Etude des marchés d'assurance non-vie à l'aide d'équilibres de ...

tel-00703797, version 2 - 7 Jun 2012
3.2. Methods to solve nonlinear equations
where J denotes the Jacobian ∗ JacF . Let M N k (h) = F (zk) + J(zk)h be the model function.
At the kth iterate, the Newton step consists in solving the system M N k (dk) = 0. We get the
following equation
J(zk)dk = −F (zk). (3.12)
Note that there is no guarantee that J(zk) is nonsingular.
Another method consists in replacing the Jacobian by an approximate matrix, which will
be always invertible. The direction dk solves
Hkdk = −F (zk), (3.13)
where Hk is updated by a so-called quasi-Newton scheme. This is equivalent to a model
function M QN
k (h) = F (zk) + Hkh. A quasi-Newton scheme needs an iterative process to
update the approximate Jacobian from Hk to Hk+1. The choice of the matrix is large, since
there are n 2 terms. We can set first that
F (zk) = M QN
k+1 (zk+1 − zk) ⇔ Hk+1sk = yk,
where sk = zk+1 − zk and yk = F (zk+1) − F (zk). The latter equation is called the secant
equation. We could have considered a scheme to approximate directly the inverse of the
Jacobian by Wk with the secant equation sk = Wk+1yk. But still, we have an underdetermined
system n equations for n × n matrix.
Having no other property on the Jacobian of F (e.g. symmetry or positiveness), we generally
consider the matrix that makes the smallest possible change to the preceding Jacobian
according to the Frobenius norm † , see (Nocedal and Wright, 2006, Chap. 11). For an implementation
point of view, the smallest possible change feature reduces the linear algebra
computation work from O n 3 to O n 2 cost.
If we consider minimizing minH ||H −Hk||F for all matrix H verifying the secant equation,
we get the (good) Broyden scheme
Hk+1 = Hk + yk − Hksk
s T k sk
s T k ⇔ Wk+1 = Wk + (sk − Wkyk)y T k Wk
sT k Wkyk
,
using the Sherman-Morrison formula. Otherwise, if we minimize minW ||W − Wk||F for all
matrix W verifying the secant equation, then we will obtain the (bad) Broyden scheme
Hk+1 = Hk + (sk − Hkyk)y T k Hk
y T k Hksk
⇔ Wk+1 = Wk + sk − Wkyk
yT k yk
y T k .
According to Broyden (1965), this method appears often unsatisfactory in practice, so it will
be discused no further.
∗. The Jacobian of a function f is defined as usual by
⎛
∂f1 . . .
∂x1 ⎜ .
Jacf(x) = ⎜ .
⎝
.
. . .
And the ∇f denotes the transpose of the Jacobian.
∂f b
∂x1
∂f i
∂x j
∂f1
∂xa
.
∂f b
∂xa
⎞
⎟
⎠ (x).
†. The Frobenius norm (also called the Euclidean norm) for matrix A is defined as ||A||F =
i,j |aij|2 .
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