The LinearMatrixConsistency procedure

This procedure should be used in conjunction with RegularMatrix. It should be used only if the matrix includes rows or columns in which a variable appears with degree 1 in at least 2 elements of a row or a column. Its syntax is:

 
 LinearMatrixConsistency(A,VAR,vars,row,context,rohn,
                        Func,FuncKA,FuncAK,name)

The parameters are:

• A: the considered matrix
• VAR: a list of the variable name that appears in the matrix
• vars: a list of variable names that appears linearly in the row or column of A
• row: 1 if the procedure look at the row of A, 2 for the column
• context: see description
• rohn: 1 if the Rohn consistency procedure is used in the program
• Func, FuncKA, FuncAK: the name of C++ procedures to compute respectively the matrix, the left conditioned matrix and the right conditioned matrix. If this procedure is used in conjunction with RegularMatrix these names are we have Func="F", FuncKA="AKA", FuncAK="AAK".
• name: the name of the C++ procedure that will be created in the file name.C

Linearity has to be understood in a very loose sense. For example if a row of the matrix is

 
x    xy    x+y

this row may be considered as linear in x in which case we have 3 linear elements in the row. If the row is considered as linear in y we have 2 linear elements in the row. Note that the linearity is checked according to the order provided in Vars.

This procedure may be used for the matrix A, the conditioned matrix KA or the conditioned matrix AK. Context is used to indicate the choice according to the following code:

• 0: for A only
• 1: for KA only
• 2: for AK only
• 3: for A, KA, KA
• 4: for AK, KA
• 5: for A, KA
• 6: for A, AK

The complexity of this procedure is approximately $\prod (2^m_i)$ where $m_i$ is is the number of linear terms in the elements of row or column i of A. Hence it should not be be used with a large Vars. The use of Rohn matrix may also be expensive as it requires to calculate $2^(2n-1)$ scalar determinant, with $n$ the dimension of the matrix A.

For example we may use

 
LinearMatrixConsistency(A,[x,y],[x],1,5,"F","AKA","","Simp"):

In this example we will consider the row of matrix A and its elements that are linear in x. The procedure will be used for both the matrix A and the conditioned matrix KA (hence this procedure should be used with the parameter cond of RegularMAtrix set to 1 or 3. Note that we assume here that the conditioning matrix is ${\tt A}^{-1}(Mid(X))$.