The Jacobian matrix is declared sparse if more than half values are zeros.

Example for the system B1.

Note that a procedure for the evaluation of the Jacobian matrix (d f/d y)(y(t)) must be constructed, especially for the problem classification (for y=y₀), but also if we apply an implicit numerical method which uses a Newton like procedure to solve the implicit equations. Therefore, we must find the mathematical expressions for all derivatives (d f_i/d y_j), i,j=1, ... , n. We can do this task transforming the Polish sequence associated to an f_i in an expression tree (each leaf is a number or a dependent variable, and each internal node is an arithmetic operator or a mathematical function), applying recursively the derivation rules on this tree, and, simultaneously creating, a new derivation tree; the final (stored) result is a Polish sequence of the derivative obtained from the derivation tree. The evaluation of the Jacobian matrix for a given n-dimensional vector is not a problem if we have a procedure for evaluating Polish sequences.