An analytical method for computing the coefficients of the trivariate polynomial approximation

7

An Analytical Method for Computing the Coefficients of the

Trivariate Polynomial Approximation

Süleyman ŞAFAK

Division of Mathematics, Faculty of Engineering, Dokuz Eylül University, 35160 Tınaztepe, Buca, İzmir, Turkey.

suleyman.safak@deu.edu.tr

(Geliş/Received: 24.02.2012; Kabul/Accepted: 23.07.2012)

Abstract

This paper deals with computing the coefficients of the trivariate polynomial approximation (TPA) of large distinct points given on 3. The TPA is formulated as a matrix equation using Kronecker and Khatri-Rao products of the matrices. The coefficients of the TPA are computed using the generalized inverse of the matrix. It is seen that the trivariate polynomial approximation can be investigated as the matrix equation and the coefficients of the TPA can be computed directly from the solution of the matrix equation.

Keywords: Polynomial approximation, Trivariate polynomial, Generalized inverse of a matrix, Matrix equation

Üç Değişkenli Polinom Yaklaşımının Katsayılarının Hesabı İçin Bir

Analitik Yöntem

Özet

Bu makale, 3de verilen çok sayıda farklı noktanın üç değişkenli polinom yaklaşımındaki katsayıların hesaplaması ile ilgilidir. Üç değişkenli polinom yaklaşımı, matrislerin Kronecker ve Khatri - Rao çarpımlarını kullanarak bir matris denklemi olarak formüle edilmiştir. Bu polinomun katsayıları bir matrisin genelleştirilmiş tersi kullanılarak hesaplanmıştır. Üç değişkenli polinom yaklaşımın bir matris denklemi olarak incelenebileceği ve bu polinomun katsayılarının doğrudan bu matris denkleminin çözümünden hesaplanabileceği görülmüştür.

Anahtar kelimeler: Polinom yaklaşımı, Üç değişkenli polinom, Matrisin genelleştirlmiş tersi, Matris denklemi.

1.Introduction

One of the most interesting mathematical studies is that finding the best polynomial approximation of the given data or the function [1-11]. Multivariable approximation and interpolation has been an active research area in applied mathematics for many years, and has had impact on various engineering and scientific applications in mathematical modeling, in computations with large scale data, in signal analysis and image processing [4, 6, 8, 10-12].

The polynomial approximation is investigated using the different forms and algorithms. Also, it is solvable either numerically or using a computer algebra package [1, 3, 7].

The polynomial approximations in two or several variables is the most commonly investigated problem by researchers [2, 4, 5, 10]. There are many research for the developments of least squares polynomial approximations to sets of data and the polynomial approximations in several variables have been investigated by the different methods [1, 8-11, 13, 14].

In this study, we formulate the trivariate polynomial approximation (TPA) as a matrix equation using Kronecker and Khatri-Rao products of the matrices and compute the coefficients of the trivariate polynomial approximation using the generalized inverse matrices. It is seen that the trivariate polynomial approximation can be investigated as the matrix

8 equation and the coefficients of the TPA can be computed directly from the solution of the matrix equation.

2.Basic definitions and theorems

In this section, we give some basic definitions and theorems associated with the trivariate polynomial, Kronecker product, Khatri-Rao product and generalized inverse of a matrix. Further details and proofs can be found elsewhere [1-3, 5, 6, 14-19].

Definition 1. Given a hyper surface uh(x,y,z)

on 4 _{to approximate over a region that is} gridded by ( , , ) l k l l y z x , 0ls, 0k_lr_lon 3  . Assumed that ( , , ) l l l l k llk h x y z u  data are

given for the function of three variables at the     s l l r s 0 ) 1 ( ) 1

( distinct points in the

region on 3, there is a hyper surface on 4 of the form k j i m i n j r k ijk z y x a z y x p      0 0 0 ) , , ( (1)

of degree, not exceding mnr, namely r

n m_{y z}

x that passes through each point in the solid region, where (m1)(n1)(s1) and

} { minr_i

r fori0,1,2,,r. We say that Eq.(1) is a trivariate polynomial approximation satisfiying ( , , ) ( , , ) l l l l k k l l y z h x y z x p  for all s l 

0 and 0k_l r_l. It is clear that



x y z



h , , at any point



xˆ,yˆ,zˆ



, which is not in solid region on 3_{, can be estimated by}



x y z



p



x y z



h ˆ,ˆ,ˆ  ˆ,ˆ,ˆ [5-7].

Now we define the trivariate polynomial approximation for general distinct points. Given

) , , (x y z f

u to approximate over a solid rectangular region that is gridded by (x_i,y_j,z_k) on



3, 0im, 0 jn and 0kr.

Assumed that f_ijk  f(x_i,y_j,z_k)data are given for the function of three variables at the(m1)(n1)(r1) distinct points in the

solid rectangular region, there is a hyper surface on 4 of the form k j i p i q j t k ijk

x

y

z

b

z

y

x

p



  



0 0 0

)

,

,

(

(2)

degree at pqt, namely xpyqzt that deviates as little as possible from f in the solid rectangular region, where pm, qn and

r

t . Also, we say that Eq.(2) is a trivariate polynomial approximation which satisfies

) , , ( ) , , (x_i y_j z_k f x_i y_j z_k p  at minimum error

for all 0i p, 0 jq and 0kt [5-7]. Definition 2. Let A[a_ij] be an mn matrix and B[b_ij] be a pqmatrix, then the Kronecker product of A and B , AB, which is an mpnq matrix is defined by

] [Ab_ij B

A  . (3) Let A[A_ij] be partinoned with A of _ij order m_in_j as the (i ),j thblock matrix and let

] [B_kl

B partinoned with B of order _kl p_kq_l as the (k ),l thblock matrix where, m_i m,

n_j n, p_k p and  q_l q. The Khatri– Rao product of the matrices A and B is defined as ij ij ij B A B A [  ] , (4) where A_ij B_ijis order of m_ip_in_jq_j and

B

A of order



m_ip_i







n_jq_j



[15, 19, 20]. Definition 3. Let A be an arbitrary

m



n

complex matrix. The generalized inverse Aof A uniquely determined as the nm matrix, which is simultaneously satisfied the following system of four matrix equations:

  _{AA A} A , AAAA,   _{ AA} AA )* ( , (AA)*AA. (5) If A is a real matrix, A*AT where A * denotes the conjugate transpose of A [14, 16-18].

9 Theorem 1. A necessary and sufficient condition for the equation XBYC to have a solution is that

C Y CY XX  

in which case the general solution is

    _{ } X CY W X XWYY B ,

where W is an arbitrary matrix and X is the  generalized inverse of X [14,18].

3.Trivariate polynomial approximation In the this section, we first formulate a matrix equation to calculate the coefficients of the trivariate polynomial approximation defined in Eqs.(1) and (2) using Kronecker and Khatri-Rao products, and then investigate the solution of this matrix equation using the generalized inverse of a matrix. Let ] 1 [ x x2  xm  x , ] 1 [ y y2  yn  y , (6) ] 1 [ _{z z}2 _{ z}r  z and ] [ _{0 1 m} T _{A A A} A   , (7) where              nr r r n n a a a a a a a a a A 0 01 00 1 0 011 001 0 0 010 000 0                     nr r r n n a a a a a a a a a A 1 11 10 1 1 111 101 0 1 110 100 1        , ...,              mnr r m r m mn m m mn m m m a a a a a a a a a A        1 0 1 11 01 0 10 00 , m A A

A₀, ₁,, are(r1)(n1)matrices and A is the (m1)(n1)(r1) matrix. Using (6) and (7), we can state the trivariate polynomial defined in Eq.(1) as a matrix equation as follows:





_A T z y x p( , , ) yx z , (8) where  is the Kronecker product. Eq.(8) is the matrix equation form of Eq. (1).

The polynomial approximation in three variables defined in Eq.(8) can be found such

that the equation must satisfy

) , , ( ) , , ( l l l l k k l l y z h x y z x

p  for all 0ls, and

l

l r

k  

0 at minimum error. For this purpose, we can compute the matrix A from a matrix equation. Let                 s s M x y x y x y        0 0 0 0 0 0 1 1 0 0 ,              s Z Z Z Z        0 0 0 0 0 0 1 0 , (9)              s F u u u        0 0 0 0 0 0 1 0 , where ] 1 [ ] 1 [ 2 l l2 lm n l l l l l x x x y y y     x y and ] [ _{0 1} l l l ll llr ll l  u u  u u for l0,1,,s and

10                r r r r r l l l l l l l z z z z z z Z        1 0 1 0 1 1 1 ,

where M is the (s1)(s1)(m1)(n1) matrix of rank (s1), Z is the ( 1)( 1) ( 1) 0       s l l r s r

matrix of rank (s1)(r1), F is the

) 1 ( ) 1 ( 0      s i l r

s matrix and u is the _l 1(r_l 1) vector.

Using (9) and properties of Kronecker product of matrices, we can formulate the coefficients of Eq.(1) as the matrix equation for all l0,1,2,,s as follows: H Z I A M(  _s1)  , (10) where I_s1is the (s1)(s1) identity matrix. We can compute the matrix A from Eq.(10) and we can find Eq.(1) or Eq.(6) using the matrix A which is approximated with minimum error. For this purpose, we can solve Eq.(10) using the generalized inverses of matrices. This solution is known as the best solution.

Theorem 2. Eq.(10) has a solution and its general solution is



uZ y x W



W x y x x y y A i i i i T i T i T i i T i i       ) ( ) ( ) )( ( 1 (11)

for l0,1,2,,s, where W is an arbitrary matrix.

Proof. Let AI_s1 be solution of (10). Since

, ) ( ) ( 1 1 Z HZ MM Z ZZ I A M MM Z I A M H s s           

Eq.(10) has a solution, where MM I and I

ZZ . Using Theorem 1 and Eq.(11), we obtain       _   ) )( ( ) ( T i i T i i T i T i x x y y x y Diag M , for i0,1,2,,s ,



  



_ s Z Z Z Diag Z ₀ , ₁ , , and W M M I HZ M I A _s1  (   ) .

Eq.(11) are the approximate solutions of (10). In this study, we also compute the best approximate solution of (10). For this, we can rewrite Eq.(10), using properties of Kronecker product, Kahatri-Rao product and generalized inverse of the matrices, as

U A V V_ys _xs)  ( , (12) where                   s sZ Z Z U u u u  1 1 0 0 ,                  s xs V x x x x  2 1 0 ,                  s ys V y y y y  2 1 0 ,

 is Khatri–Rao product, V_ysV_xs is the ) 1 )( 1 ( ) 1 (s  m n matrix of rank ) 1 )( 1 (m n , U is the (s1)(r1) matrix, the rank of Z for _l l0,1,2,,s is r1,

1 ) (  _ T l l T l l Z ZZ Z and Z_lZ_l I_r1.

Using Theorem 1, we solve the Eq.(12) and find the unique solution matrix A which is the best solution of (12). This leads to following result.

Theorem 3. The best solution of Eq.(12) is

U V V V V V V A[( _ysT  _xsT)( _ys _xs)1]( _ys _xs) _{. (13)} Proof Let V_ysV_xsV . Since the rank of the matrix V is (m1)(n1), V(VV)1VT and VV I_{(m1)(n1)}. Then we have,

11 U VV VA VV VA U     .

Thus we see that Eq.(12) has a solution. In which case, we obtain the general solution as

h

p A

A

A  ,

where A_p VU and Ah WVVW 0 are

the particular and homogenous solutions, respectively and W is an arbitrary matrix. If

A

A_p  , then Eq.(12) has a unique solution. Thus the proof is completed.

Also, we can construct the matrix equation to find the trivariate polynomial for

) , , ( _{i j k} ijk f x y z

f  data which are given for the function of three variables at the

) 1 )( 1 )( 1

(m n r distinct points in the solid rectangular region. Let ] [B₀ B₁ B_t B  , (14) where              0 0 1 0 0 10 110 010 00 100 000 0 pq q q p p b b b b b b b b b B                     1 1 1 1 0 11 111 011 01 101 001 1 pq q q p p b b b b b b b b b B        , ... ,              pqt qt qt t p t t t p t t t b b b b b b b b b B        1 0 1 11 01 0 10 00 , ] 1 [ x x2  xp  x , ] 1 [ y y2  yq  y , (15) ] 1 [ z z2  zt  z and B_k is an (q1)(p1) matrix.

Note that b_ijk are the elements of the ) 1 )( 1 ( ) 1 (q  p t matrix B defined in Eq.(2), where 0i p, 0 jq and

t k 

0 . Using (14) and (15), we can state TPA defined in Eq.(2) as a matrix equation as follows: T t T I B z y x p( , , ) y (x  1)z (16)

Also, we can express the coefficients of the trivariate plynomial which is the best approximate as a matrix equation such that Eq.(2) must satisfy at minimum error

) , , ( ) , , (xi yj zk f xi yj zk

p  in the solid rectangular

region. Let                  m x V x x x x  2 1 0 ,                  n y V y y y y  2 1 0 ,                  r z V z z z z  2 1 0 (17) and



F F Fm



F _{0 1}  , (18) where              nr n n r r f f f f f f f f f F 0 1 0 0 0 01 011 010 00 001 000 0                     nr n n r r f f f f f f f f f F 1 1 1 0 1 11 111 110 10 101 100 1        ,... ,              mnr mn mn r m m m r m m m m f f f f f f f f f F        1 0 1 11 10 0 01 00 ] 1 [ _{i i}2 _ip i  x x  x x , ] 1 [ _{j j}2 _jq j  y y  y y ,

12 ] 1 [ _{k k}2 _kt k  z z  z z and F is an _i (n1)(r1) matrix.

Note that f(x_i,y_j,z_k) f_ijk are the elements of the matrix F defined in Eq.(18) where 0im, 0 jn and 0kr. Using (17) and (18), we can formulate the coefficients of Eq.(2) as follows:



V I



V I



F B V T _zT _m m t T x y ( 11 1)  1  , (19) where 1_m1is the (m1)1vector whose entries are 1.

We can find the matrix B from Eq.(19). This leads to following results.

Corollary 1. Let



V_xT(I_t11_mT_1)



be the matrix defined in Eq.(19). Then its generalized inverse is















1



1 1 1 1 1 ) ( ) (              r x T x m t x T m t T x I V V I V I V 1 1 . (20)

Proof Observing the rank of the ) 1 )( 1 ( ) 1 )( 1 (p t  t m matrix defined in Eq.(19) is (p1)(t1), and using the generalized inverse and Khatri-Rao product of the matrices, we easily prove it.

Theorem 4. Eq.(19) has a unique solution as follows:







( )



[( )1 ₁] 1 1 1        _{   }  T _{x t} x m t x T m z y FV I V I V V I V B 1 (21)

Proof Using Theorem 1, Eq.(19) has a solution. Since ranks of the matrices V , _x V and _y

V

z are

1  p ,q1 and t1 respectively, V_xV_xI_p1, 1   _ q y y V I

V and V_zV_zI_t1. In which case we obtain the general solution as BB_pB_h, where





 ( )[( ) 1 ₁], 1 1 1        _{   }  t x T x m t x T m z y p I V V I V I V F V B 1



 

 





0 ) ( ) ( 1 1 1 1 1 1                   m T z T m t T x T m t T x m T z y y h I V I V I V I V W V V W B 1 1

W is an arbitrary matrix, and B_pand B are the _h particular and homogenous solutions,

respectively. If B_pB, then (19) has a unique solution. Thus the proof is completed.

Theorem 5. Let B and _k F be submatrices _i defined in (14) and (18). Then, the solution of Eq.(19) is

  



1



1



0 ) (     _{ }  T _{x r} x i T z m i i y F V V V I V B x . (22)

Proof To prove the theorem, we can use Theorem 4 and the matrices defined in (14) and (18). Putting (14) and (18) in (21) and using Kronecker and Khatri-Rao products, then Eq.(22) is obtained.

4. Conclusions

We consider the trivariate polynomial approximation (TPA) of given distinct points on

3

 _{and formulate the TPA as a matrix equation} using Kronecker and Khatri-Rao products of the matrices. We conclude that the coefficients of the TPA can be computed directly from the matrix equation by the use of generalized inverse matrices. It is seen that the TPA can be investigated as the matrix equation and the coefficients of it can be computed directly from this matrix equation.

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