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| A '''gradient''' is a vector of partial derivatives. It describes the direction of steepest ascent for a differentiable function. | A '''gradient''' is a vector of [[Calculus/PartialDerivative|partial derivatives]]. It describes the direction of steepest ascent for a [[Calculus/Derivative|differentiable]] function. |
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| The gradient of function ''f'' is notated as ''∇f''. In terms of [[Calculus/PartialDerivatives|partial derivatives]], the gradient of ''f(x,,1,,, x,,2,,, ... x,,n,,)'' is: | The gradient of function ''f'' is notated as ''∇f''. In terms of [[Calculus/PartialDerivative|partial derivatives]], the gradient of ''f(x,,1,,, x,,2,,, ... x,,n,,)'' is: |
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| Note the assumption; it is not negligible. For example, ''(xy)/(x^2^ + y^2^)'' is partially derivable but is itself not totally derivable at point ''p = [0 0]''. Furthermore, it is not derivable if rotated; the [[LinearAlgebra/Basis|basis]] must be [[LinearAlgebra/Orthonormalization|orthonormal]]. | Note the assumption; it is not negligible. For example, ''(xy)/(x^2^ + y^2^)'' is partially derivable but is itself not totally derivable at point ''p = [0 0]''. Furthermore, it is not derivable if rotated; the [[LinearAlgebra/Basis|basis]] must be [[LinearAlgebra/Orthogonality|orthonormal]]. Also note that, technically, this is the [[LinearAlgebra/Transposition|transpose]] of the gradient; vectors are by default columns. === Relation to Jacobian Matrices === A [[Calculus/JacobianMatricesAndDeterminants|Jacobian matrix]] is a square matrix of partial derivatives. Consider a vector-valued function, such as a [[Calculus/ParametricEquation|parameterized]] function, but for simplicity let ''F(X) = <f,,1,,(X), f,,2,,(X)>'' where ''X'' is a vector as ''X = <x,,1,,``, x,,2,,``, ... x,,n,,>''. A gradient can be calculated for each of the component function. {{attachment:jacobian.svg}} |
Gradient
A gradient is a vector of partial derivatives. It describes the direction of steepest ascent for a differentiable function.
Notation
The gradient of function f is notated as ∇f. In terms of partial derivatives, the gradient of f(x1, x2, ... xn) is:
At a given point p, as long as the function f is differentiable at p, the gradient vector is:
Note the assumption; it is not negligible. For example, (xy)/(x2 + y2) is partially derivable but is itself not totally derivable at point p = [0 0]. Furthermore, it is not derivable if rotated; the basis must be orthonormal.
Also note that, technically, this is the transpose of the gradient; vectors are by default columns.
Relation to Jacobian Matrices
A Jacobian matrix is a square matrix of partial derivatives. Consider a vector-valued function, such as a parameterized function, but for simplicity let F(X) = <f1(X), f2(X)> where X is a vector as X = <x1, x2, ... xn>. A gradient can be calculated for each of the component function.
Usage
By setting a gradient to 0, critical points (local minima, local maxima, and inflections) can be calculated.
More generally, gradient descent can be used to estimate minima.