Divided difference schemes

The question of continuity of the limit function itself was addressed by using difference schemes. Closely related is the divided difference scheme, which is based on the definition of the derivative at a point. 

We can keep going, taking higher and higher divided difference schemes, until we find a scheme whose difference scheme is not contractive. The last scheme which was continuous gives a lower bound on the Holder continuity of the original scheme. 

The procedure is identical for dual schemes. Indeed, if the original scheme is a primal scheme, its first divided difference scheme is a dual scheme. 

It is convenient to make a common notation for all arities by defining the symbol24 a = (1 — za)/a( 1 — z). Then the divided difference scheme is constructed by dividing by a. It is computationally convenient then to replace the condition that the /,XJ norm of the difference scheme be less than 1 by the equivalent condition that the Ly , norm of the divided difference scheme be less than the arity. 

The first divided difference scheme is [—1,1, 8, 8,1, —1]/8, whose difference scheme is [—1, 2, 6, 2, —1]/8. The largest row sum is 8/8 which is not strictly less than 1, and so we cannot assert from this that the first derivative is continuous. 

The divided difference scheme is four times the difference scheme ... 

Clearly this property can be extended to any number of steps taken at once, and to divided difference schemes. 

The next twist of the plot links the two strands encountered so far, using the ideas of the divided difference schemes, expressed in terms of -transforms, to make the upper bound eigenanalysis dramatically easier. 

The original scheme and its divided difference scheme are clearly closely linked, and an interesting question to ask is whether their eigenfactorisations are related in any transparent and useful way. 

Note that the unit eigencolumn vanishes in a puff of smoke, because its first differences are all zero. Yes, a column of zeroes is an eigenvector, but it is the trivial one, not to be considered beside the real ones. The number of eigencomponents of the divided difference scheme is therefore one less than the number in the original scheme. 

Once the eigenvalues are determined, it is always possible to get the eigenvectors from the original subdivision matrix. It is also possible to determine them in parallel with the eigenvalues by working back down the chain of divided difference schemes. 

Dual schemes are handled in exactly the same way. Each taking of a divided difference scheme switches either from primal to dual or the reverse, and so both primals and duals are intimately involved in any scheme. 

If the dth divided difference scheme Sd of some power of a scheme S has an /oo norm less than the arity of that power, then the difference scheme of the (d— l)th divided difference has an norm less than one, and so the (d— l)th divided difference scheme is contractive, and therefore converges. The limit curve of the original scheme has continuity of the (d— l)th derivative. 

If, however, we look at the second divided difference scheme 2((1 + z2)/2)2, we find a more complicated story. 

The second divided difference scheme is 3[1, l]/2 which does not have any factors of a and so we cannot make any statement about the second derivative continuity. The second derivative is probably not continuous. 

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