Cubic Spline of Hermit

One calls cubic Spline of Hermit a spline of degree 3, named thus in homage to Charles Hermite, and whose each Polynôme $P\_i\; (X)$ is in the following form:

$P\_i\; \backslash \; in\; Vect\; \backslash \; \{h\_\; \{00\},\; h\_\; \{10\},\; h\_\; \{01\},\; h\_\; \{11\}\; \backslash \}\; \backslash ,$

with

$h\_\; \{00\}\; (T)\; =\; 2t^3-3t^2+1\; \backslash ,\; \backslash !$

$h\_\; \{10\}\; (T)\; =\; t^3-2t^2+t\; \backslash ,\; \backslash !$

$h\_\; \{01\}\; (T)\; =\; -2t^3+3t^2\; \backslash ,\; \backslash !$

$h\_\; \{11\}\; (T)\; =\; t^3-t^2\; \backslash ,\; \backslash !$

to give the polynomial ace

$\backslash \; mathbf\; \{p\}\; (T)\; =\; h\_\; \{00\}\; (T)\; \backslash \; mathbf\; \{p\}\; \_0\; +\; h\_\; \{10\}\; (T)\; \backslash \; mathbf\; \{m\}\; \_0\; +\; h\_\; \{01\}\; (T)\; \backslash \; mathbf\; \{p\}\; \_1\; +\; h\_\; \{11\}\; (T)\; \backslash \; mathbf\; \{m\}\; \_1.$

Under this writing, it is possible to see that the polynomial p checks:

$p\; (0)\; =\; \backslash \; mathbf\; \{p\}\; \_0\; \backslash ,\; \backslash !$

$p\; (1)\; =\; \backslash \; mathbf\; \{p\}\; \_1\; \backslash ,\; \backslash !$

$p\text{'}\; (0)\; =\; \backslash \; mathbf\; \{m\}\; \_0\; \backslash ,\; \backslash !$

$p\text{'}\; (1)\; =\; \backslash \; mathbf\; \{m\}\; \_1\; \backslash ,\; \backslash !$

The cubic splines of Hermit are thus a convenient manner to build a polynomial of degree possible bottom interpolating a function in 2 points with its tangent S.

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