξ3 = (− 3/2, −1/2) ξ4 = ( 3/2, −1/2)T
0
-0.5
-1
-1
-0.5
0
0.5
1
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3.1 Fixed Point Iteration in Rn
Definition. [4.1] Let g : D ⊂ Rn → Rn, D closed (cf. p. 105), x0 ∈ D
and g(D) ⊂ D.
The iteration
xk+1 = g(xk )
is called fixed point iteration or simple iteration.
and similar to the scalar case we define
Definition. [4.2] Let g : D ⊂ Rn → Rn, D closed (cf. p. 105).
If there is a constant L < 1 such that
kg(x) − g(y)k ≤ Lkx − yk
then g is called contractive or contractive.
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3.2 Contractivity and norms
Contractivity depends on the choice of a norm.
Here a function, which is contractive in one norm, but not in another
g(x) =
3/4 1/3
x
0 3/4
It follows
kg(x) − g(y)k = kA(x − y)k ≤ kAkkx − yk
Thus L = kAk.
But kAk1 = kAk∞ =
13
12
and kAk2 = 0.9350.
g is contractive in the 2-norm and dissipative and the others.
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3.3 Contractivity and norms (Cont.)
We start a fixed point iteration in the last example with x = (1, 1)T and
get the following norms:
2
1
1.5
different norms
of fixed point iterates
2
1
inf
0.5
0
0 1 2 3 4 5 6 7 8 9 10
(see also ”equivalence of norms”)
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3.4 Fixed Point Theorem in Rn
Theorem. [4.1] Let g : D ⊂ Rn → Rn, D closed, g(D) ⊂ D.
If there is a norm such that g is contractive, then g has a unique fixed
point ξ ∈ D and the fixed point iteration converges.
Let J(x) be the Jacobian (functionalmatrix → flerdim) of g.
If kJ(ξ)k < 1 then fixed point iterations converges in a neighborhood of ξ.
(Th. 4.2)
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