lecture notes on polynomial interpolation

lecture notes on polynomial interpolation

1

Summary of notation and terminonlogy

•

multi-evaluation mapping

:

ev

𝐱

:

𝒫

m

→

ℝ

n

,

𝐱

∈

ℝ

n

. Here

𝒫

m

is the

vector space

of polynomials of
degree

m

or less.

•

interpolation mapping

:

pol

𝐱

:

ℝ

n

→

𝒫

n

-

1

,

𝐱

∈

ℝ

n

where

x

1

,

…

,

x

n

are
distinct;

•

Vandermonde matrix

and polynomial

;

•

overdetermined

and

underdetermined

linear system

;

•

overdetermined and underdetermined

interpolation

problem

2

The polynomial interpolation problem

Definition 1

.

Let

𝐱

∈

ℝ

n

be given. Define

ev

𝐱

:

𝒫

m

→

ℝ

n

, the multi-evaluation mapping, to be the

linear
transformation

given by

ev

𝐱

:

p

→

[

p

⁢

(

x

1

)

⋮

p

⁢

(

x

n

)

]

,

p

∈

𝒫

m

.

Problem 2

(Polynomial interpolation).

Given

(

x

1

,

y

1

)

,

…

,

(

x

n

,

y

n

)

∈

ℝ

2

to find a polynomial

p

∈

𝒫

m

such that

p

⁢

(

x

i

)

=

y

i

,

i

=

1

,

…

,

n

. Equivalently, given

𝐱

,

𝐲

∈

ℝ

n

to find the

preimage

ev

𝐱

-

1

⁡

(

𝐲

)

.

Theorem 3

.

Fix

x

∈

R

n

. The multi-evaluation mapping

ev

x

:

P

n

-

1

→

R

n

is an

isomorphism

if and only
if the

components

x

1

,

…

,

x

n

are distinct.

3

Vandermonde matrix, polynomial, and determinant

Definition 4

.

For a given

𝐱

∈

ℝ

n

, the following matrix

𝖵𝖬

⁢

(

𝐱

)

=

𝖵𝖬

⁢

(

x

1

,

…

,

x

n

)

=

[

1

x

1

x

1

2

…

x

1

n

-

1

1

x

2

x

2

2

…

x

2

n

-

1

1

x

3

x

3

2

…

x

3

n

-

1

⋮

⋮

⋮

⋱

⋮

1

x

n

x

n

2

…

x

n

n

-

1

]

is called
the

Vandermonde matrix

. The
expression,

𝖵

⁢

(

𝐱

)

=

𝖵

⁢

(

x

1

,

…

,

x

n

)

=

∏

1

≤

i

<

j

≤

n

(

x

j

-

x

i

)

is called the Vandermonde polynomial. Note: we define

𝖵

⁢

(

x

1

)

=

1

; the usual convention is that the empty product is equal to

1

.

Proposition 5

.

The Vandermonde matrix is the transformation matrix of

ev

x

with the
monomial basis

[

1

,

x

,

…

,

x

n

]

as the input basis and the

standard basis

[

e

1

,

…

,

e

n

+

1

]

as the output basis.

Theorem 6

.

The Vandermonde polynomial gives the

determinant

of the Vandermonde
matrix:

|

1

x

1

…

x

1

n

-

1

1

x

2

…

x

2

n

-

1

⋮

⋮

⋱

⋮

1

x

n

…

x

n

n

-

1

|

=

∏

1

≤

i

<

j

≤

n

+

1

(

x

j

-

x

i

)

,

(1)

or more succinctly,

det

⁡

𝖵𝖬

⁢

(

𝐱

)

=

𝖵

⁢

(

𝐱

)

.

Proof.

We will prove

formula

(

1

) by

induction

on

n

. Note
that

𝖵𝖬

⁢

(

x

1

)

=

[

1

]

,

𝖵

⁢

(

x

1

)

=

1

.

Evidently then, the
formula works for

n

=

1

. Next, suppose that we believe the formula
for a given

n

. We show that the formula is valid for

n

+

1

. For

x

1

,

…

,

x

n

∈

ℝ

and a variable

x

, and consider the

n

th

degree polynomial

p

⁢

(

x

)

=

det

⁡

𝖵𝖬

⁢

(

x

1

,

…

,

x

n

,

x

)

=

|

1

x

1

…

x

1

n

-

1

x

1

n

1

x

2

…

x

2

n

-

1

x

2

n

⋮

⋮

⋱

⋮

⋮

1

x

n

…

x

n

n

-

1

x

n

n

1

x

…

x

n

-

1

x

n

|

By the properties of determinants,

x

1

,

…

,

x

n

are roots of

p

⁢

(

x

)

. Taking the

cofactor expansion

along the bottom row, we
see that the coefficient of

x

n

is

𝖵

⁢

(

x

1

,

…

,

x

n

)

.
Therefore,

p

⁢

(

x

)

=

𝖵

⁢

(

x

1

,

…

,

x

n

)

⁢

(

x

-

x

1

)

⁢

⋯

⁢

(

x

-

x

n

)

=

𝖵

⁢

(

x

1

,

…

,

x

n

,

x

)

,

as was to be shown.
∎

4

Lagrange interpolation formula

Let

x

1

,

…

,

x

n

∈

ℝ

be distinct. We know that

ev

𝐱

:

𝒫

n

-

1

→

ℝ

n

is

invertible

. Let

pol

𝐱

:

ℝ

n

→

𝒫

n

-

1

denote the

inverse

. In principle,
this inverse is described by the inverse of the Vandermonde matrix.
Is there another way to solve the interpolation problem? For

i

=

1

,

…

,

n

let us define the polynomial

p

i

⁢

(

x

)

=

∏

j

≠

i

x

-

x

j

x

i

-

x

j

.

These

n

-

1

degree polynomials have been “engineered” so that

p

i

⁢

(

x

i

)

=

1

and
so that

p

i

⁢

(

x

j

)

=

0

for

i

≠

j

Theorem 7

(Lagrange interpolation formula).

Let

x

1

,

…

,

x

n

∈

R

be distinct. Then,

pol

𝐱

⁡

(

𝐲

)

=

y

1

⁢

p

1

+

⋯

+

y

n

⁢

p

n

.

5

Underdetermined and overdetermined interpolation

Definition 8

.

Let

T

:

𝒰

→

𝒱

be a linear transformation of finite dimensional
vector spaces. A linear problem is an equation of the form

T

⁢

(

𝐮

)

=

𝐯

,

where

𝐯

∈

𝒱

is given, and

𝐮

∈

𝒰

is
the unknown. To be more precise, the problem is to determine
the preimage

T

-

1

⁢

(

𝐯

)

for a given

𝐯

∈

𝒱

. We say that the
problem is overdetermined if

dim

⁡

𝒱

>

dim

⁡

𝒰

, i.e. if there
are more equations than unknowns. The linear problem is said to be
underdetermined if

dim

⁡

𝒱

<

dim

⁡

𝒰

, i.e., if there are more
variables than equations.

Remark 9

.

By the

rank-nullity theorem

, an
underdetermined linear problem will either be

inconsistent

, or will
have multiple solutions. Thus, an underdetermined system arises when

T

is not

one-to-one

; i.e., the kernel

ker

⁡

(

T

)

is non-trivial. An
overdetermined linear system is inconsistent, unless

𝐯

satisfies

a
number linear compatibility constraints, equations that describe the image

Im

⁡

(

T

)

. To put it another way, an overdetermined system arises
when

T

is not onto.

Definition 10

.

Let

x

1

,

…

,

x

n

∈

ℝ

be distinct and let

ev

𝐱

:

𝒫

m

→

ℝ

n

be the corresponding multi-evaluation
mapping. Let

𝐲

∈

ℝ

n

be given. The linear equation

ev

𝐱

⁡

(

p

)

=

𝐲

,

p

∈

𝒫

m

is called an underdetermined interpolation problem if

m

≥

n

.
If

m

≤

n

-

2

, we call the above equation an overdetermined
interpolation problem. Note that if

m

=

n

-

1

, the interpolation
problem is “just right”; there is exactly one solution, namely the
polynomial

pol

𝐱

⁡

(

𝐲

)

given by the

Lagrange interpolation formula

.

Proposition 11

(Underdetermined interpolation).

Let

x

1

,

…

,

x

n

∈

R

be distinct. Define

q

⁢

(

x

)

=

(

x

-

x

1

)

⁢

⋯

⁢

(

x

-

x

n

)

to be the

n

𝑡ℎ

degree polynomial with the

x

i

as its roots.
Suppose that

m

≥

n

. Then,
the multi-evaluation mapping

ev

x

:

P

m

→

R

n

is not one-to-one. A basis for the
kernel is given by

q

⁢

(

x

)

,

x

⁢

q

⁢

(

x

)

,

…

,

x

m

-

n

⁢

q

⁢

(

x

)

. Let

y

1

,

…

,

y

n

∈

R

be given. The solution
set to the interpolation problem

ev

x

⁡

(

p

)

=

y

, is given by

ev

𝐱

-

1

⁡

(

𝐲

)

=

{

r

+

s

⁢

q

:

s

∈

𝒫

m

-

n

}

,

where

r

=

pol

x

⁡

(

y

)

is the

n

-

1

𝑠𝑡

degree polynomial given by the Lagrange
interpolation formula. To put it another way, the general solution
of the equation

ev

𝐱

⁡

(

p

)

=

𝐲

is given by

p

=

r

+

s

⁢

q

,

s

∈

𝒫

n

-

m

⁢

free.

Proposition 12

(Overdetermined interpolation).

Let

x

1

,

…

,

x

n

∈

R

be distinct.
Suppose that

m

≤

n

-

2

. Then,
the multi-evaluation mapping

ev

x

:

P

m

→

R

n

is not onto. If

m

=

n

-

2

, then

y

∈

R

4

belongs to

Im

⁡

(

ev

x

)

if and only if

|

1

x

1

…

x

1

m

y

1

1

x

2

…

x

2

m

y

2

⋮

⋮

⋱

⋮

⋮

1

x

n

-

1

…

x

n

-

1

m

y

n

-

1

1

x

n

…

x

n

m

y

n

|

=

0

More generally, for any

m

≤

n

-

2

, the interpolation problem

ev

x

⁡

(

p

)

=

y

,

y

∈

R

n

has a solution if and only
if

rank

⁡

M

⁢

(

x

,

y

)

=

m

+

1

where

M

⁢

(

𝐱

,

𝐲

)

=

[

1

x

1

…

x

1

m

y

1

1

x

2

…

x

2

m

y

2

⋮

⋮

⋱

⋮

⋮

1

x

m

+

1

…

x

m

+

1

m

y

m

+

1

⋮

⋮

⋮

⋮

⋮

1

x

n

…

x

n

m

y

n

]

Remark 13

.

The compatibility constraints on

y

1

,

…

,

y

n

for an overdetermined system amount to the condition
that

𝐲

lie in the image of

ev

𝐱

, or what is

equivalent

,
belong to the

column space

of the corresponding transformation matrix.
We can therefore obtain the constraint equations by row reducing the
augmented matrix

M

⁢

(

𝐱

,

𝐲

)

to

echelon form

. The back entries of
the last

n

-

m

-

1

rows will hold the constraint equations.
Equivalently, we can solve the interpolation problem for

(

x

1

,

y

1

)

,

…

,

(

x

m

,

y

m

)

to obtain a

p

=

y

1

⁢

p

1

+

y

m

+

1

⁢

p

m

+

1

∈

𝒫

m

. The additional equations

y

i

=

y

1

⁢

p

1

⁢

(

x

i

)

+

⋯

+

y

m

+

1

⁢

p

m

+

1

⁢

(

x

i

)

,

i

=

m

+

2

,

…

,

n

are the desired compatibility constraints on

y

1

,

…

,

y

n

.

Title

lecture notes on polynomial interpolation

Canonical name

LectureNotesOnPolynomialInterpolation

Date of creation

2013-03-22 16:51:59

Last modified on

2013-03-22 16:51:59

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

7

Author

rmilson (146)

Entry type

Topic

Classification

msc 41A05

Related topic

EvaluationHomomorphism

Related topic

LagrangeInterpolationFormula

Related topic

VanderMondeMatrix

Defines

multi-evaluation mapping

Defines

interpolation mapping

Defines

polynomial interpolation

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