Direct methods are constructed for solving quadratic programming problems with the tridiagonal Jacoby M-matrix and simple constraints. These methods are employed in block relaxation and splitting algorithms for solving two-dimensional mesh obstacle problems. The convergence of the algorithms is investigated and the results of numerical experiments are presented.We consider a difference approximation of the obstacle problem min<-|Vw| 2 dx-/udx>, X = {u^W ( 2\^)\u(x)^Q 9 xeQ}.(0.1)As a result, we arrive at a quadratic programming problem with a sparse symmetric positive definite matrix. To solve this problem in a one-dimensional case where the matrix is tridiagonal, we construct a direct method which requires O(N) arithmetic operations with N unknowns, i.e. an asymptotically (in N) optimal method. In a multidimensional case, we employ block relaxation and splitting methods based on successive solution of one-dimensional mesh problems. The quadratic programming problem minU(Aj^-(/,jok K = {yeR N |y,>0, i= 1,...,N} (0.2) y*K t 2 3 with the symmetric positive matrix A can be solved by well-known methods. Among them we can single out conjugate gradient methods developed for the sparse matrix [11] as well. In fact, these methods consist in determining a set of indices CD' = {i \ y t > 0 } at which the equalities (Ay -f\ = 0 hold. The conjugate gradient method is used to solve at each step a subsystem of the simultaneous equations Ay =/. Conjugate gradient methods converge in a finite number of iterations only due to the finite number of subspaces in the space R N but efficient estimates of the number of iterations, required for attaining a prescribed accuracy, are unknown. Methods are also available ([2,10] etc.) for solving a complementarity system [equivalent to (0.2)] tt^O, (Ay-f^O, 3^3;-A = 0, i=l,...,N (0.2') which exploit the fact that the matrix A belongs to classes (P) and (Z)[4], i.e. A is an M-matrix [16]. Unlike the conjugate gradient methods, these methods provide for a special construction of approximations to the set ω', with the exact solution y to problem (0.2) found at a step q < N.
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