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`
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.

** Next:** Same curvature condition.
** Up:** Spline interpolation
** Previous:** Interpolation and Continuity.
** Contents**
Adrian Sandu
2001-08-26