A Method of Extended Normal Equations for Tikhonov's Regulatization Problems with Differentiation Operator

This article is devoted to a new method of ill-conditioned linear algebraic systems solving with the help of differentiation operator. These problems appear while solving the first kind integral Fredholm equations. The most difficult thing about this method is that differential operator discrete analogue matrix is rank deficiency matrix. The generalized singular value decomposition methods are used to solve those problems. The approach has high computational complexity. This also leads to additional computational error. Our method is based on the original regularized problem transformation into equivalent augmented regularized normal equation system using differential operator discrete analogue. The problem of spectrum matrix investigation of augmented regularized normal equation system with rank deficiency differential operator discrete analogue matrix is very relevant nowadays. Accurate eigenvalue spectrum research for this problem is impossible. That is why we estimated spectrum matrix bounds. Our estimation is based on a well-known Courant–Fisher theorem. It is shown that estimated spectrum matrix bounds are rather accurate. The comparison between the proposed method and standard method based on the solving of normal system of equations is done. As shown in the paper, the condition number of normal method matrix is bigger than the condition number of augmented normal equations method matrix. In conclusion test problems description is given which proves our theoretical background.