Scientific Researchof the Instituteof Mathematicsand Computer Science
THE POLYNOMIAL INTERPOLATION FOR TECHNICAL EXPERIMENTS
Grzegorz Biernat, Anita Ciekot
Institute of Mathematics and Computer Science, Czestochowa University of Technology, Poland email: gbiernat@imi.pcz.pl
Abstract. In the article there are the interpolation of the empirical processes by polynomials of several variables.
Introduction
The interpolation formulas by polynomials of several variables are the un- known in the interpolation methods ([1]). In the empirical processes the measure- ment steps are constant. Using the Kronecker tensor product of matrices [2, 3]
the interpolation formula was given in the theorem 1.
1. Kronecker’s tensor product
Let A
1( )n1⊗ A
2( )n2⊗ ... ⊗ A
k( )nkbe the Kronecker tensor product of matrices
( ) ( ) ( )nk
k n
n
A A
A
11,
22,..., of degrees respectively n
1, n
2,..., n
k.
Proposition 1. The determinant of the Kronecker tensor product is given by the formula
( ) ( ) ( )
( A1n1 A
2n2 ... A
knk ) ( det A1( )n1 )
nˆ1n2...nk( det A2( )n2 )
n1nˆ2...nk... ( det A
k( )nn )
n1n2...nˆk
)
nˆ1n2...nk( det A2( )n2 )
n1nˆ2...nk... ( det A
k( )nn )
n1n2...nˆk
det ⊗ ⊗ ⊗ =
where the symbol ^ omit above given numbers.
Proof. This follows from the definition of the Kronecker tensor product by induc-
tion on degrees n
1, n
2,..., n
k[2, 3].
G. Biernat, A. Ciekot 20
Let
( )
( ) ( )
( ) ( ( ) ) ( ( ) )
( ) ( )
∆ +
∆ +
∆ +
∆
− +
∆
− +
∆
− +
∆ +
∆ +
∆ +
=
−
−
−
−
i i
i i
i i
ii i
i
i
p i i i p
i i i i
i i
p i i i p
i i i i
i i
p i i p
i i i
i
p i p
i i
p i
p X p
X p
X
p X p
X p
X
X X
X
X X
X
X
0 1
0 0
0 1 0
0
0 1
0 0
0 1
0 0
1
1 1
1 1
1 1
L L
M M
M M
L L
be a matrix of degree p
i+ 1 where ∆
i> 0 . Proposition 2. We have
( )
2
13
2... ( 1 )
2 ( 21)det
i i i
i i
p p i i i
p p p
i
p p
X
− +
−
− ∆
=
Proof. It result immediately of the Vandermonde determinant [2, 3].
From both above propositions we obtain
Proposition 3. The next tensor determinant is given by the fornula
( ) ( ) ( )
( )
( )
( ) ( ) ( ) ( )∏
=+ + + +
−
−
− ∆
=
=
⊗
⊗
⊗
k
i
p p p p
p i i i
p p p k p
p
k i i
i i
i k
p p
X X
X
1
1 ...
1 ˆ ...
1 2
1 2
2 1 2
1
1 2
1
1 ...
3 2 ...
det
2. Polynomial interpolation
Consider the empirical process dependant on finite quantity of parameters with given initial conditions and measurement steps and know results. Thus let parameters X
1, X
2,..., X
ktake initial positive values X
01, X
02,..., X
0kand the values X
01+ I
1∆
1, X
02+ I
2∆
2,..., X
0k+ I
k∆
kwith fixed positive steps
∆
k∆
∆
1,
2,..., , where 0 ≤ I
1≤ p
1, 0 ≤ I
2≤ p
2,..., 0 ≤ I
k≤ p
k. The results W
I1I2...Ikat all steps X
01+ I
1∆
1, X
02+ I
2∆
2,..., X
0k+ I
k∆
kare know. So and the problem of polynomial interpolation follows to the question about to determine of coeffi- cients of the polynomial
( ) ∑
≤
≤
≤
≤
≤
≤
=
k k
k k
p i
p i
p i
i k i i i i i
k
a X X X
X X X W
0 0 0
2 1 ...
2 1
2 2
1 1
2 1 2
1
...
,..., ,
M
The polynomial interpolation for technical experiments 21
for it
( ) ( ) ( )
kk k
k
k II I
p i
p i
p i
i k k k i i
i i
i
X I X I X I W
a
...0 0 0
0 2
2 02 1 1 01
... 12
2 2
1 1
2 1
2
1
+ ∆ + ∆ ... + ∆ =
∑
≤
≤
≤
≤
≤
≤ M
(*)
where 0 ≤ I
1≤ p
1, 0 ≤ I
2≤ p
2,..., 0 ≤ I
k≤ p
k. Let A
ii ...ik2
1
be the i
1i
2... i
k-th algebraic replacement of the matrix
( ) ( ) ( )pk
k p
p
X X
X
1 1⊗
2 2⊗ ... ⊗ of linear sysytems (*) by a column vector
Ik
I
W
I ...2
1
.
Theorem 1. The linear systems of equations (*) has the unique solution and
( ) ( ) ( )
(
k k)
k p
k p
p
i i i i
i
i
X X X
a A
⊗
⊗
= ⊗
...
det
det
2 1
2 1 2
1
2 1
...
...
for 0 ≤ i
1≤ p
1, 0 ≤ i
2≤ p
2,..., 0 ≤ i
k≤ p
k.
Proof. Take the order corresponding to the tensor product
k k
k p p pp p
p
a a
a a
a
00...0 00...1 00... 0 ... ...2 1 2
,..., ,...,
,..., ,
Then the determinant of the system (1) is equal with it at the proposition 3 and the Cramer’s formulas are used.
3. Example
In the standard process of the nitriding under glow discharge ([4]) the parame- ters were as follow: temperature T
0= 793 K, T
1= 823 K, T
2= 853 K, pressure
Pa, 150
p
0= p
1= 300 Pa and time of the treatment t
0= 5 h, t
1= 10 h, t
2= 15 h.
After nitriding surface layers were characterized by surface microhardness meas- urements
k j ipt
H
Tfor 0 ≤ i ≤ 2 , 0 ≤ j ≤ 1 and 0 ≤ k ≤ 2 .
So which the steps ∆T = 30 by temperature, ∆p = 150 by pressure and ∆t = 5 by time we obtain the next polynomial inerpolation formula for the surface micro- hardness
( ) ∑ ( )
≤
≤ ≤
≤≤
∆
≤∆
= ∆
2 0
1 0
2 0 18 9
18
det
4 , 1 ,
k j i
k j i
ijk
T p t
t A p t T
p T H
where A denote the ijk -th algebraic replacement of the matrix
ijkG. Biernat, A. Ciekot 22
⊗
⊗
2 2 2
2 1 1
2 0 0
1 0 2
2 2
2 1 1
2 0 0
1 1 1 1
1 1
1 1
t t
t t
t t p
p T
T T T
T T
by a column vector
k j ipt
H
T. (In this case the linear system (1) has the ( 18 × 18 )
type). After the normalization
10 T = ∆ T
∆ and
10 p = ∆ p
∆ we obtain
( ) ∑ ( )
≤
≤ ≤
≤≤
⋅
≤=
2 0
1 0
2 0 18 9
18
det
5 19 3 4 , 1 ,
k j i
k j i
ijk
T p t
A t
p T H
where A denote the ijk-th algebraic replacement of the matrix
ijk
⊗
⊗
2 2 2
2 2 2
15 15 1
10 10 1
5 5 1 30 1
15 1 58 58 1
55 55 1
52 52 1
by a column wector H
Tipjtk.
References
[1] Kincaid D., Cheney 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 Matrices, Acad. Press 1969.
[4] Grant Nr 3T08C06726, Modyfikacja powierzchni tytanu i jego stopów drogą azotowania w wyładowaniu jarzeniowym w celu poprawy własności eksploatacyjnych warstwy wierzchniej.
