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# Tangent Linear Validation

## 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 p2.

### 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 …