Same slope condition.

The first derivative is

At the end points

where we used the relation several times. Changing in the first relation above gives

The ``same slope'' condition at all intermediate points

translates to

or, after some manipulations

These are relations to determine the unknown coefficients . Clearly, we need two more relations in order to arrive at a unique solution. Depending on these extra two relations, we obtain splines of different flavours (natural, B-splines, not-a-knot).

For ``natural spline'' the extra conditions are

For the remaining unknown values the relations () can be written as a linear system

The matrix of this system is tridiagonal and the system can be solved very efficiently.