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HOW TO SOLVE AN INFINITE SIMULTANEOUS SYSTEM OF

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Kangweon-Kyungki Math. Jour. 13 (2005), No. 2, pp. 275–284
HOW TO SOLVE AN INFINITE SIMULTANEOUS
SYSTEM OF QUADRATIC EQUATIONS
Phil Ung Chung* and Ying Zhen Lin
Abstract. In the present paper we shall introduce several operators on the reproducing kernel spaces. And using them we shall
find a solution of an infinite system of quadratic equations (1.1).
In particular we shall convert problem for finding an approximate
solution of infinite system of quadratic equations into problem for
minimizing nonnegative biquadratic polynomial.
1. Introduction
Throughout this paper we shall concern with an infinite system of
quadratic equations
(1.1)
aii,k x2i + в€ћ
i=j aij,k xi xj = bk , (k в€€ N)
x1 = x0 (constant),
в€ћ
i=1
2
в€ћ
where X = (x1 , x2 , В· В· В· ) в€€ 2 , b = (b1 , b2 , В· В· В· ) в€€ 2 ,
i,j=1 (aij,k ) <
+в€ћ, (k в€€ N), and aij,k = aji,k for all i, j в€€ N. If we informally introduce
в€ћ Г— в€ћ symmetric matrix
Ak = (aij,k )в€ћГ—в€ћ , (k в€€ N),
then (1.1) can be transformed into
(1.2)
(X, Ak X T ) 2 = bk , (k в€€ N)
x1 = x0
and vice versa, where (В·, В·) 2 denotes the standard inner product in
space.
2
Received September 22, 2005.
2000 Mathematics Subject Classification: 45L05; 44-XX.
Key words and phrases: system, quadratic equation, reproducing kernel space.
*Corresponding author.
-
276
Phil Ung Chung and Ying Zhen Lin
The purpose of the present paper is to find a solution of an infinite
system of quadratic equations (1.1). In particular we shall convert problem for finding an approximate solution of infinite system of quadratic
equations into problem for minimizing nonnegative biquadratic polynomial.
2. Preliminaries
The reproducing kernel space W21 [0, 1] is defined as the set of functions
W21 [0, 1] = {u(t)|u is absolutely continuous and u, u в€€ L2 [0, 1]},
equipped with the inner product
1
(u, v)W21 =
(u(t)v(t) + u (t)v (t)) dt
0
and with norm
u
2
W21
= (u, u)W21 .
W21 [0, 1]
The reproducing kernel of
can be given by
1
(2.3)
RО· (t) =
et+О· + e2в€’(t+О·) + e|tв€’О·| + e2в€’|tв€’О·| ,
2
2(e в€’ 1)
for each t, О· в€€ [0, 1], which satisfies the reproducing property
(2.4)
(u(t), RО· (t))W21 = u(О·)
for every u в€€ W21 [0, 1].
Let D = [0, 1] Г— [0, 1]. The reproducing kernel space W (D) is defined
as the set of functions
∂u ∂u ∂ 2 u
W (D) = u(s, t)|u is complete continuous,
, ,
в€€ L2 (D) ,
∂s ∂t ∂s∂t
equipped with the inner product
∂u ∂v ∂u ∂v
∂2u ∂2v
(u, v)W (D) =
uv +
+
+
dsdt
∂s ∂s
∂t ∂t ∂s∂t ∂s∂t
D
and with norm
u 2W (D) = (u, u)W (D) .
The reproducing kernel of W (D) can be given by
(2.5)
K(Оѕ,О·) (s, t) = RОѕ (s)RО· (t),
where RОѕ (t) is given by (2.3) [1].
How to solve an infinite simultaneous system of quadratic equations
277
3. Linear operators on reproducing kernel spaces
In this section we shall introduce several operators, which will be
needed in the later parts of our discussion. Throughout the present
paper we shall choose and fix a countable dense subset
(3.6)
T = {t1 , t2 , В· В· В· }
of the interval [0, 1], and put
(3.7)
def
П†i (t) = Rti (t), i в€€ N.
Lemma 3.1. The sequence of functions {П†i (t)}в€ћ
i=1 constitutes a complete system of W21 [0, 1].
Proof. Let u(t) в€€ W21 [0, 1]. Since (u(В·), П†i (В·))W21 = u(ti ) for each i в€€
N, we have (u(В·), П†i (В·)) = 0 if and only if u(ti ) = 0 if and if u(t) = 0,
which proves our assertion.
Using Gram-Schmidt process, we orthonormalize {П†i (t)}в€ћ
i=1 to obtain
1
в€ћ
ВЇ
an orthonormal system {П†i (t)}i=1 for W2 [0, 1],
i
def
П†ВЇi (t) =
О±il П†l (t),
l=1
where О±il are the orthonormal coefficients.
We shall define an operator ПЃ : 2 в€’в†’ W21 [0, 1] by
в€ћ
(3.8)
def
xi П†ВЇi (t)
ПЃX =
i=1
for each X = (x1 , x2 , В· В· В· ) в€€ 2 . It is easy to show that ПЃ is one-to-one
and norm preserving. It is noteworthy that (1.2) can be converted into
(3.9)
u(В·), (ПЃAk ПЃв€’1 u)(В·)
W21
= bk , (k в€€ N)
where u(t) = ПЃX.
Again we shall define an operator AЛњk : W21 [0, 1] в€’в†’ W21 [0, 1] by
def
AЛњk u (t) = ПЃAk ПЃв€’1 u (t),
kв€€N
for each u(t) в€€ W21 [0, 1]. Thus (3.9) can be converted into
u(В·), (AЛњk u)(В·)
W21
= bk ,
kв€€N
278
Phil Ung Chung and Ying Zhen Lin
that is,
1
u(t)(AЛњk u)(t) + u (t)(AЛњk u) (t) dt = bk .
0
Let I and D be the identity and differential operators on W21 [0, 1]
respectively. For each k в€€ N, we shall define an operator Hk : W (D) в€’в†’
R by
(3.10)
1
def
Hk v =
(В·)
AЛњk I (в€—) + (DAЛњk )(В·) D(в€—) v(в€—, В·) (t)dt,
v в€€ W (D),
0
where ” · ” and ” ∗ ” denote the variables corresponding to function
respectively.
4. Operator equation associated with (1.2)
We shall introduce an operator L : W (D) в†’ W21 [0, 1] defined by
в€ћ
(4.11)
def
(Hk v)П†ВЇk (t),
(Lv)(t) = ПЃ((H1 v, H2 v, В· В· В· )) =
v в€€ W (D)
k=1
where Hk and ПЃ are given by (3.10) and (3.8) respectively. In fact, since
L is a composition of bounded linear operators I, AЛњk , D, ПЃ and integral,
we have L is also bounded linear operator.
Lemma 4.1. Let X = (x1 , x2 , В· В· В· ) в€€
have
(4.12)
u(t1 ) = x1 П†1
2
, and let ПЃX = u(t). Then we
W21 .
1
Proof. Since {П†ВЇi (t)}в€ћ
i=1 is an orthonormal system of W2 [0, 1], we have,
by virtue of (2.4), (3.7), and (3.8),
u(t1 ) = (u(В·), П†1 (В·))W 1
2
ВЇ
= П†1 u(В·), П†1 (В·)
W21
в€ћ
xi П†ВЇi (В·), П†ВЇ1 (В·)
= П†1
i=1
= x1 П†1
W21 .
W21
How to solve an infinite simultaneous system of quadratic equations
279
def
Theorem 4.2. Let X = (x1 , x2 , В· В· В· ) в€€ 2 and x1 = x0 , and let
ПЃX = u(t). Then X is a solution of (1.2) if and only if u(s)u(t) в€€ W (D)
is a solution of
(4.13)
(Lv)(t) = f (t),
def
в€ћ
ВЇ
k=1 bk П†k (t).
where f (t) =
Proof. Suppose that X is a solution of (1.2). Then we have
(Lu(в€—)u(В·))(t)
= ПЃ((H1 u(в€—)u(В·), H2 u(в€—)u(В·), В· В· В· ))
1
u(t)(ПЃA1 ПЃв€’1 u)(t) + u (t)(ПЃA1 ПЃв€’1 u) (t) dt, В· В· В·
=ПЃ
0
=ПЃ
(u(В·), (ПЃA1 ПЃв€’1 u)(В·))W21 , В· В· В·
=ПЃ
(X, A1 X T ) 2 , В· В· В·
= ПЃ ((b1 , В· В· В· ))
= f (t).
Conversely suppose that u(s)u(t) is a solution of (4.13). Then we
have
ПЃ((u(t), (ПЃA1 ПЃв€’1 u)(t))W21 , В· В· В· )) = ПЃ((b1 , b2 , В· В· В· )).
Since ПЃ is one-to-one and norm preserving, we obtain, by virtue of
Lemma 4.1,
((X, A1 X T ), В· В· В· ) = (b1 , В· В· В· ),
x1 = x 0 .
Hence our assertion is proved.
5. Direct sum decomposition of W (D)
Using the adjoint operator Lв€— of L defined by (4.11), we define П€i (s, t)
by
(5.14)
def
П€i (s, t) = (Lв€— П†i )(s, t) = (Lв€— Rti )(s, t), (i в€€ N).
Lemma 5.1. A function П€i (s, t), defined above, can be expressed by
в€ћ
(5.15)
в€ћ
в€ћ
aij,k П†ВЇi (s)П†ВЇj (t) П†ВЇk (ti ), (i в€€ N)
П€i (s, t) =
k=1
i=1 j=1
280
Phil Ung Chung and Ying Zhen Lin
Proof. By virtue of (2.4), (2.5), (3.7), (3.10), and (4.11), we have
П€i (s, t) = ((Lв€— П†i )(в€—, В·), Rs (в€—)Rt (В·))W (D)
= (П†i ( ), (LRs (в€—)Rt (В·))( ))W21
= (LRs (в€—)Rt (В·))(ti )
в€ћ
Rs (В·), (AЛњk Rt )(В·)
=
k=1
в€ћ
W21
П†ВЇk (ti )
ПЃAk ПЃв€’1 Rt (s)П†ВЇk (ti ).
=
k=1
On the other hand, since
в€ћ
в€ћ
П†ВЇk (t)П†ВЇk (s)
(Rt (В·), П†ВЇk (В·))W21 П†ВЇk (s) =
Rt (s) =
k=1
k=1
we have
ПЃв€’1 Rt (s) = (П†ВЇ1 (t), П†ВЇ2 (t), В· В· В· ) в€€
2
,
hence
в€ћ
в€ћ
aij,k П†ВЇj (t) П†ВЇi (s).
в€’1
(ПЃAk ПЃ Rt )(s) =
i=1
j=1
Thus we obtain the desired result
в€ћ
в€ћ
в€ћ
aij,k П†ВЇj (t)П†ВЇi (s) П†ВЇk (ti ).
П€i (s, t) =
k=1
i=1 j=1
By Gram-Schmidt process, we obtain an orthonormal system
{П€ВЇi (s, t)}в€ћ
i=1
of W (D) such that
i
(5.16)
def
П€ВЇi (s, t) =
ОІik П€k (s, t),
k=1
where ОІik are orthonormal coefficients.
How to solve an infinite simultaneous system of quadratic equations
281
вЉҐ
Let S be the closure of span {П€ВЇi (s, t)}в€ћ
i=1 and let S be the orthogonal complement of S in W (D). We choose a countable dense subset
B = {(s1 , t1 ), (s2 , t2 ), В· В· В· } of D. It is easy to show that
def
ПЃj (s, t) = Rsj (s)Rtj (t),
jв€€N
constitutes a basis of the space W (D). Again we orthonormalize
{П€ВЇ1 , П€ВЇ2 , В· В· В· , ПЃ1 , ПЃ2 , В· В· В· }
to obtain
ПЃВЇj (s, t) =
jв€’1
ВЇ ВЇ
ПЃj (s, t) в€’ в€ћ
ВЇm )ВЇ
ПЃm
k=1 (ПЃj , П€k )П€k в€’
m=1 (ПЃj , ПЃ
, j в€€ N,
в€ћ
jв€’1
ПЃj (s, t) в€’ k=1 (ПЃj , П€ВЇk )П€ВЇk в€’ m=1 (ПЃj , ПЃВЇm )ВЇ
ПЃm W (D)
that is,
j
в€ћ
(5.17)
def
ОІjk П€ВЇk (s, t) +
ПЃВЇj (s, t) =
в€—
ОІjm
ПЃm (s, t), j в€€ N.
m=1
k=1
Hence we have W (D) = S вЉ• S , and {П€ВЇ1 , П€ВЇ2 , В· В· В· , ПЃВЇ1 , ПЃВЇ2 , В· В· В· } constitutes
an orthonormal basis for W (D).
вЉҐ
6. A separated type solution of (Lv)(t) = f (t)
Theorem 4.2 tells us that finding a solution of (1.2) is equivalent to
finding a separated type solution of (4.13).
Theorem 6.1. Let О» = (О»1 , О»2 , В· В· В· ) and (О±1k , О±2k , В· В· В· ) be arbitrary
constant in 2 for each k в€€ N. With the same notation of (5.16),
в€ћ
(6.18)
в€ћ
i
О±ik f (tk )П€ВЇk (s, t) +
v(s, t) =
i=1 k=1
О»j ПЃВЇj (s, t)
j=1
def
is a solution of (4.13), where f (t) =
в€ћ
ВЇ
k=1 bk П†k (t).
Proof. Taking L of both sides of (6.18), we have
в€ћ
в€ћ
i
О±ik f (tk )(LП€ВЇi )(t) +
(Lv)(t) =
i=1 k=1
О»j (LВЇ
ПЃj )(t).
j=1
Let T be the same set as (3.6). For every tl в€€ T , we have
(LВЇ
ПЃj )(tl ) = (LВЇ
ПЃj , П†l )W21 = (ВЇ
ПЃj , Lв€— П†l )W (D) = (ВЇ
ПЃj , П€l )W (D) = 0.
282
Phil Ung Chung and Ying Zhen Lin
Hence
в€ћ
i
О±ik f (tk )(LП€ВЇi )(tl )
(Lv)(tl ) =
i=1 k=1
в€ћ
i
О±ik f (tk )(LП€ВЇi , П†l )W21
=
i=1 k=1
в€ћ
i
О±ik f (tk )(П€ВЇi , П€l )W (D) .
=
i=1 k=1
Multiplying both sides of the above equality by ОІnl and summing with
respect to l, (1 ≤ l ≤ n), we have, in the view of (5.16),
в€ћ
n
i
О±ik f (tk )(П€ВЇi , П€ВЇn )W21
ОІnl (Lv)(tl ) =
i=1 k=1
n
l=1
=
О±nk f (tk ).
k=1
We claim that(Lv)(tm ) = f (tm ) holds for all m в€€ N. For n = 1, it
is easy to show that (Lv)(t1 ) = f (t1 ). For induction, we assume that
(Lv)(tn ) = f (tn ) holds for n ≤ m. Since
m+1
m+1
О±m+1,k f (tk )
О±m+1,l (Lv)(tl ) =
l=1
k=1
and
m
m+1
О±m+1,l f (tl ) + О±m+1,m+1 (Lv)(tm+1 ) =
l=1
О±m+1,k f (tk ),
k=1
we have
(Lv)(tm+1 ) = f (tm+1 ).
Hence (Lv)(tm ) = f (tm ) holds for every tm в€€ T . Since T is dense in
[0, 1], we conclude (Lv)(t) = f (t) holds for all t в€€ [0, 1]. Therefore our
assertion is proved.
Lemma 6.2. If v(s, t) of (6.18) is expressible as a separated type
u(s)u(t), then we have
(i) v(t1 , t) = x0 П†1 u(t)
(ii) v(t, t) = u2 (t)
How to solve an infinite simultaneous system of quadratic equations
283
Theorem 6.3. If v(s, t) of (6.18) is expressible as a separated type
u(s)u(t), then we have
(6.19)
в€ћ
1
u(t) =
x0 П†1
def
where f (t) =
в€ћ
i
О±ik f (tk )П€ВЇi (t1 , t) +
i=1 k=1
О»j ПЃВЇj (t1 , t) ,
j=1
в€ћ
ВЇ
k=1 bk П†k (t).
Proof. We have, by virtue of (6.18),
в€ћ
в€ћ
i
О±ik f (tk )П€ВЇi (s, t) +
u(s)u(t) =
i=1 k=1
О»j ПЃВЇj (s, t).
j=1
Putting s = t1 and dividing both sides of the above by u(t1 ), we have
the required result by Lemma 6.1.
Remark : If we take partial sum of (6.19) to get an approximation
unm (t) of u(t), then we have
1
unm (t) =
x0 П†1
n
m
i
ОІik f (tk )П€ВЇi (t1 , t) +
i=1 k=1
О»j ПЃВЇj (t1 , t)
j=1
for each m, n в€€ N. In order to obtain unm , we have to determine the
values of О»1 , В· В· В· , О»m . To do so, it suffices to find О»1 , В· В· В· , О»m so that they
def
may minimize G = vnm (t, t) в€’ u2nm (t) 2W 1 , where vnm (t, t) is a partial
2
sum of (6.18) in correspondence with unm (t). Fortunately G is a biquadratic polynomial with respect to О»1 , В· В· В· , О»m , of which optimization
problem is familiar to us. In the present paper we converted problem for
finding an approximate solution of infinite system of quadratic equations
into problem for minimizing biquadratic polynomial. Running Mathematica 4.2 for a concrete example, it can be easily confirmed that our
result is effective.
References
[1] Ming Gen Cui & Bo Ying Wu. Reproducing Kernel Space and Numerical Analysis.
Ke Xue Chu Ban She, Bei Jing, 2004
284
Phil Ung Chung and Ying Zhen Lin
[2] T.Y. Li, T. Sauer, J.A. Yorke. Numerical solution of a class of deficient polynomial
system. SIAM Nuerical Math. 51(1991), 481–500
[3] T.Y.Li. On Chow mallet-Paret and Yorke homotopy for solving systems of polynomials. Bulletin of the Institute of Mathematics Acad.Sin. 11(1983), 433–437
Phil Ung Chung
Department of Mathematics
Kangwon National University
Chunchon, Kangwon 200–701, Korea
E-mail : puchung@kangwon.ac.kr
Ying Zhen Lin
Department of Mathematics
Harbin Institute of Technology, Weihai Campus
Weihai, San Dong 264209, P.R. of China
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