`ihessian`

Hessian of a multivariate, scalar-valued function using the complex-step and central difference approximations.

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Syntax
Description
Input/Output Parameters
Note
Example
See also

Syntax

H = ihessian(f,x0)
H = ihessian(f,x0,hi,hc)

Description

H = ihessian(f,x0) numerically evaluates the Hessian of $f$ with respect to $\mathbf{x}$ at $\mathbf{x}=\mathbf{x}_{0}$ using a hybrid of complex-step and central difference approximations with default step sizes of $h_{i}=10^{-200}$ (an absolute step size) and $h_{c}=\varepsilon^{1/3}$ (a relative step size), respectively, where $\varepsilon$ is the machine zero.

H = ihessian(f,x0,hi,hc) numerically evaluates the Hessian of $f$ with respect to $\mathbf{x}$ at $\mathbf{x}=\mathbf{x}_{0}$ using a hybrid of complex-step and central difference approximations with user-specified step sizes $h_{i}$ and $h_{c}$ , respectively.

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

hi $h_{i}$ (OPTIONAL) step size for complex-step approximation (defaults to $h_{i}=10^{-200}$ ) 1×1
double

hc $h_{c}$ (OPTIONAL) relative step size for central difference approximation (defaults to $h_{c}=\varepsilon^{1/3}$ ) 1×1
double

Output H $\mathbf{H}(\mathbf{x}_{0})$ Hessian of $f$ with respect to $\mathbf{x}$ , evaluated at $\mathbf{x}=\mathbf{x}_{0}$ n×n
double

Note

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

Example

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

$\mathbf{H}(\mathbf{x}_{0})=\left.\pmatrix{20x_{1}^{3}x_{2} & 5x_{1}^{4} +3\sin^{2}{x_{2}}\cos{x_{2}} \cr 5x_{1}^{4}+3\sin^{2}{x_{2}}\cos{x_{2}} & 6x_{1}\sin{x_{2}}\cos^{2}{x_{2}}-3x_{1}\sin^{3}{x_{2}}}\right|_{(x_{1}, x_{2})=(5,8)}=\pmatrix{20(5)^{3}(8) & 5(5)^{4}+3\sin^{2}{(8)}\cos{(8)} \cr 5(5)^{4}+3\sin^{2}{(8)}\cos{(8)} & 6(5)\sin{(8)}\cos^{2}{(8)} -3(5)\sin^{3}{(8)}}$

Approximating the Hessian,

f = @(x) x(1)^5*x(2)+x(1)*sin(x(2))^3;
x0 = [5;8];
H = ihessian(f,x0)

H =

   1.0e+04 *

    2.0000    0.3125
    0.3125   -0.0014

Calculating the error,

error = H-[20*5^3*8,5*5^4+3*sin(8)^2*cos(8);5*5^4+3*sin(8)^2*cos(8),...
    6*5*sin(8)*cos(8)^2-3*5*sin(8)^3]

error =

   1.0e-05 *

    0.1036    0.0011
    0.0011    0.0049

	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
	`hi`	$h_{i}$	(OPTIONAL) step size for complex-step approximation (defaults to $h_{i}=10^{-200}$ )	1×1 double
	`hc`	$h_{c}$	(OPTIONAL) relative step size for central difference approximation (defaults to $h_{c}=\varepsilon^{1/3}$ )	1×1 double
Output	`H`	$\mathbf{H}(\mathbf{x}_{0})$	Hessian of $f$ with respect to $\mathbf{x}$ , evaluated at $\mathbf{x}=\mathbf{x}_{0}$	n×n double

ihessian