cderivative

Derivative of a univariate, vector-valued function using the central difference approximation.

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Contents

Syntax

df = cderivative(f,x0)
df = cderivative(f,x0,h)

Description

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

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

Input/Output Parameters

Variable Symbol Description Format
Input f univariate, vector-valued function () 1×1
function_handle
x0 evaluation point 1×1
double
h (OPTIONAL) relative step size (defaults to $h=\varepsilon^{1/3}$) 1×1
double
Output df derivative of with respect to , evaluated at m×1
double

Note

Example #1: Derivative of a scalar-valued function.

Approximate the derivative of $f(x)=x^{3}$ at $x=2$ using the cderivative function, and compare the result to the true result of $f'(2)=12$.

Approximating the derivative,

f = @(x) x^3;
df = cderivative(f,2)

df =

   12.0000

Calculating the error,

error = df-12

error =

   3.3153e-10

Example #2: Derivative of a vector-valued function.

Approximate the derivative of

$$\mathbf{f}(t)=\pmatrix{\sin{t}\cr\cos{t}}$$

at $t=1$ using the cderivative function, and compare the result to the true result of

$$\left.\frac{d\mathbf{f}}{dt}\right|_{t=1}=\pmatrix{\cos{(1)}\cr-\sin{(1)}}$$

Approximating the derivative,

f = @(t) [sin(t);cos(t)];
df = cderivative(f,1)

df =

    0.5403
   -0.8415

Calculating the error,

error = df-[cos(1);-sin(1)]

error =

   1.0e-10 *

   -0.1420
    0.1959

See also

fderivative | iderivative


Published with MATLAB® R2022a