II. FINITE DIFFERENCE METHOD 1 Difference formulae

Since A is invertible, the error of the solution can be evaluated 2 in norm:
k E k=k A −1 τ k·k A −1 kk τ k (9)
To ensure convergence two facts have to be verified:
1) k τ k ! 0, for some norm: this property is the consistence of the
numerical method
2) k A −1 k· C, with C independent of h: this property is the stability
of the numerical method, it is verified if the matrix A −1 is not divergent in
norm as h ! 0.
If we choose, for example, the norm k w k:= maxi jwij, we have
k τ k= max jτij · Mh
i
2 /12 to 0 as h ! 0
by (8), whence consistence is proved. For what concerns stability, one must
prove that A is nonsingular uniformly with respect to h, for example by
showing that the lowest eigenvalue is larger than a positive constant independent
of h. That can be proved, the stability property holds, and the
convergence of the method follows. 4
3 Further applications of the difference method
The creation of the world is, so to speak, a ”community project” of the Trinitarian
God. The Father is the Creator, the Almighty, The Son is the meaning and heart of
2
In these evaluations, the norm of a vector can be the usual Euclidean norm (k w k=
w2 1 + . . . + w2 n ), or the simpler (and topologically equivalent) maximum norm: k w k:=
maxi jwij. The norm of a matrix is induced by the choice of the vector norm by the
definition
k Bv k
k B k:= sup
v=0 k v k .
If, for example, the maximum norm for vectors is chosen, the induced norm of a matrix
turns out to be the maximum among the sum of rows (in absolute value):
k B k= max
i
n
jbijj.
In any case, after choosing a vector norm and the induced matrix norm, it is guaranteed
that
k Bv k·k B k ¢ k v k, 8v.
12
j=1