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ACTA UNIVERSITATIS APULENSIS
ORDERING OF SOME SUBSETS FROM IR d USED IN THE
MULTIDIMENSIONAL INTERPOLATION THEORY
by
Crainic Nicolae
Abstract: The ordering of the elements of a subset of IN d , d ≥ 1 , with the order
relations defined at the beginning of this article and implicitly obtaining some expressions for
them, corresponding to the defined order relations, is useful for the multidimensional
interpolation theory, especially when it is necessary to exemplify some general theoretical
notions. This is the reason for presenting expressions for ordered subsets from IN 2 , IN 3 and
in general from IN d .
Next we present the most usual order relations encountered in the
interpolation theory and used throughout this article.
1. We say that i = (i1 , i2 ,..., id ) ∈ IN d
is in the relation “«” with
j = ( j1 , j 2 ,..., j d )∈ IN d and we write i « j , when i1 < j1 , or i1 = j1 and i2 < j2 ,…,
or i1 = j1 ,…, id −1 = j d −1 and id ≤ j d .
2. We say that i = (i1 , i2 ,..., id )∈ IN d
is in the relation "p" with
j = ( j1 , j 2 ,..., j d )∈ IN d and we write i p j , when i < j or i = j and i « j .
Let be the set S = {i = (i1 , i2 ) / 0 ≤ i1 ≤ n,0 ≤ i2 ≤ m j , j = 0, n) } from IN 2 ,
denoted by S n;m0 ,...,mn , where n and 0 ≤ m0 ≤ m1 … ≤ mn are given natural numbers.
It is obvious that we would rather write out the set ( S n;m0 ,...,mn , «) than the set
( S n;m0 ,...,mn , p) whose last terms are difficult to be written. Thus we have:
n
mj
n
( S n;m0 ,...,mn , «)= UU{( j , k )} = U {( j ,0), ( j ,1),..., ( j , m j )} =
j =0 k =0
j =0
= {(0,0), (0,1),..., (0, m0 ), (1,0), (1,1),..., (1, m1 ),..., (n,0), (n,1),..., (n, m n )}
(1)
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
If in the set S n;m0 ,...,mn we take m0 = n , m j = n − j ,
j = 1, n , and
respectively m0 = m1 = ... = mn = n 2 , n = n1 , we obtain some of the most usual
forms of some subsets from S ⊂ IN 2 , that is the triangle Tn2 , respectively the
rectangle Rn2 , where n = (n1 , n 2 )∈ IN 02 . More concrete, these sets are
{
}
Tn2 = i ∈ IN 02 / i ≤ n ,
{
}
Rn2 = Rn21 ,n2 = i ∈ IN 02 / 0 ≤ i j ≤ n j , j = 1,2
and they are depicted in Figure 1 a and 1 b.
n2
n
n-1
n-2
2
1
1
0
0
0
1
2
n-1
n
0
(a)
1
2
n1
(b)
Figure 1
It is quite obvious that the grid of points from Figure 1 a is made out
of the parallel lines
∆2k = U{( j , k − j )}, k = 0, n .
k
j =0
It follows the next explicit expression of triangle Tn2 ordered by relation
" p"
(Tn2 , p) = UU{( j , k − j )} = U{(0, k ), (1, k − 1),..., (k ,0)}
n
k
k =0 j =0
n
k =0
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
= {(0,0), (0,1), (1,0),..., (0, k ), (1, k − 1),..., ( k ,0),..., (0, n), (1, n − 1),..., (n,0)} .
(2)
Moreover, if we consider the relation (1) and the fact that Tn2 = S n;n , n −1,..., 0 ,
we have:
n n− j
n
j =0 k =0
j =0
( Tn2 ,«)= UU {( j , k )} = U {( j ,0), ( j ,1),..., ( j , n − j )} =
= {(0,0), (0,1),..., (0, n), (1,0), (1,1),..., (1, n − 1),..., (n − 1,0), (n − 1,1),..., (n,0)} .
(3)
From the same relation (1) and the fact that Tn2 = S n1 ;n2 ,n2 ,...,n2 , we obtain
n1
n1
j =0
j =0 k =0
n2
( Rn2 ,«)= U {( j ,0), ( j ,1),..., ( j , n 2 )} = UU {( j , k )} =
= {(0,0), (0,1),..., (0, n 2 ), (1,0), (1,1),..., (1, n 2 ),..., ( n1 ,0), (n1 ,1),..., ( n1 , n 2 )} .
(4)
An explicit expression of ( Rn2 , p) is more difficult because it depends on
the position of the integers n1 and n2 with respect to each other. But for n1 = n 2 we
have:
( Rn2,n , p) = (Tn2 , p) U{(1, n), (2, n − 1),..., (n,1), (2, n), (3, n − 1),..., (n,2),..., (n, n)} .
We keep also in mind the following expressions of the ordered subsets from
2
IN :
U{(0, j ).(1, j − 1),..., ( j,0)} = U U{(i, j − i)} =
n
n
2
n
2
k
( T \ T , p )=
j = k +1
j
j = k +1 i = 0
{(0, k + 1), (1, k ),..., (k + 1,0), (0, k + 2), (1, k + 1),..., (k + 2,0),..., (0, n), (1, n − 1),..., (n,0)}
=
,
( Rn2,n \ Tn2 , p )= , {(1, n), (2, n − 1),..., (n,1)(2, n), (3, n − 1),..., (n,2),..., (n, n)} and
( Rn2,n \ Tn2 ,«)=
{(1, n), (2, n − 1), (2, n), (3, n − 2), (3, n − 1), (3, n),..., (n,0), (n,1),..., (n, n)} .
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
In general, extensions of the two dimensional case are used as examples in
IN , d ≥ 3 , among which we exemplify the simplex (Figure 2, in IN 03 )
{
d
}
Tnd = i ∈ IN 0d / i ≤ n ,
{
respectively the hyper-parallelipiped (Figure 4, in IN 03 )
}
Rnd = Rnd1 ,n2 ,...,nd = i ∈ IN 0d / 0 ≤ i j ≤ n j , j = 1, d .
S ⊂T .
d
n
Proposition: If S is an arbitrary set from IN d , and n = max i , then
i∈S
Proof: For any i ∈ S , with i ≤ max i = n , we have i ∈ Tnd .
i∈S
Next we will build successively an expression for the simplex (Tnd , p) .
Figure 2
The grid of points from the zero x-coordinate plane, Figure 2 (plane yOz), is
made out of the parallel lines
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
∆2k = U{( j , k − j )}, k = 0, n ,
k
j =0
their traversal order being ( ∆2k , p )=( ∆2k ,«), and of each of them, from left to right, in
descendant order. It follows that ∆2k given in the previous relation is ( ∆2k , p ). As
2
each element of the set ∆ k has the sum of indexes k -constant it follows that
( ∆2k , p )=( ∆2k ,«). From the same relation we have that
k −i
∆2k −i = U{( j , k − i − j )} , i = 0, k .
j =0
If we translate every line ∆2k −i , i = 1, k in the i = 1, k x-coordinate plane,
obviously parallel with the initial yOz plane, we obtain the ordered triangle ∆3k from
IN 03 , that is:
( ∆3k , p )= U ∆2i , k −i =
k
i =0
k k −i
UU{(i, j, k − i − j )} =
i =0 j =0
= U{(i,0, k − i ), (i,1, k − i − 1),..., (i,0, k − i ), (i, k − i,0)} =
k
i =0
= {(0,0, k ), (0,1, k − 1),..., (0, k ,0), (1,0, k − 1), (1,1, k − 2),..., (1, k − 1,0),..., (k ,0,0)} ,
depicted in Figure 2, whose vertices are the points of coordinates (k ,0,0), (0, k ,0) and
(0,0, k ) . To denote the plane of which ∆2k is a part of after translation, we added the i
index to the line ∆2k thus obtaining ∆2k −i . Actually i becomes the x-coordinate for all
points from the translated i x-coordinate plane. The traversal order of the lines ∆2i , k −i
is ∆20 ,k , ∆21,k −1 ,… , ∆2k , 0 . It follows that:
n
(Tn3 , p) = U
k =0
∆
3
k
k k −i
UUU{(i, j, k − i − j )}
n
= k =0 i =0
j =0
(5)
which is the same to (Tn3 , p) , taking into consideration the traversal order of the
triangles ∆3k , k = 0, n .
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
An explicit form of the ordered set (Tn3 , p) is:
(Tn3 , p) =
{(0,0,0), (0,0,1), (0,1,0), (1,0,0), (0,0,2), (0,1,1), (0,2,0), (1,0,1), (1,1,0), (2,0,0),...,
(0,0, n), (0,1, n − 1),..., (0, n,0), (1,0, n − 1), (1,1, n − 2),..., (1, n − 1,0),..., (n,0,0))} .
Proceeding similarly to the previous situation, we obtain
k −t
( ∆3k −t , p )= U ∆2i , k −t −i =
i =0
respectively
( ∆3t ,k −t , p )=
U U{(i, j, k − t − i − j )} ,
i =0 j =0
k −t k −t − i
U U{(t , i, j, k − t − i − j )} ,
i =0 j =0
from where we obtain that
k k −t k −t − i
∆ = U ∆3t ,k −t = UU
k
4
k
k −t k − t −i
t =0
U{(t , i, j, k − t − i − j )} ,
t =0 i =0 j =0
respectively
k k −t k −t −i
(Tn4 , p) = U ∆4k = UUU
n
n
k =0
U{(t , i, j, k − t − i − j )} .
k =0 t =0 i =0 j =0
Now the generalization to the d-dimensional space becomes obvious,
namely:
k k −i1 k − ( i1 + i2 )
∆dk = U U
U
i1 = 0 i2 = 0
i3 = 0
k − ( i1 +...+ id − 2 )
U{(i , i ,..., i
...
id −1 = 0
1
2
d −1
, k − (i1 + i2 + ... + id −1 )} ,
(6)
respectively
k k −i1 k − ( i1 + i2 )
(Tnd , p) = U ∆dk =UU U
n
k =0
n
k = 0 i1 = 0 i2 = 0
U
i3 = 0
k − ( i1 +...+ id − 2 )
...
U{(i , i ,..., i
id −1 = 0
1
2
d −1
, k − (i1 + i2 + ... + id −1 )}
(7)
If in addition we consider also the results of the previous proposition, then
we obtain the following theorem:
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
Theorem: An arbitrary set ( S , p) from IN d can be written in the form
( S , p) = U ∆'dkt ,
n
t =1
where k n = max i , ∆'dkt ⊂ ∆dk are given by (6).
(8)
i∈S
Example: S = {(0,0), (0,2), (1,1), (1,2)} = {(0,0)} ∪ {(0,2), (1,1)} ∪ {(1,2)} =
= ∆'2k1 ∪ ∆' 2k 2 ∪ ∆' 2k3 =
∆'2k3 = {(1,2)} ⊂ ∆23 .
U∆
3
t =1
'2
kt
,
where
∆'2k1 = ∆20 ,
∆'2k 2 = {(0,2), (1,1)} ⊂ ∆22 ,
and
In what follows, we will build in successive steps another expression for the
simplex Tnd using partially the relations " p" and “«”. Eventually we obtain a total
order relation defined on Tnd , which we denote by “∠”.
Figure 3
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
The grid of points from the zero x-coordinate plane, Figure 3 (plane yOz),
UU{( j, k − j )} , made out of the
n
according to relation (.3), is the triangle (Tn2 , p) =
parallel lines ∆2k =
k
k =0 j =0
U{( j, k − j )}, k = 0, n , their traversal order being
k
j =0
∆20 , ∆21 , …
, ∆ , and of each of them, from left to right, in descending order. It follows that
2
n
n −i k
(Tn2−i , p) = UU{( j , k − j )}, i = 0, n .
k =0 j =0
If we translate each triangle (Tn2−i , p) , with i = 1, n , in the i = 1, n xcoordinate plane, obviously parallel to the initial plane of the triangle, it results the
ordered tetrahedron
(Tn3 , ∠)= U Ti ,2n −i
n
i =0
from Figure 3., which is different both from (Tn3 , p) and ( Tn3 ,«). We added the
index i to the triangle Tn2−i to indicate the x-coordinate of the plane of which Tn2−i is
part of after translation. In fact i becomes x-coordinate for all points from the i xcoordinate translated plane. The traversal order of the triangles Ti ,2n −i is T02,n , T1,2n −1 , …
, Tn2, 0 . It follows that:
n n −i k
(T , ∠)= UUU{(i, j , k − j )} .
3
n
i =0 k =0 j =0
An explicit form of the ordered set (Tn3 , ∠) is:
(Tn3 , ∠)=
{(0,0,0), (0,0,1), (0,1,0), (0,0,2), (0,1,1), (0,2,0),..., (0,0, n), (0,1, n − 1),..., (0, n,0),
(1,0,0), (1,0,1), (1,1,0), (1,0,2), (1,1,1), (1,2,0),..., (1,0, n − 1), (1,1, n − 2),...,
..., (1, n − 1,0,..., (n,0,0))}
Continuing the used procedure and generalizing we can obtain rather easily that in ddimensional space
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
n n − i1 n − ( i1 + i 2 )
(Tnd , ∠)= U U
i1 = 0 i 2 = 0
U
i3 = 0
n − ( i1 + ...+ i d −3 ) n − ( i1 + ... + i d − 2 )
U
...
U
i d −2 = 0
k =0
k
U{(i , i ,..., i
i d −1 = 0
1
d −1
2
, k − id −1 )} .
(9)
An explicit expression for the simplex (Tnd , «) can be obtained starting from
n n− j
the triangle ( Tn2 ,«)= UU {( j , k )} given by relation (3), lying in the plane yOz from
j =0 k =0
Figure 3. The elements of the triangle ( Tn2 ,«) are the grid of points lying in the
mentioned plane, whose explicit form is obtained if we traverse Tn2 from bottom to top
and from left to right. We can also write that
n − i n −i − j
( Tn2−i ,«)= U
U{( j, k )} .
j =0 k =0
Translating each triangle (Tn2−i , p) , i = 1, n in the i = 1, n x-coordinate
plane, we obtain the ordered tetrahedron
n
(Tn3 , «)= U Ti ,2n −i .
i =0
The i index added to Tn2−i , indicates the plane of which Tn2−i is part of after
translation and becomes the x-coordinate for all points from this plane. The traversal
order of triangles Ti ,2n −i is also T02,n , T1,2n −1 , … , Tn2, 0 . It follows that:
n n − i n −i − j
(T , «)= UU
3
n
U{(i, j, k )}
i =0 j =0 k =0
Eventually, for the d -dimensional space we obtain that:
n n − i1 n − ( i1 + i 2 )
(Tnd , «)= U U
i1 = 0 i 2 = 0
U
i3 = 0
n − ( i1 + ... + i d − 2 ) n − ( i1 + ... + i d −1 )
...
U
i d −1 = 0
U{(i , i ,..., i
id = 0
1
2
d −1
, id )}
(10)
Next we build an expression for the hyper-parallelipiped Rnd from the d dimensional space using the total order relation “«”.
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
Figure 4
The grid of points from the zero x-coordinate plane, Figure 4 (plane yOz),
n1
n2
according to relation (4), is the rectangle ( Rn2 ,«)=( Rn21 , n2 ,«)= UU {( j , k )} , its
j =0 k =0
rectangle Rn2 by successive translations, in the i = 1, n 3 x-coordinate planes,
obviously parallel to the initial plane of the rectangle, we obtain the ordered
parallelepiped
traversal order being from bottom to top and from left to right. If we “multiply” the
n3
( Rn3 ,«)= U Ri2,( n1 ,n2 )
i =0
from Figure 4. The i index added to Rn2 indicates the plane of which this is a part of
after translation and becomes x-coordinate for all points from the i x-coordinate
plane. The traversal order of the rectangles Rn2 is R02,( n1 ,n2 ) , R12,( n1 , n2 ) , …, Rn23 ,( n1 , n2 ) . It
follows that:
n3
( Rn33 , n1 , n 2 ,«)=
n1 n 2
UUU{(i, j, k )} .
i =0 j =0 k =0
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
Generalizing to the d -dimensional space we obtain:
n1
( Rnd1 , n 2 ,..., n d ,«)=
n d −1
n2
U U... U
i1 = 0 i 2 = 0
nd
U{(i , i ,..., i )}
i d −1 = 0 i d = 0
1
2
d
(11)
Remark: An expression for ( Rnd1 , n2 ,...,nd , p) can be obtained applying the relation (8)
from the previous theorem.
As we pointed out in the beginning, the notions presented in this article are
useful in the field of multidimensional interpolation. We illustrate this by two
examples.
Example1:
The totally ordered set S =( Rnd1 , n 2 ,..., n d ,«), given by relation (11), generates
the following multidimensional polinom
P( z ) = ∑ a i x1i1 x 2i2 ...x did = ∑ ∑ ... ∑ ai1 ,i2 ,...,id x1i1 x2i2 ...xdid ,
n1
i∈S
n2
i1 = 0 i 2 = 0
nd
id = 0
and in this situation
∂ α1 +...+α d
P( z ) =
∂x1α1 ...∂x 2α d
=
∑α ∑α
n1
i1 =
n2
1 i2
=
2
... ∑ ai1 ,i2 ,...,id
nd
i d =α d
i1!
i2!
id !
...
x1i1 −α1 x2i2 −α 2 ...xdid −α d .
(i1 − α1 )! (i2 − α 2 )! (id − α d )!
Example 2:
If S = Tn2 , then P ( z ) = P ( x, y ) =
∑ ai x
i∈S
i1!
i2 !
∂
P( z ) = ∑ a i
x
α1
α2
(i1 − α 1 )! (i2 − α 2 )!
∂x ∂y
i∈S
with α ≤ i .
α1 +α 2
i1
i1 −α1
y i2 , and
y
i2 −α 2
, for any α = (α 1 , α 2 ) ∈ S
If S = (Tn2 , p) and taking into consideration the relation (2), then P ( x, y )
= ∑∑ a j,k - j x j y k − j ,
n
k
k =0 j =0
∂ α1 +α 2
P (z) =
∂x α1 ∂y α 2
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
∑
∑α a
α α
n
=
k=
1+ 2
k −α 2
j=
j ,k − j
1
If
S
(k − j )!
j!
x
( j − α 1 )! (k − j − α 2 )!
j −α1
y
k − j −α 2
= ( Tn2 ,«) and taking into account the relation (3), then
P ( x, y ) = ∑∑ a j,k x j y k , and
n n− j
j =0 k =0
∂ α1 +α 2
P ( x, y ) =
∂x α1 ∂y α 2
∑α ∑α a
n
j=
n− j
1k =
2
j, k
j!
k!
x j −α 1 y k −α 2 .
( j − α1 )! (k − α 2 )!
REFERENCES
[1] Lorentz, G. G., Solvability of multivariate interpolation, J. reine angew. Math., 398
(1989), 101-104.
[2] Rudolf A. Lorentz, Multivariate Birkhoff Interpolation, Springer – Verlag Berlin
Heidelberg 1992.
[3] Bloom, T. and Calvi, J.P., A continuity property of multivariate Lagrange
interpolation, Mathematics of computation, Volume 66, Number 220, 1997.
[4] Wang, R. H., Piecewise algebraic curves and its branches, The Seventh Conf. Of
Chinese Mathematical Society, Beijing, May 18-21, 1995.
Author:
Crainic Nicolae„1 Decembrie 1918” University of Alba Iulia, str. N. Iorga, No. 13,
Departament of Mathematics and Computer Science
ORDERING OF SOME SUBSETS FROM IR d USED IN THE
MULTIDIMENSIONAL INTERPOLATION THEORY
by
Crainic Nicolae
Abstract: The ordering of the elements of a subset of IN d , d ≥ 1 , with the order
relations defined at the beginning of this article and implicitly obtaining some expressions for
them, corresponding to the defined order relations, is useful for the multidimensional
interpolation theory, especially when it is necessary to exemplify some general theoretical
notions. This is the reason for presenting expressions for ordered subsets from IN 2 , IN 3 and
in general from IN d .
Next we present the most usual order relations encountered in the
interpolation theory and used throughout this article.
1. We say that i = (i1 , i2 ,..., id ) ∈ IN d
is in the relation “«” with
j = ( j1 , j 2 ,..., j d )∈ IN d and we write i « j , when i1 < j1 , or i1 = j1 and i2 < j2 ,…,
or i1 = j1 ,…, id −1 = j d −1 and id ≤ j d .
2. We say that i = (i1 , i2 ,..., id )∈ IN d
is in the relation "p" with
j = ( j1 , j 2 ,..., j d )∈ IN d and we write i p j , when i < j or i = j and i « j .
Let be the set S = {i = (i1 , i2 ) / 0 ≤ i1 ≤ n,0 ≤ i2 ≤ m j , j = 0, n) } from IN 2 ,
denoted by S n;m0 ,...,mn , where n and 0 ≤ m0 ≤ m1 … ≤ mn are given natural numbers.
It is obvious that we would rather write out the set ( S n;m0 ,...,mn , «) than the set
( S n;m0 ,...,mn , p) whose last terms are difficult to be written. Thus we have:
n
mj
n
( S n;m0 ,...,mn , «)= UU{( j , k )} = U {( j ,0), ( j ,1),..., ( j , m j )} =
j =0 k =0
j =0
= {(0,0), (0,1),..., (0, m0 ), (1,0), (1,1),..., (1, m1 ),..., (n,0), (n,1),..., (n, m n )}
(1)
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
If in the set S n;m0 ,...,mn we take m0 = n , m j = n − j ,
j = 1, n , and
respectively m0 = m1 = ... = mn = n 2 , n = n1 , we obtain some of the most usual
forms of some subsets from S ⊂ IN 2 , that is the triangle Tn2 , respectively the
rectangle Rn2 , where n = (n1 , n 2 )∈ IN 02 . More concrete, these sets are
{
}
Tn2 = i ∈ IN 02 / i ≤ n ,
{
}
Rn2 = Rn21 ,n2 = i ∈ IN 02 / 0 ≤ i j ≤ n j , j = 1,2
and they are depicted in Figure 1 a and 1 b.
n2
n
n-1
n-2
2
1
1
0
0
0
1
2
n-1
n
0
(a)
1
2
n1
(b)
Figure 1
It is quite obvious that the grid of points from Figure 1 a is made out
of the parallel lines
∆2k = U{( j , k − j )}, k = 0, n .
k
j =0
It follows the next explicit expression of triangle Tn2 ordered by relation
" p"
(Tn2 , p) = UU{( j , k − j )} = U{(0, k ), (1, k − 1),..., (k ,0)}
n
k
k =0 j =0
n
k =0
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
= {(0,0), (0,1), (1,0),..., (0, k ), (1, k − 1),..., ( k ,0),..., (0, n), (1, n − 1),..., (n,0)} .
(2)
Moreover, if we consider the relation (1) and the fact that Tn2 = S n;n , n −1,..., 0 ,
we have:
n n− j
n
j =0 k =0
j =0
( Tn2 ,«)= UU {( j , k )} = U {( j ,0), ( j ,1),..., ( j , n − j )} =
= {(0,0), (0,1),..., (0, n), (1,0), (1,1),..., (1, n − 1),..., (n − 1,0), (n − 1,1),..., (n,0)} .
(3)
From the same relation (1) and the fact that Tn2 = S n1 ;n2 ,n2 ,...,n2 , we obtain
n1
n1
j =0
j =0 k =0
n2
( Rn2 ,«)= U {( j ,0), ( j ,1),..., ( j , n 2 )} = UU {( j , k )} =
= {(0,0), (0,1),..., (0, n 2 ), (1,0), (1,1),..., (1, n 2 ),..., ( n1 ,0), (n1 ,1),..., ( n1 , n 2 )} .
(4)
An explicit expression of ( Rn2 , p) is more difficult because it depends on
the position of the integers n1 and n2 with respect to each other. But for n1 = n 2 we
have:
( Rn2,n , p) = (Tn2 , p) U{(1, n), (2, n − 1),..., (n,1), (2, n), (3, n − 1),..., (n,2),..., (n, n)} .
We keep also in mind the following expressions of the ordered subsets from
2
IN :
U{(0, j ).(1, j − 1),..., ( j,0)} = U U{(i, j − i)} =
n
n
2
n
2
k
( T \ T , p )=
j = k +1
j
j = k +1 i = 0
{(0, k + 1), (1, k ),..., (k + 1,0), (0, k + 2), (1, k + 1),..., (k + 2,0),..., (0, n), (1, n − 1),..., (n,0)}
=
,
( Rn2,n \ Tn2 , p )= , {(1, n), (2, n − 1),..., (n,1)(2, n), (3, n − 1),..., (n,2),..., (n, n)} and
( Rn2,n \ Tn2 ,«)=
{(1, n), (2, n − 1), (2, n), (3, n − 2), (3, n − 1), (3, n),..., (n,0), (n,1),..., (n, n)} .
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
In general, extensions of the two dimensional case are used as examples in
IN , d ≥ 3 , among which we exemplify the simplex (Figure 2, in IN 03 )
{
d
}
Tnd = i ∈ IN 0d / i ≤ n ,
{
respectively the hyper-parallelipiped (Figure 4, in IN 03 )
}
Rnd = Rnd1 ,n2 ,...,nd = i ∈ IN 0d / 0 ≤ i j ≤ n j , j = 1, d .
S ⊂T .
d
n
Proposition: If S is an arbitrary set from IN d , and n = max i , then
i∈S
Proof: For any i ∈ S , with i ≤ max i = n , we have i ∈ Tnd .
i∈S
Next we will build successively an expression for the simplex (Tnd , p) .
Figure 2
The grid of points from the zero x-coordinate plane, Figure 2 (plane yOz), is
made out of the parallel lines
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
∆2k = U{( j , k − j )}, k = 0, n ,
k
j =0
their traversal order being ( ∆2k , p )=( ∆2k ,«), and of each of them, from left to right, in
descendant order. It follows that ∆2k given in the previous relation is ( ∆2k , p ). As
2
each element of the set ∆ k has the sum of indexes k -constant it follows that
( ∆2k , p )=( ∆2k ,«). From the same relation we have that
k −i
∆2k −i = U{( j , k − i − j )} , i = 0, k .
j =0
If we translate every line ∆2k −i , i = 1, k in the i = 1, k x-coordinate plane,
obviously parallel with the initial yOz plane, we obtain the ordered triangle ∆3k from
IN 03 , that is:
( ∆3k , p )= U ∆2i , k −i =
k
i =0
k k −i
UU{(i, j, k − i − j )} =
i =0 j =0
= U{(i,0, k − i ), (i,1, k − i − 1),..., (i,0, k − i ), (i, k − i,0)} =
k
i =0
= {(0,0, k ), (0,1, k − 1),..., (0, k ,0), (1,0, k − 1), (1,1, k − 2),..., (1, k − 1,0),..., (k ,0,0)} ,
depicted in Figure 2, whose vertices are the points of coordinates (k ,0,0), (0, k ,0) and
(0,0, k ) . To denote the plane of which ∆2k is a part of after translation, we added the i
index to the line ∆2k thus obtaining ∆2k −i . Actually i becomes the x-coordinate for all
points from the translated i x-coordinate plane. The traversal order of the lines ∆2i , k −i
is ∆20 ,k , ∆21,k −1 ,… , ∆2k , 0 . It follows that:
n
(Tn3 , p) = U
k =0
∆
3
k
k k −i
UUU{(i, j, k − i − j )}
n
= k =0 i =0
j =0
(5)
which is the same to (Tn3 , p) , taking into consideration the traversal order of the
triangles ∆3k , k = 0, n .
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
An explicit form of the ordered set (Tn3 , p) is:
(Tn3 , p) =
{(0,0,0), (0,0,1), (0,1,0), (1,0,0), (0,0,2), (0,1,1), (0,2,0), (1,0,1), (1,1,0), (2,0,0),...,
(0,0, n), (0,1, n − 1),..., (0, n,0), (1,0, n − 1), (1,1, n − 2),..., (1, n − 1,0),..., (n,0,0))} .
Proceeding similarly to the previous situation, we obtain
k −t
( ∆3k −t , p )= U ∆2i , k −t −i =
i =0
respectively
( ∆3t ,k −t , p )=
U U{(i, j, k − t − i − j )} ,
i =0 j =0
k −t k −t − i
U U{(t , i, j, k − t − i − j )} ,
i =0 j =0
from where we obtain that
k k −t k −t − i
∆ = U ∆3t ,k −t = UU
k
4
k
k −t k − t −i
t =0
U{(t , i, j, k − t − i − j )} ,
t =0 i =0 j =0
respectively
k k −t k −t −i
(Tn4 , p) = U ∆4k = UUU
n
n
k =0
U{(t , i, j, k − t − i − j )} .
k =0 t =0 i =0 j =0
Now the generalization to the d-dimensional space becomes obvious,
namely:
k k −i1 k − ( i1 + i2 )
∆dk = U U
U
i1 = 0 i2 = 0
i3 = 0
k − ( i1 +...+ id − 2 )
U{(i , i ,..., i
...
id −1 = 0
1
2
d −1
, k − (i1 + i2 + ... + id −1 )} ,
(6)
respectively
k k −i1 k − ( i1 + i2 )
(Tnd , p) = U ∆dk =UU U
n
k =0
n
k = 0 i1 = 0 i2 = 0
U
i3 = 0
k − ( i1 +...+ id − 2 )
...
U{(i , i ,..., i
id −1 = 0
1
2
d −1
, k − (i1 + i2 + ... + id −1 )}
(7)
If in addition we consider also the results of the previous proposition, then
we obtain the following theorem:
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
Theorem: An arbitrary set ( S , p) from IN d can be written in the form
( S , p) = U ∆'dkt ,
n
t =1
where k n = max i , ∆'dkt ⊂ ∆dk are given by (6).
(8)
i∈S
Example: S = {(0,0), (0,2), (1,1), (1,2)} = {(0,0)} ∪ {(0,2), (1,1)} ∪ {(1,2)} =
= ∆'2k1 ∪ ∆' 2k 2 ∪ ∆' 2k3 =
∆'2k3 = {(1,2)} ⊂ ∆23 .
U∆
3
t =1
'2
kt
,
where
∆'2k1 = ∆20 ,
∆'2k 2 = {(0,2), (1,1)} ⊂ ∆22 ,
and
In what follows, we will build in successive steps another expression for the
simplex Tnd using partially the relations " p" and “«”. Eventually we obtain a total
order relation defined on Tnd , which we denote by “∠”.
Figure 3
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
The grid of points from the zero x-coordinate plane, Figure 3 (plane yOz),
UU{( j, k − j )} , made out of the
n
according to relation (.3), is the triangle (Tn2 , p) =
parallel lines ∆2k =
k
k =0 j =0
U{( j, k − j )}, k = 0, n , their traversal order being
k
j =0
∆20 , ∆21 , …
, ∆ , and of each of them, from left to right, in descending order. It follows that
2
n
n −i k
(Tn2−i , p) = UU{( j , k − j )}, i = 0, n .
k =0 j =0
If we translate each triangle (Tn2−i , p) , with i = 1, n , in the i = 1, n xcoordinate plane, obviously parallel to the initial plane of the triangle, it results the
ordered tetrahedron
(Tn3 , ∠)= U Ti ,2n −i
n
i =0
from Figure 3., which is different both from (Tn3 , p) and ( Tn3 ,«). We added the
index i to the triangle Tn2−i to indicate the x-coordinate of the plane of which Tn2−i is
part of after translation. In fact i becomes x-coordinate for all points from the i xcoordinate translated plane. The traversal order of the triangles Ti ,2n −i is T02,n , T1,2n −1 , …
, Tn2, 0 . It follows that:
n n −i k
(T , ∠)= UUU{(i, j , k − j )} .
3
n
i =0 k =0 j =0
An explicit form of the ordered set (Tn3 , ∠) is:
(Tn3 , ∠)=
{(0,0,0), (0,0,1), (0,1,0), (0,0,2), (0,1,1), (0,2,0),..., (0,0, n), (0,1, n − 1),..., (0, n,0),
(1,0,0), (1,0,1), (1,1,0), (1,0,2), (1,1,1), (1,2,0),..., (1,0, n − 1), (1,1, n − 2),...,
..., (1, n − 1,0,..., (n,0,0))}
Continuing the used procedure and generalizing we can obtain rather easily that in ddimensional space
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
n n − i1 n − ( i1 + i 2 )
(Tnd , ∠)= U U
i1 = 0 i 2 = 0
U
i3 = 0
n − ( i1 + ...+ i d −3 ) n − ( i1 + ... + i d − 2 )
U
...
U
i d −2 = 0
k =0
k
U{(i , i ,..., i
i d −1 = 0
1
d −1
2
, k − id −1 )} .
(9)
An explicit expression for the simplex (Tnd , «) can be obtained starting from
n n− j
the triangle ( Tn2 ,«)= UU {( j , k )} given by relation (3), lying in the plane yOz from
j =0 k =0
Figure 3. The elements of the triangle ( Tn2 ,«) are the grid of points lying in the
mentioned plane, whose explicit form is obtained if we traverse Tn2 from bottom to top
and from left to right. We can also write that
n − i n −i − j
( Tn2−i ,«)= U
U{( j, k )} .
j =0 k =0
Translating each triangle (Tn2−i , p) , i = 1, n in the i = 1, n x-coordinate
plane, we obtain the ordered tetrahedron
n
(Tn3 , «)= U Ti ,2n −i .
i =0
The i index added to Tn2−i , indicates the plane of which Tn2−i is part of after
translation and becomes the x-coordinate for all points from this plane. The traversal
order of triangles Ti ,2n −i is also T02,n , T1,2n −1 , … , Tn2, 0 . It follows that:
n n − i n −i − j
(T , «)= UU
3
n
U{(i, j, k )}
i =0 j =0 k =0
Eventually, for the d -dimensional space we obtain that:
n n − i1 n − ( i1 + i 2 )
(Tnd , «)= U U
i1 = 0 i 2 = 0
U
i3 = 0
n − ( i1 + ... + i d − 2 ) n − ( i1 + ... + i d −1 )
...
U
i d −1 = 0
U{(i , i ,..., i
id = 0
1
2
d −1
, id )}
(10)
Next we build an expression for the hyper-parallelipiped Rnd from the d dimensional space using the total order relation “«”.
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
Figure 4
The grid of points from the zero x-coordinate plane, Figure 4 (plane yOz),
n1
n2
according to relation (4), is the rectangle ( Rn2 ,«)=( Rn21 , n2 ,«)= UU {( j , k )} , its
j =0 k =0
rectangle Rn2 by successive translations, in the i = 1, n 3 x-coordinate planes,
obviously parallel to the initial plane of the rectangle, we obtain the ordered
parallelepiped
traversal order being from bottom to top and from left to right. If we “multiply” the
n3
( Rn3 ,«)= U Ri2,( n1 ,n2 )
i =0
from Figure 4. The i index added to Rn2 indicates the plane of which this is a part of
after translation and becomes x-coordinate for all points from the i x-coordinate
plane. The traversal order of the rectangles Rn2 is R02,( n1 ,n2 ) , R12,( n1 , n2 ) , …, Rn23 ,( n1 , n2 ) . It
follows that:
n3
( Rn33 , n1 , n 2 ,«)=
n1 n 2
UUU{(i, j, k )} .
i =0 j =0 k =0
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
Generalizing to the d -dimensional space we obtain:
n1
( Rnd1 , n 2 ,..., n d ,«)=
n d −1
n2
U U... U
i1 = 0 i 2 = 0
nd
U{(i , i ,..., i )}
i d −1 = 0 i d = 0
1
2
d
(11)
Remark: An expression for ( Rnd1 , n2 ,...,nd , p) can be obtained applying the relation (8)
from the previous theorem.
As we pointed out in the beginning, the notions presented in this article are
useful in the field of multidimensional interpolation. We illustrate this by two
examples.
Example1:
The totally ordered set S =( Rnd1 , n 2 ,..., n d ,«), given by relation (11), generates
the following multidimensional polinom
P( z ) = ∑ a i x1i1 x 2i2 ...x did = ∑ ∑ ... ∑ ai1 ,i2 ,...,id x1i1 x2i2 ...xdid ,
n1
i∈S
n2
i1 = 0 i 2 = 0
nd
id = 0
and in this situation
∂ α1 +...+α d
P( z ) =
∂x1α1 ...∂x 2α d
=
∑α ∑α
n1
i1 =
n2
1 i2
=
2
... ∑ ai1 ,i2 ,...,id
nd
i d =α d
i1!
i2!
id !
...
x1i1 −α1 x2i2 −α 2 ...xdid −α d .
(i1 − α1 )! (i2 − α 2 )! (id − α d )!
Example 2:
If S = Tn2 , then P ( z ) = P ( x, y ) =
∑ ai x
i∈S
i1!
i2 !
∂
P( z ) = ∑ a i
x
α1
α2
(i1 − α 1 )! (i2 − α 2 )!
∂x ∂y
i∈S
with α ≤ i .
α1 +α 2
i1
i1 −α1
y i2 , and
y
i2 −α 2
, for any α = (α 1 , α 2 ) ∈ S
If S = (Tn2 , p) and taking into consideration the relation (2), then P ( x, y )
= ∑∑ a j,k - j x j y k − j ,
n
k
k =0 j =0
∂ α1 +α 2
P (z) =
∂x α1 ∂y α 2
Crainic Nicolae - Ordering of some subsets from IR d used in the multidimensional
interpolation theory
∑
∑α a
α α
n
=
k=
1+ 2
k −α 2
j=
j ,k − j
1
If
S
(k − j )!
j!
x
( j − α 1 )! (k − j − α 2 )!
j −α1
y
k − j −α 2
= ( Tn2 ,«) and taking into account the relation (3), then
P ( x, y ) = ∑∑ a j,k x j y k , and
n n− j
j =0 k =0
∂ α1 +α 2
P ( x, y ) =
∂x α1 ∂y α 2
∑α ∑α a
n
j=
n− j
1k =
2
j, k
j!
k!
x j −α 1 y k −α 2 .
( j − α1 )! (k − α 2 )!
REFERENCES
[1] Lorentz, G. G., Solvability of multivariate interpolation, J. reine angew. Math., 398
(1989), 101-104.
[2] Rudolf A. Lorentz, Multivariate Birkhoff Interpolation, Springer – Verlag Berlin
Heidelberg 1992.
[3] Bloom, T. and Calvi, J.P., A continuity property of multivariate Lagrange
interpolation, Mathematics of computation, Volume 66, Number 220, 1997.
[4] Wang, R. H., Piecewise algebraic curves and its branches, The Seventh Conf. Of
Chinese Mathematical Society, Beijing, May 18-21, 1995.
Author:
Crainic Nicolae„1 Decembrie 1918” University of Alba Iulia, str. N. Iorga, No. 13,
Departament of Mathematics and Computer Science