`cjacobian`

Jacobian of a multivariate, vector-valued function using the central difference approximation.

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Syntax
Description
Input/Output Parameters
Note
Example
See also

Syntax

J = cjacobian(f,x0)
J = cjacobian(f,x0,h)

Description

J = cjacobian(f,x0) numerically evaluates the Jacobian of $\mathbf{f}$ with respect to $\mathbf{x}$ at $\mathbf{x}=\mathbf{x}_{0}$ using the central difference approximation with a default relative step size of $h=\varepsilon^{1/3}$ , where $\varepsilon$ is the machine zero.

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

Input/Output Parameters

Variable Symbol Description Format

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

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

h $h$ (OPTIONAL) relative step size (defaults to $h=\varepsilon^{1/3}$) 1×1
double

Output J $\mathbf{J}(\mathbf{x}_{0})=\dfrac{\partial\mathbf{f}}{\partial\mathbf{x}}\bigg\rvert_{\mathbf{x}=\mathbf{x}_{0}}$ Jacobian of $\mathbf{f}$ with respect to $\mathbf{x}$ , evaluated at $\mathbf{x}=\mathbf{x}_{0}$ m×n
double

Note

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

Example

Approximate the Jacobian of

$\mathbf{f}(\mathbf{x})=\pmatrix{x_{1} \cr 5x_{3} \cr 4x_{2}^{2}-2x_{3} \cr x_{3}\sin{x_{1}}}$

at $\mathbf{x}=\mathbf{x}_{0}=(5,6,7)^{T}$ using the cjacobian function, and compare the result to the true result of

$\mathbf{J}(\mathbf{x}_{0})=\pmatrix{1 & 0 & 0 \cr 0 & 0 & 5 \cr 0 & 48 & -2 \cr 7\cos{(5)} & 0 & \sin{(5)}}$

Approximating the Jacobian,

f = @(x) [x(1);5*x(3);4*x(2)^2-2*x(3);x(3)*sin(x(1))];
x0 = [5;6;7];
J = cjacobian(f,x0)

J =

    1.0000         0         0
         0         0    5.0000
         0   48.0000   -2.0000
    1.9856         0   -0.9589

Calculating the error,

error = J-[1,0,0;0,0,5;0,48,-2;7*cos(5),0,sin(5)]

error =

   1.0e-09 *

   -0.0009         0         0
         0         0   -0.0289
         0    0.1248   -0.0471
   -0.4423         0    0.0087

NOTE: The function and its corresponding Jacobian are from an example on Wikipedia.

	Variable	Symbol	Description	Format
Input	`f`	$\mathbf{f}(\mathbf{x})$	multivariate, vector-valued function ( $\mathbf{f}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ )	1×1 function_handle
	`x0`	$\mathbf{x}_{0}$	evaluation point	n×1 double
	`h`	$h$	(OPTIONAL) relative step size (defaults to $h=\varepsilon^{1/3}$)	1×1 double
Output	`J`	$\mathbf{J}(\mathbf{x}_{0})=\dfrac{\partial\mathbf{f}}{\partial\mathbf{x}}\bigg\rvert_{\mathbf{x}=\mathbf{x}_{0}}$	Jacobian of $\mathbf{f}$ with respect to $\mathbf{x}$ , evaluated at $\mathbf{x}=\mathbf{x}_{0}$	m×n double

cjacobian