As we have proved in [3], the function makeTab generates a multiset subdivision (see the output of the function makeTab in the previous section).  

Parameters:  2;

Univariate with Univariate

By using the subdivisions of a multiset with all equal elements, we are able to compute the univariate Faà di Bruno's formula, that is to expand

For example in order to expand  we call MFB(2) which gives  as output

Remark: the function MFB uses the function makeTab as we show in the following example.

Calling makeTab(2) we have The first block  has two elements: its cardinality gives the index of  . The second block  has one element: its cardinality gives the index of . The first element  in the first block  has degree 1: its degree gives the index of . Since in   we have two equal elements of degree 1, we get  This term must be multiplied with the integer (the multiplicity) that appears in each block (for example: 1 in  or n in ). The second block  contains only one element with degree 2. This gives just  At the end, the summation of the two terms returns .

Univariate with Multivariate

When in the multiset there are two or more different elements, the result of a call to MFB is different from the previous one. For esample MFB(1,1) generates the following output:  that represents the expansion of .

Indeed if in we replace

Remark: Similarly to what it happens for the univariate/univariate case,  the function MFB calls the function makeTab as we show in the following example.

Calling makeTab(1,1) we obtain

The first block  has one element: its cardinality gives the index of  The second block  has two elements: its cardinality gives the index of . In the blocks, two variables  are involved. This explains why we use two indexes in   elements. In particular we have:  

 we have , and from the block  we get . At the end, the summation of the two terms returns .

Univariate with Univariate 

Univariate with Multivariate 

The procedure joint provides a way to multiply the factors of formula (22) in [1].

Parameters:

Example:

If we need to multiply MFB(1,0) with MFB(1,1) we must follow the following steps:

   1) Call MFB(1,0) and

   2) For each output, make the following evalutions:   where i=1..n.

        So from MFB(1,0) = , by using the previous evaluations, we have

        From MFB(1,1) =

   3) Compute the following cefficients:

in the example and

   4) Multiply the two previous terms as follows:

This procedure gives a list of all subvectors having the same lenght of a vector V and such that their summation returns V.

Parameters: mkT(V, n),  n2;

Note: the output of

Example: Suppose to call mkt([1,1],2).

1) First the function makeTab is called: makeTab(1,1)=

2) From this list and for each block, we consider only the first subblock, that is

3) Since the first block  has cardinality one, we set  as we have asked for two blocks (that is two subvectors, note that the second parameter is n=2).  So we have  [1,0],[0,1]

4) The result is the list

5) At the end, in order to have all lists, we need to permute the subblocks for each block, that is:

This function is used in the master function.

(Umbral Multivariate Fa`a di Bruno’s Formula)

All previous steps are combined in this procedure. The UMFB function recalls the MFB function for the univariate and the multivariate case (parameter n=1) and allows us to construct the multivariate-multivariate case (parameter n>1).

Parameters: UMFB(V, n), V=

Esample: UMFB([1,3],2) =

Pseudocode:

     UMFB(V, n)
         if n = 1 then return( MFB(V) );
         S := 0;

         for v in mkT( V, n ) do
               S := S + joint(v) {with  }

        next for

        return(S);

      end:

Univariate with Univariate  we have the same result of MFB function.

Univariate with Multivariate

Multivariate with Multivariate