The polynomial tensor interpolation

THE POLYNOMIAL TENSOR INTERPOLATION

Grzegorz Biernat, Anita Ciekot

Institute of Mathematics, Czestochowa University of Technology, Poland, ciekot@imi.pcz.pl

Abstract. In this paper the tensor interpolation by polynomials of several variables is con- sidered.

Introduction

The formulas of tensor interpolation by polynomials of several variables are the unknow in the interpolation methods ([1]). Using the Kronecker tensor product of matrices ([2, 3]) the polynomial tensor interpolation formula was given in tis arti- cle.

1. The Kronecker product of matrices

The Kronecker product of two matrices A=a_ij and B=   of degrees re-b^ij spectivly m and n we define as a matrix given in block form as:

11 12 1

21 22 2

1 2

[ ]

m

m ij

m m mm

a B a B a B

a B a B a B

A B a B

a B a B a B

 

 

 

⊗ = = 

 

 

K K

M M O M

K

we denote the Kronecker product of A and B by A⊗B. The Kronecker product is also known as the direct product or the tensor product.

Some properties for Kronecker products of two matrices:

1. The matrices A⊗Band B⊗Aare orthogonaly similar, which means that square matrix U exists and ^B⊗^A=^Ut

(

^A⊗^B

)

^U, U^tU =I.

2. If A,B,C,D are square matrices such that the products AC and BD exist, then

C D

Α Β

( ⊗ )( ⊗ ) exist and

(A⊗B C)( ⊗D)=AC⊗BD (the “Mixed Product Rule”)

3.

( ) ( )

α^{A ⊗} β^{B =}αβ

(

^A⊗B

)

4.

(

^A⊗B

)

^{t =At ⊗Bt}

5. If A and B are invertible matrices, then

(

^A⊗B

)

^{−1=A−1⊗B−1}

6. ^det

(

^A⊗B

) (

^{= detA}

) (

^{n detB}

)

^m

7. the

(

a⊗b

)

rs element of the matrix A⊗B is given by the product

(

a⊗b

)

rs =aijbkl

where ^r=

( )

^{i−1n+k , s=}

(

^j−1

)

^n+l.

Next, we consider the quadratic matrices

1 1 i j1 1

A = ( ) a  ,...,Ak = ( ) ak i jk k of de- grees respectivly n ,..,n and define the tensor product inductively: _{1 k}

(

k

)

k

k A A A A

A A

A1⊗ 2 ⊗...⊗ = 1⊗ 2 ⊗...⊗ ₋1 ⊗ for k≥3.

Some properties for Kronecker products:

1. The matrices A_{σ( )1} ⊗A_{σ( )2} ⊗...⊗A_{σ( )k} and A₁⊗A₂ ⊗...⊗A_k, where σ ^is any determine permutation of numbers 1,...,k, are orthogonaly similar. It means that square matrix U _σ exists and A_{σ( )1} ⊗A_{σ( )2} ⊗...⊗A_{σ( )k} =

( )

σ

σ A A ... A U

U^t ⊗ ⊗ ⊗ _k

= 1 2 , U_σ^tU_σ =I.

2.

(

A₁⊗A₂ ⊗...⊗Ak

)(

B₁⊗B₂ ⊗...⊗Bk

) (

= A₁B₁

) (

⊗ A₂B₂

)

⊗...⊗

(

AkBk

)

pro- vided the dimensions of the matrices are such that the various expressions exist (the “Mixed Product Rule”).

3.

(

α₁A₁

) (

⊗ α₂A₂

)

⊗...⊗

(

αkAk

)

=α₁α₂...αk

(

A₁⊗A₂ ⊗...⊗Ak

)

4.

( )

k^t

t t t

k A A A

A A

A₁⊗ ₂ ⊗...⊗ = ₁ ⊗ ₂ ⊗...⊗ 5. if A ,..., A are invertible matrices, then: _{1 k}

( )

2^{1 1}

1 1 1 2

1⊗A ⊗...⊗Ak ⁻ =A⁻ ⊗A⁻ ⊗...⊗Ak⁻

A

6. ^det

(

^{A1 ⊗A2 ⊗...⊗Ak}

) (

^{= detA1}

)

^nˆ1n2...nk

(

^detA2

)

^{n1nˆ2...nk...}

(

^detAk

)

^n1n2...nˆk

where ˆn is omission. _j

7. the

(

a₁⊗a₂ ⊗...⊗ak

)

ij element of the matrix A₁⊗A₂ ⊗...⊗A_k is given by the product

(

a a ... ak

) ( ) ( )

ij a ij a i j ...

( )

ak i_kj_k

2 2 1

1 2

1 2

1⊗ ⊗ ⊗ = , where:

(

i

)

nn nk

(

i

)

nn nk

(

ik

)

nk ik

i= ₁−1 ˆ_{1 2}... + ₂−1 ˆ₁ˆ₂... +...+ ₋₁−1 +

2. One of the property for Vandermonde’s matrix Consider the Vandermonde’s matrix:

( )





















=

= ₊

+

p p p

p p

p p

p

X X

X X

X X

X X X X V V

K M O M M

K K

1 1 1 ,...,

,

, ^{1 1}

0 0

2 1 0 1 1

of degree p+1.

The algebraic complement of the matrix V_p+1, has the form

( )

^{i j p j}

(

^{i p}

)

^p

(

^{i p}

)

ij X X X V X X X

D = −1 ⁺ τ _{− 0},..., ˆ ,..., det ₀,..., ˆ ,...,

where τ^p−^j

(

^{X0,...,Xˆi,...,Xp}

)

design the fundamental symmetric τ_{p j}− polynomial of the rank p− j of variables X ,..., X ,..., X , and the symbol ₀ ˆ_{i p} X means omitting ˆ_i the variable X (_i τ₀ =0). Similarly for Vandermonde’s determinant detV . _p

This property we easly obtain on the one hand by evolving determinant

( )

∏

≤ <≤

+ = −





















= _{k l p} l k

p p p

p p

p X X

X X

X X

X X

V 0

1 1

0 0

1

1 1 1 det det

K M O M M

K K

according to i-th row, and on the other hand by sorting its value according toX _i variable power.

In particular

( ) ( )

i

p i j

p j i

p

ij X X X

V D

Π

= − ⁻

+ +

,..., ,..., ˆ 1

det

0

1

τ where

( ) ( ) (

1

)(

1

) (

0

)

0 X X X_p X_i ... X_i X_i X_i X_i ... X_i X

i l i k

p l

k l k

i = − = − − − −

Π _{+ −}

≠

≠<≤

∏

≤

3. The polynomial tensor interpolation The coefficients matrix

[

aj_...jk

]

1 of the polynomial interpolation

( )

⁼

∑

_{≤ ≤ ≤ ≤}

k k

k

p k

j p j

j k j j j

k a X X

X X

W 1 0 ,...,0 ... 1

1 1

1

1 ...

,...,

or

are unknow. The results matrix

[ ( ) ( )

^{w1 i1...wk ik}

]

⁼

^{[ ]}

^wi1...ik and the nodes matrix

( ) ( )

[

^{X1 i1×...× Xk ik}

]

⁼

^[

^X1i1...Xkik

^]

are know. The coefficients of the polynomial are determined from the linear system

[ ] [ ]

( ) [ ] [ ]

k k

k

k j j i i

j i j

i X veca vecw

X _{... ...}

1 1

1

1 ⊗...⊗ =

where the nodes matrix is the Kronecker product of Vandermonde’s matrices

[ ]

k

[ ]

^kk

j p ki

j

p Xi V X

V ₊1= 1¹ ,..., ₊1=

1 1 , and the operator “vec” put coefficient matrix

[

^aj1...jk

]

and results matrix

[ ]

^wi1...ik in columns, attributing multiindexes j ...₁ j_k and ik

i ...₁ properly positions

( )( ) ( ) ( )( ) ( )

( )

1 1 2 2 1 2

1

1 1 1 1 1 1

1 1

k k

k k k

ˆ ˆ ˆ

j j p p ... p j p p ... p

... j₋ p j

= + + + + + + + +

+ + + +

and

( )( ) ( ) ( )( ) ( )

( )

1 1 2 2 1 2

1

1 1 1 1 1 1

1 1

k k

k k k

ˆ ˆ ˆ

i i p p ... p i p p ... p

... i₋ p i

= + + + + + + + +

+ + + +

of the

jk

aj_...

1 element in coefficients column and

ik

wi_...

1 element in results column.

It means that we select ordering

1

2 1 1 1

00 00 00 01 00 0 0 1 00

0

k k k k

k k k k

... , ... , ..., ... p , ... p , ..., ...p p , ..., p ...p p , ..., p ...p p

−

− −

with sequence shown above.

Then the searching coefficients column has a form

[ ]

^{aj jk}

[ ]

^Xji

[ ]

^Xkijkk

⁽

^{DV DVk}

⁾

^tvec

^{[ ]}

^{wi ik}

vec _...

1

... _{1 1}

1 1

1 ...

det ... 1 det

1 ⊗ ⊗

=

where:

[ ]

k

[ ]

^kk

j p ki

k j

p X i V V X

V

V1= ₊1= 1¹ ,..., = ₊1=

1

1 .

According to property 7’ we obtain the formula for coefficients

( ) ( )

k j V i j V i p

i p

i i i

j

j V

D V D w

a ^{k k k}

k

k k

k det

...

det ₁

0 ,...,

0 ...

...

1 1 1

1

1 1

1 ⁼

∑

_{≤ ≤ ≤ ≤}

and because fractions shown above has got known form then:

( ) ( )

( )

1 1 1 1

1 1 1 1

1

10 1 1

0 0

1

0

1

k k k k

k k k k

k

p j i p

I J

j ... j i p ,..., i p i ...i

i

p j k ki kp

ki

X ,..., X ,..., Xˆ

a w

X ,..., Xˆ ,..., X ...

++ + −

≤ ≤ ≤ ≤

−

= − τ

Π τ

Π

∑

where: I⁺ =i₁+...+i_k, J⁺ = j₁ +...+ j_k and

(

^{1 1}

) (

^{1 1 1}

)(

^{1 1 1}

) (

^{1 10}

)

1_i1= X _p1−X_i1... X _i1 −X _i1 X_i1−X _i1 ... X_i1−X

Π _{+ −}

(

^{kp ki}

) (

^{... ki 1 ki}

)(

^{ki ki 1}

) (

^{... ki k0}

)

ki X X X X X X X X

k k

k k k

k k

k = − − − −

Π _{+ −}

And now the polynomial coefficient we can obtain numerically.

References

[1] Kincaid D., Chnej W., Numerical Analysis, Mathematics of Scientific Computing, The Univer- sity of Texas at Austin, 2002.

[2] Gantmacher F.R., The theory of matrices, Vol. 1, 2, Chelsea 1974.

[3] Lancaster P., Theory of Matries, Acad. Press 1969.

[4] Biernat G., Ciekot A., The polynomial interpolation for technical experiments, Scientific Re- search of the Institute of Mathematics and Computer Science, Czestochowa University of Tech- nology 2007, 1(6), 19-22.