`fgradient`

Gradient of a multivariate, scalar-valued function using the forward difference approximation.

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Syntax
Description
Input/Output Parameters
Note
Example
See also

Syntax

g = fgradient(f,x0)
g = fgradient(f,x0,h)

Description

g = fgradient(f,x0) numerically evaluates the gradient of $f$ with respect to $\mathbf{x}$ at $\mathbf{x}=\mathbf{x}_{0}$ using the forward difference approximation with a default relative step size of $h=\sqrt{\varepsilon}$ , where $\varepsilon$ is the machine zero.

g = fgradient(f,x0,h) numerically evaluates the gradient of $f$ with respect to $\mathbf{x}$ at $\mathbf{x}=\mathbf{x}_{0}$ using the forward difference approximation with a user-specified relative step size $h$ .

Input/Output Parameters

Variable Symbol Description Format

Input f $f(\mathbf{x})$ multivariate, scalar-valued function ( $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ ) 1×1
function_handle

x0 $\mathbf{x}_{0}$ evaluation point n×1
double

h $h$ (OPTIONAL) relative step size (defaults to $h=\sqrt{\varepsilon}$ ) 1×1
double

Output g $\mathbf{g}(\mathbf{x}_{0})=\nabla f(\mathbf{x}_{0})$ gradient of $f$ with respect to $\mathbf{x}$ , evaluated at $\mathbf{x}=\mathbf{x}_{0}$ n×1
double

Note

This function requires evaluations of $f(\mathbf{x})$ .

Example

Approximate the gradient of $f(\mathbf{x})=x_{1}^{5}+\sin^{3}{x_{2}}$ at $\mathbf{x}=\mathbf{x}_{0}=(5,8)^{T}$ using the fgradient function, and compare the result to the true result of

$\nabla f(\mathbf{x}_{0})=\pmatrix{3125\cr3\sin^{2}{(8)}\cos{(8)}}$

Approximating the gradient,

f = @(x) x(1)^5+sin(x(2))^3;
x0 = [5;8];
g = fgradient(f,x0)

g =

   1.0e+03 *

    3.1250
   -0.0004

Calculating the error,

error = g-[3125;3*sin(8)^2*cos(8)]

error =

   1.0e-03 *

    0.1119
   -0.0004

	Variable	Symbol	Description	Format
Input	`f`	$f(\mathbf{x})$	multivariate, scalar-valued function ( $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ )	1×1 function_handle
	`x0`	$\mathbf{x}_{0}$	evaluation point	n×1 double
	`h`	$h$	(OPTIONAL) relative step size (defaults to $h=\sqrt{\varepsilon}$ )	1×1 double
Output	`g`	$\mathbf{g}(\mathbf{x}_{0})=\nabla f(\mathbf{x}_{0})$	gradient of $f$ with respect to $\mathbf{x}$ , evaluated at $\mathbf{x}=\mathbf{x}_{0}$	n×1 double

fgradient