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Interface jam.operator.BiLinearOperatorHessian
- public interface interface BiLinearOperatorHessian
- extends JamObject, BiLinearOperatorAdjoint, hasHessian
BiLinearOperatorHessian is a BiLinearOperatorAdjoint
in which the derivatives, gradients, and hessian have been implemented.
The derivative of a bilinear operator B(x,y) at a point [x0 y0]'
(where we consider B(x,y) as a non-linear (unary) operator of [x y]')
is the row operator:
[B(.,y0) B(x0,.)]
That is:
Derivative at [x0 y0]' acting on [x y]' = B(x,y0) + B(x0,y)
The Hessian of a bilinear operator considered as a unary operator is
very simple:
Hessian([x y]', [v w]') = B(x,w) + B(v,y)
The Hessian is independent of its place of evaluation ([x0 y0]').
In fact, not surprisingly, the full Taylor expansion of B ends with
the Hessian, i.e. there are no terms higher than second order.
The bilinear property of B gives us the Taylor expansion immediately:
B(x+x0, y+y0) = B(x0,y0) + {B(x,x0) + B(x0,y)} + (1/2){2B(x,y)}
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