Kantorovitch’s theorem

Kantorovitch’s theorem

Let

𝐚

0

be a point in

ℝ

n

,

U

an open neighborhood of

𝐚

0

in

ℝ

n

and

𝐟

:

U

→

ℝ

n

a

differentiable mapping

, with its

derivative

[

𝐃𝐟

⁢

(

𝐚

0

)

]

invertible

. Define

𝐡

0

=

-

[

𝐃𝐟

⁢

(

𝐚

0

)

]

-

1

⁢

𝐟

⁢

(

𝐚

0

)

,

𝐚

1

=

𝐚

0

+

𝐡

0

,

U

0

=

{

𝐱

|

|

𝐱

-

𝐚

1

|

≤

|

𝐡

0

|

}

.

If

U

0

⊂

U

and the derivative

[

𝐃𝐟

⁢

(

𝐱

)

]

satisfies

the

http://planetmath.org/node/765

Lipschitz condition

|

[

𝐃𝐟

⁢

(

𝐮

1

)

]

-

[

𝐃𝐟

⁢

(

𝐮

2

)

]

|

≤

M

⁢

|

𝐮

1

-

𝐮

2

|

for all points

𝐮

1

,

𝐮

2

∈

U

0

, and if the

inequality

|

𝐟

⁢

(

𝐚

𝟎

)

|

⁢

|

[

𝐃𝐟

⁢

(

𝐚

𝟎

)

]

-

1

|

2

⁢

M

≤

1

2

is satisfied, the equation

𝐟

⁢

(

𝐱

)

=

𝟎

has a unique
solution in

U

0

, and Newton’s method with initial guess

𝐚

0

converges

to it. If we replace

≤

with

<

, then it can be shown
that Newton’s method

http://planetmath.org/node/793

superconverges! If you want an
even stronger version, one can replace

|

…

|

with the norm

||

…

||

.

Logic behind the theorem:

Let’s look at the useful part of the

theorem

:

|

𝐟

⁢

(

𝐚

𝟎

)

|

⁢

|

[

𝐃𝐟

⁢

(

𝐚

𝟎

)

]

-

1

|

2

⁢

M

≤

1

2

.

It is a

product

of three distinct properties of your

function

such
that the product is less than or equal to a certain number, or
bound. If we call the product

R

, then it says that

𝐚

0

must
be within a ball of radius

R

. It also says that the solution

𝐱

is within this same ball. How was this ball defined?

The first term,

|

𝐟

⁢

(

𝐚

𝟎

)

|

, is a measure of how far the
function is from the domain; in the Cartesian plane, it would be how
far the function is from the x-axis. Of course, if we’re solving for

𝐟

⁢

(

𝐱

)

=

𝟎

, we want this value to be small, because it
means we’re closer to the axis. However a function can be annoyingly
close to the axis, and yet just happily curve away from the axis. Thus
we need more.

The second term,

|

[

𝐃𝐟

⁢

(

𝐚

𝟎

)

]

-

1

|

2

is a little more
difficult. This is obviously a measure of how fast the function is
changing with respect to the domain (x-axis in the plane). The larger
the derivative, the faster it’s approaching wherever it’s going
(hopefully the axis). Thus, we take the

inverse

of it, since we want
this product to be

less than

a number. Why it’s squared though,
is because it is the denominator where a product of two terms of like
units is the numerator. Thus to conserve units with the numerator, it
is multiplied by itself. Combined with the first term, this also seems
to be enough, but what if the derivative changes sharply, but it
changes the wrong way?

The third term is the Lipschitz ratio

M

. This measures sharp changes
in the

first derivative

, so we can be sure that if this is small, that
the function won’t try to curve away from our goal on us too sharply.

By the way, the number

1

2

is unitless, so all the units on
the left side cancel. Checking units is essential in applications,
such as physics and engineering, where Newton’s method is used.

Title

Kantorovitch’s theorem

Canonical name

KantorovitchsTheorem

Date of creation

2013-03-22 11:58:09

Last modified on

2013-03-22 11:58:09

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

25

Author

stevecheng (10074)

Entry type

Theorem

Classification

msc 49K10

Synonym

Kantorovitch inequality

Related topic

LipschitzCondition

Related topic

NewtonsMethod

Related topic

Superconvergence

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