Residual error estimation for the linear difference equations system having constant coefficients under two-point boundary values

Günümüzde bilgisayar teknolojisindeki gelişmelere paralel olarak matematiksel problemlerin bilgisayarla çözülmesi önem kazanmıştır. Bilgisayarlar sadece reel sayıların bir alt kümesi olan rasyonel sayılarla çalıştığından bilgisayarla yapılan hesaplamalarda bazı hesap hataları meydana gelir. Bu çalışmada aşağıdaki ayrılabilir sınır şartlı iki-nokta sınır değer problemi için rezidü hata tahminine dikkat çekiyoruz: $\biggl \{ \begin {array} {ll} x(n+1)=Ax(n) \\ Lx(n_0)= \phi; Rx(n_1)= \psi; \{n:n,n_o,n_1\in Z,n_o\leq n \leq n_1\} \end {array}$ Burada A, L ve R matrisleri sırasıyla NxN, kxN ve (N-k)xN tipinde reel matrisler, $\phi$ ve $\psi$ sırasıyla N ve N-k bileşenli reel kolon vektörleridir. Bilindiği gibi bu problemin rezidü vektörü f(n) = y(n+l) - Ay(n) şeklinde verilebilir. Burada $Ly(n_0) = \tilde{\phi}$, $Ry(n_1)=\tilde{\psi}$ dir. Böylece aşağıdaki problemi elde ederiz: $\biggl \{ \begin {array} {ll} y(n+1)=Ay(n)+f(n) \\ Ly(n_0)= \tilde {\phi}; Ry(n_1)= \tilde {\psi}; \{n:n,n_o,n_1\in Z,n_o\leq n \leq n_1\} \end {array}$ Burada y(n) çözümü verilen problemin bilgisayarda hesaplanan çözümüdür.

Sabit katsayılı lineer iki nokta sınır değerli fark denklem sistemi için kalıntı hata tahmini

Nowadays solving mathematical problems with computer have been taken important affords parallel to the developments of computer technologies. Since computers are using only rational subsets of the real numbers usually have some calculating errors. In this study taking attention to these we deal with the following two-point boundary value problem (TPBVP) with seperated boundary conditions specially with the residual error estimation : $\biggl \{ \begin {array} {ll} x(n+1)=Ax(n) \\ Lx(n_0)= \phi; Rx(n_1)= \psi; \{n:n,n_o,n_1\in Z,n_o\leq n \leq n_1\} \end {array}$ Where A NxN matrix, L kxN matrix and R (N-k)xN matrix are real matrices, $\phi$ and $\psi$ are real column vectors of N and N-k orderly. It is known that the residue of this problem can be given as f(n) = y(n+1) - Ay(n) where $Ly(n_0) = \tilde{\phi}$, $Ry(n_1)=\tilde{\psi}$. Therefore we obtain the following problem: $\biggl \{ \begin {array} {ll} y(n+1)=Ay(n)+f(n) \\ Ly(n_0)= \tilde {\phi}; Ry(n_1)= \tilde {\psi}; \{n:n,n_o,n_1\in Z,n_o\leq n \leq n_1\} \end {array}$ Here y(n) is the computed solution by computers of the given problem.

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