1549301742-The_Theory_of_Difference_Schemes__Samarskii

50 Basic Concepts of the Theory of Difference Schemes

3. If A*= A:'.'.: 0, then all the eigenvalues of A are nonnegative.


4. Any vector x E Rn is representable by the eigenvectors of any
   operator A = A*:
   n
   and llx/1^2 = ~
   k=l


5. Let A* =A> 0 (A is self-adjoint and nonnegative). Then

and

\II XII '.S II Ax II < Anll x II for all XE H,

where A 1 :'.'.: 0 and An :'.'.: 0 are, respectively, the smallest and greatest

eigenvalues of A. The norm of a self-adjoint nonnegative operator in the

space Rn is equal to its greatest eigenvalue: 11 A II = An-

6. If self-adjoint operators A and B are commuting (AB =BA), then
   they possess a common system of eigenvectors.


7. Let self-adjoint operators A and B be commuting (AB = BA).
   Then the operator AB possesses the same system of eigenvectors as the
   operators A and Band AAB (k) =AA (k) AB (k) , k = 1, 2, ... , n, where AA (k) , An (k) ,
   and A~~ are the kth eigenvalues of A, B and AB= BA, respectively. By
   the same token,
   dk) - dk) + ,(k)
   /\A+B - /\A /\B ·

2.2 DIFFERENCE APPROXIMATION OF ELEMENTARY

DIFFERENTIAL OPERATORS

I. Grids and grid functions. The composition of a difference scheme ap-
proximating a differential equation of interest amouts to performing the
following operations:

• to substitute the domain of discrete variation of an argument for
  the domain of continuous variation;


• to replace a differential operator by some difference operator and
  impose difference analogs of boundary conditions and initial data.
  Following these procedures, we are led to a system of algebraic equations,
  thereby reducing numerical solution of an initial (linear) differential equa-
  tion to solving an algebraic syste1n.