Last edited [[Timestamp]]
= ''' Tangent Linear Validation ''' =

[[PageOutline]]

== Principle ==
The tangent validation test checks that the Tangent Linear model '''L''' is a first order approximation of the model '''M'''. We can then write a second order Taylor expansion as follows:

{{{
M(X0+p.dX) = M(X0) + L(p.dX) + e
}}}
with,
{{{
X0: initial state vector
dX: perturbation state vector
p: scale factor
e: residu 
}}}
when the factor '''p''' tends to zero we have:
{{{
-1- M(X0+p.dX) - M(X0) tends to L(p.dX)
and
-2- e is equivalent to O(p^2), means Norm(e,.) <= K.p^2 (with K a constant)
}}}

This means that for a given '''p0''', for p smaller than p0 the error of the Linear Tangent with respect to the model is decreasing as p^2.

=== First order validation ===
Let's note 
{{{
Np = M(X0+p.dX) - M(X0)
Lp = L(p.dX)
}}}
the first order validation checks:
{{{
for a given p0, for all p>p0:
Ep = Norm(Np,.)/Norm(Lp,.) tends to 1
}}}

=== Second order validation ===
the second order validation ensure that the residu is controlled. We check then:
{{{
for a given p0, for all p>p0:
Rp = Norm(Np - Lp,.) behaves as p^2
}}}

note that we can also perform the following equivalent tests:
|| test || expression             || behaviour ||
||b     ||Norm((Np - Lp)/Lp,.)    || p         ||
||c     ||Norm((Ep - 1,.)/p       || constant  ||
||d     ||Norm((Ep - Lp/(p.Lp),.) || constant  ||

== Implementation ==

... work in progress ...