gradient

Gradient vector of scalar function

Syntax

g = gradient(f,v)

g = gradient(f)

gM = gradient(fM,vM)

Description

g = gradient(f,v) finds the gradient vector of the scalar function f with respect to vector v in Cartesian coordinates. The input f is a function of symbolic scalar variables and the vector v specifies the scalar differentiation variables.

g = gradient(f) finds the gradient vector of the scalar function f with respect to a vector constructed from all symbolic scalar variables found in f. The order of variables in this vector is defined by symvar.

gM = gradient(fM,vM) finds the gradient vector of the scalar function fM with respect to vector vM in Cartesian coordinates. The input function fM is a function of symbolic matrix variables and the vector vM is a symbolic matrix variable of size 1-by-N or N-by-1.

Examples

Find Gradient of Function

The gradient of a scalar function f with respect to the vector v is the vector of the first partial derivatives of f with respect to each element of v.

Find the gradient vector of f(x,y,z) with respect to vector [x,y,z]. The gradient is a vector with these components.

syms x y z
f(x,y,z) = 2*y*z*sin(x) + 3*x*sin(z)*cos(y);
gradient(f,[x,y,z])

ans(x, y, z) = 
$$\left(\begin{array}{c}3\hspace{0.17em}\cos \left(y\right)\hspace{0.17em}\sin \left(z\right)+2\hspace{0.17em}y\hspace{0.17em}z\hspace{0.17em}\cos \left(x\right)\\ 2\hspace{0.17em}z\hspace{0.17em}\sin \left(x\right)-3\hspace{0.17em}x\hspace{0.17em}\sin \left(y\right)\hspace{0.17em}\sin \left(z\right)\\ 2\hspace{0.17em}y\hspace{0.17em}\sin \left(x\right)+3\hspace{0.17em}x\hspace{0.17em}\cos \left(y\right)\hspace{0.17em}\cos \left(z\right)\end{array}\right)$$

Plot Gradient of Function

Find the gradient of a function f(x,y), and plot it as a quiver (velocity) plot.

Find the gradient vector of f(x,y) with respect to vector [x,y]. The gradient is vector g with these components.

syms x y
f = -(sin(x) + sin(y))^2;
g = gradient(f,[x,y])

g = 
$$\left(\begin{array}{c}-2\hspace{0.17em}\cos \left(x\right)\hspace{0.17em}\left(\sin \left(x\right)+\sin \left(y\right)\right)\\ -2\hspace{0.17em}\cos \left(y\right)\hspace{0.17em}\left(\sin \left(x\right)+\sin \left(y\right)\right)\end{array}\right)$$

Now plot the vector field defined by these components. MATLAB® provides the quiver plotting function for this task. The function does not accept symbolic arguments. First, replace symbolic variables in expressions for components of g with numeric values. Then use quiver.

[X, Y] = meshgrid(-1:.1:1,-1:.1:1);
G1 = subs(g(1),[x y],{X,Y});
G2 = subs(g(2),[x y],{X,Y});
quiver(X,Y,G1,G2)

Gradient of Matrix Multiplication

Use symbolic matrix variables to define a matrix multiplication that returns a scalar.

syms X Y [3 1] matrix
A = Y.'*X

A = $$Y^{\mathrm{T}}\hspace{0.17em}X$$

Find the gradient of the matrix multiplication with respect to $$X$$.

gX = gradient(A,X)

gX = $$Y$$

Find the gradient of the matrix multiplication with respect to $$Y$$.

gY = gradient(A,Y)

gY = $$X$$

Gradient of Multivariable Function

Find the gradient of the multivariable function

$$f(x)={\sin }^2(x_{1,1})+{\sin }^2(x_{1,2})+{\sin }^2(x_{1,3})$$

with respect to the vector $$x=[x_{1,1},x_{1,2},x_{1,3}]$$.

Use symbolic matrix variable to express the function $$f$$ and its gradient in terms of the vector $$x$$.

syms x [1 3] matrix
f = sin(x)*sin(x).'

f = $$\sin \left(x\right)\hspace{0.17em}{\sin \left(x\right)}^{\mathrm{T}}$$

g = gradient(f,x)

g = $$2\hspace{0.17em}\left(\cos \left(x\right)\odot {\mathrm{I}}_3\right)\hspace{0.17em}{\sin \left(x\right)}^{\mathrm{T}}$$

To show the gradient in terms of the elements of $$x$$, convert the result to a vector of symbolic scalar variables using symmatrix2sym.

g = symmatrix2sym(g)

g = 
$$\left(\begin{array}{c}2\hspace{0.17em}\cos \left(x_{1,1}\right)\hspace{0.17em}\sin \left(x_{1,1}\right)\\ 2\hspace{0.17em}\cos \left(x_{1,2}\right)\hspace{0.17em}\sin \left(x_{1,2}\right)\\ 2\hspace{0.17em}\cos \left(x_{1,3}\right)\hspace{0.17em}\sin \left(x_{1,3}\right)\end{array}\right)$$

Alternatively, you can convert $$f$$ and $$x$$ to symbolic expressions of scalar variables and use them as inputs to the gradient function.

g = gradient(symmatrix2sym(f),symmatrix2sym(x))

g = 
$$\left(\begin{array}{c}2\hspace{0.17em}\cos \left(x_{1,1}\right)\hspace{0.17em}\sin \left(x_{1,1}\right)\\ 2\hspace{0.17em}\cos \left(x_{1,2}\right)\hspace{0.17em}\sin \left(x_{1,2}\right)\\ 2\hspace{0.17em}\cos \left(x_{1,3}\right)\hspace{0.17em}\sin \left(x_{1,3}\right)\end{array}\right)$$

Input Arguments

f — Scalar function of symbolic scalar variables

symbolic expression | symbolic function

Scalar function, specified as a symbolic expression or symbolic function that is a function of symbolic scalar variables.

Data Types: sym | symfun

v — Vector with respect to which you find gradient vector

symbolic vector

Vector with respect to which you find gradient vector, specified as a symbolic vector. By default, v is a vector constructed from all symbolic scalar variables found in f. The order of variables in this vector is defined by symvar.

If v is a scalar, gradient(f,v) = diff(f,v). If v is an empty symbolic object, such as sym([]), then gradient returns an empty symbolic object.

Data Types: sym

fM — Scalar function

symbolic expression of symbolic matrix variables

Scalar function, specified as a symbolic expression that is a function of symbolic matrix variables.

Data Types: symmatrix

vM — Vector with respect to which you find gradient vector

symbolic matrix variable

Vector with respect to which you find gradient vector, specified as a symbolic matrix variable of size 1-by-N or N-by-1.

Data Types: symmatrix

Output Arguments

g — Gradient vector

symbolic expression | symbolic function

Gradient vector, returned as a symbolic expression or symbolic function that is a function of symbolic scalar variables.

Data Types: sym | symfun

gM — Gradient vector

symbolic expression

Gradient vector, returned as a symbolic expression that is a function of symbolic matrix variables.

Data Types: symmatrix

More About

Gradient Vector

The gradient vector of f(x) with respect to the vector $$x=\left(x_1,x_2,\dots ,x_n\right)$$ is the vector of the first partial derivatives of f.

$${\nabla }_{x}f(x)=\left(\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\dots ,\frac{\partial f}{\partial x_n}\right)$$

Version History

Introduced in R2011b

R2021b: Compute gradient of symbolic matrix variables

The gradient function accepts input arguments of type symmatrix. For examples, see Gradient of Matrix Multiplication and Gradient of Multivariable Function.