G Kalogeropulos - The investigation of the numerical solution of the linear syngular systems - страница 1

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ВІСНИК ЛЬВІВ. УН-ТУ Серія прикл. матем. інформ. 2009. Віт. 15. C. 149-156

VISNYKLVIV UNIV. Ser. Appl. Math. Inform. 2009. Is. 15. P. 149-156

УДК 512.83:512.64


G. Kalogeropulos*, O.Kossak**, K. Zachkovska**

Athens National Kapodistrian University, Athens, Greece; Ivan Franko National University of Lviv Universytetska str. 1, Lviv, 79000, e-mail: olhakossak@yahoo.com

In this paper the investigation of the linear singular systems are under consideration. We assume that matrixes and vectors are rather smooth and depend on time. Thanks to the Moore-Penrose generalized inverse we can write dawn the Cauchy problem and finite difference scheme for singular systems of ordinary differential equations. The unique smooth solution of the initial problem exists. For finding solution we used the first accuracy order fully implicit finite difference Euler schemes (backward Euler schemes).

Key words: linear systems, systems of the differential equations, Moore-Penrose generalized inverse matrix, backward Euler schemes, matrix regular pencil.


Nowadays linear systems are wildly used in different spheres of science. Very often matrices of the linear differential operators are singular [1, 2]. In this paper the singular matrixes with the depending on t coefficient are under consideration. Such problem can be calculated by means of numerical methods. We assumed that matrices and vectors are rather smooth and unique smooth solution exists. In this paper the firs accuracy order fully implicit finite difference Euler schemes (backward Euler schemes) is used.

Let us consider first order linear system of the differential equations

A (t) x(t ) = B (t) x (t) + f (t), a< t <p, (1)

Cx(a) = a. (2)

Wherex(t Rn, matrices A(t Rm%n,B(t Rm%n,C(t)є Rm%n are input matrices


For the case C = E the problem (1.1) - (1.2) can be rewrite:

A (t) x(t ) = B (t) x (t) + f (t), a< t <в, (3) x(a) = a, (4)

The rows of matrix A(t) are linear dependent, but the couple of matrix A(t),B(t) construct the regular pensile [6].


Definition 1. The matrix A+ є Cmxn is called the Moore-Penrose generalized inverse of - A є Cnxm if:

AA+A = A , A+ = UA* = A*V ,

where U   V - some matrixes [3].

© Kalogeropulos G., Kossak O., Zachkovska K., 2009

The properties of the Moore-Penrose generalized inverse of - A є Cnxm :

1. (A*)+=(A+)*;

2. (A+)+ = A;

3. (AA+)* = AA+, (AA+)2 = AA+;

4. (A+A)* = A+A, (A+A)2 = A+A .

Definition 2. The couple of matrixes (A(t), B(t)) are called regular for a < t < в, if exists for t є [a, в] indeterminate c when det [B(t) - cA(t)] ^ 0 [3].

Definition   3.   The   performance   of  the   matrix A є Cn

rankA = r, r > 1 as

A = GF,

we called full -rank factorization of A, where G є Cmxr і F є Crxm [4]:

\fl1      f12      ... fin

A = \\aik





A = GF =




gm1 ..

g mr

fr1      fr 2

(r = rA

The investigation of the existence and uniqueness of the Moore-Penrose generalized inverse A+ give us the next theorem:

Theorem 1. The Moore-Penrose generalized inverse A+ can be calculated:

A+= F* (FF*)-1 (G*G)-1 G*, (5)


A = G F .

Lemma 1. When columns of matrix A є Cnxm are linearly independent or rankA = n , then E - A" A = 0 [3].

Corollary1. (from theorem 1) If columns of matrix  A є Cnxm   are linearly

independent ( rank A = n ), then A+ = (A*A)-1 A*.

Let us multiply (1) by A+, remembering that A+A= E.


Let us consider the problem (1.1)-(1.2) for C = E. Then the Cauchy problem can be

write down:

x'(t )= A+(t) B (t) x (t) + A+ (t)f (t) , (6)

x(a) = a. (7) From the uniqueness of the Cauchy problem solution (6) - (7), the problems (1)-(2) and (6)-(7)are equivalent [3]. Let us multiply (6) by

(E - A (t) A (t)


 (E - A (t) A (t )+)B (t) x (t ) = -(E - A (t) A (t )+)f (t).

If t = a:

[E- A(a)A+ (a)] B (a) a = -[E- A (a)A+ (a)] f (a).

To solve the problem (6)-(7), which is equivalent to initial       2), we will use fully implicit finite difference Euler schemes (backward Euler schemes) [5]:

= A+Bx + A+f,, i = 1,...,N; т = в-a, (8) T N

Xo = a, (9)


A+ = A+ (if), B, = B(iT), f = f(it), xt - the approximate solution of the (8)-(9) in the point t = it.

Let us show that solution xt of the scheme (8)-(9) fort 0, approximate the exact solution x(it).

Let vt = xt - x(it), then x i = vt + x(if). For vt we have got: vt =(E-tA+Bt )-1 v--1 +(E-tA+Bt )-1 x x[-(x(it) - x ((i -1) t)) + tA+Bix (iff + Tf (if)], (10)


Using formula

v0 = 0, i = 1,...,N, T--

0 N

x ((i - 1)t) = x (it)-fx' (if) + у x "(i6T), (11) where вє [0,1] and putting (11) into (10), we have got:

vt = Cv-1 +(f2l2)(Pi,


C =(E-tA+Btщ = C,x"((вт). Thanks to the smoothness of the matrixes A+, B and vector x", exist constants K and L which are independent on i = 1,..., N and т (from (0, т0)) such that

\\C\\ < 1 + Kt, \<p\ < L .

But in this case

\v,\ <(1 + Kf)vi J + f L, ||vj| = 0,

and correspondently


Let us note, as


(1 + Kt)s =(1 + KTfT/KT< ек(в-а) = M, Iv\ < LTiM < LM (в-а)т. From the last expression we can see that when т 0 :

1Ы1 0.

We can sea the convergence of the finite difference solution xt to exact solutionх(іт). Taking under consideration the Corollary1, using A+ = (A*A^) A* and (8), we have got:

N   ' (12)

A^AjX^LL = a*Bixi +      , і = 1,...,n; r =

And finely

Xi=(д*д.-та;ві )-1      +та;/і ), і=1,..., n (13)

Formula (13) is using for finding the numerical solution of the problem (1)-(2). 4. NUMERICAL EXAMPLES

Example 1. We shell consider Cauchy problem for linear system with matrixes coefficients depending on t:

A (t) x(t ) = B (t) x (t) + f (t),      а< t <в,

x (to)--

Matrixes are next:


(1     t Л


(0 0^


(    et \

A (t ) =

t  1 +12

,         в (t ) =


, f (t)=

(t - 1)et


j  1 +12 j


j t2J



0 < t < 1.

The exact solution is [3] and initial conditions correspondently are:


The results are shown in the fig1., 2., and 3. From the fig. 1, 2 we can see that exact solution coincide with approximate solution.

We can see from the fig. 1. and 2., the approximation to the exact solutions. This fact shows us the effective usage of our Cauchy scheme. On the fig. 3. we can sea that the max error is at the end of interval. Tab. 1 show us the error dependence from the amount of steps of discretization. As we can see, with the increasing of the amount of the steps of discretization the approximate solution converges to exact one.




























































1 1



















Fig. l.Time distribution of the function x1 = e'        Fig. 2.Time distribution of the function x2 = 0

Fig. 3. The error dependence from the discrimination step

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