Difference between revisions of "Gâteaux gradient"
From Encyclopedia of Mathematics
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| − | + | ''of a functional $ f $ | |
| + | at a point $ x _ {0} $ | ||
| + | of a Hilbert space $ H $'' | ||
| − | + | The vector in $ H $ | |
| + | equal to the [[Gâteaux derivative|Gâteaux derivative]] $ f _ {G} ^ { \prime } ( x _ {0} ) $ | ||
| + | of $ f $ | ||
| + | at $ x _ {0} $. | ||
| + | In other words, the Gâteaux gradient is defined by the formula | ||
| − | + | $$ | |
| + | f ( x _ {0} + h) = \ | ||
| + | f ( x _ {0} ) + | ||
| + | ( f _ {G} ^ { \prime } ( x _ {0} ), h) + \epsilon ( h), | ||
| + | $$ | ||
| − | and is simply known as the [[Gradient|gradient]]. The concept of the Gâteaux gradient may be extended to the case when | + | where $ \epsilon ( th)/t \rightarrow 0 $ |
| + | as $ t \rightarrow 0 $. | ||
| + | In an $ n $- | ||
| + | dimensional Euclidean space the Gâteaux gradient $ f _ {G} ^ { \prime } ( x _ {0} ) $ | ||
| + | is the vector with coordinates | ||
| + | |||
| + | $$ | ||
| + | \left ( | ||
| + | |||
| + | \frac{\partial f ( x _ {0} ) }{\partial x _ {1} } | ||
| + | \dots | ||
| + | |||
| + | \frac{\partial f ( x _ {0} ) }{\partial x _ {n} } | ||
| + | \right ) , | ||
| + | $$ | ||
| + | |||
| + | and is simply known as the [[Gradient|gradient]]. The concept of the Gâteaux gradient may be extended to the case when $ X $ | ||
| + | is a Riemannian manifold (finite-dimensional) or an infinite-dimensional Hilbert manifold and $ f $ | ||
| + | is a smooth real function on $ X $. | ||
| + | The growth of $ f $ | ||
| + | in the direction of its Gâteaux gradient is larger than in any other direction passing through the point $ x _ {0} $. | ||
Latest revision as of 19:42, 5 June 2020
of a functional $ f $
at a point $ x _ {0} $
of a Hilbert space $ H $
The vector in $ H $ equal to the Gâteaux derivative $ f _ {G} ^ { \prime } ( x _ {0} ) $ of $ f $ at $ x _ {0} $. In other words, the Gâteaux gradient is defined by the formula
$$ f ( x _ {0} + h) = \ f ( x _ {0} ) + ( f _ {G} ^ { \prime } ( x _ {0} ), h) + \epsilon ( h), $$
where $ \epsilon ( th)/t \rightarrow 0 $ as $ t \rightarrow 0 $. In an $ n $- dimensional Euclidean space the Gâteaux gradient $ f _ {G} ^ { \prime } ( x _ {0} ) $ is the vector with coordinates
$$ \left ( \frac{\partial f ( x _ {0} ) }{\partial x _ {1} } \dots \frac{\partial f ( x _ {0} ) }{\partial x _ {n} } \right ) , $$
and is simply known as the gradient. The concept of the Gâteaux gradient may be extended to the case when $ X $ is a Riemannian manifold (finite-dimensional) or an infinite-dimensional Hilbert manifold and $ f $ is a smooth real function on $ X $. The growth of $ f $ in the direction of its Gâteaux gradient is larger than in any other direction passing through the point $ x _ {0} $.
How to Cite This Entry:
Gâteaux gradient. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=G%C3%A2teaux_gradient&oldid=47150
Gâteaux gradient. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=G%C3%A2teaux_gradient&oldid=47150
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article