integral

Numerical integration

Syntax

q = integral(fun,xmin,xmax)

q = integral(fun,xmin,xmax,Name,Value)

Description

q = integral(fun,xmin,xmax) numerically integrates function fun from xmin to xmax using global adaptive quadrature and default error tolerances.

q = integral(fun,xmin,xmax,Name,Value) specifies additional options with one or more Name,Value pair arguments. For example, specify 'WayPoints' followed by a vector of real or complex numbers to indicate specific points for the integrator to use.

Examples

Improper Integral

Create the function $$f(x)=e^{-x^2}(\ln x{)}^2$$.

fun = @(x) exp(-x.^2).*log(x).^2;

Evaluate the integral from x=0 to x=Inf.

q = integral(fun,0,Inf)

q = 1.9475

Parameterized Function

Create the function $$f(x)=1/(x^3-2x-c)$$ with one parameter, $$c$$.

fun = @(x,c) 1./(x.^3-2*x-c);

Evaluate the integral from x=0 to x=2 at c=5.

q = integral(@(x) fun(x,5),0,2)

q = -0.4605

See Parameterizing Functions for more information on this technique.

Singularity at Lower Limit

Create the function $$f(x)=\ln (x)$$.

fun = @(x)log(x);

Evaluate the integral from x=0 to x=1 with the default error tolerances.

format long
q1 = integral(fun,0,1)

q1 = 
  -1.000000010959678

Evaluate the integral again, this time with 12 decimal places of accuracy. Set RelTol to zero so that integral only attempts to satisfy the absolute error tolerance.

q2 = integral(fun,0,1,'RelTol',0,'AbsTol',1e-12)

q2 = 
  -1.000000000000010

Complex Contour Integration Using Waypoints

Create the function $$f(z)=1/(2z-1)$$.

fun = @(z) 1./(2*z-1);

Integrate in the complex plane over the triangular path from 0 to 1+1i to 1-1i to 0 by specifying waypoints.

q = integral(fun,0,0,'Waypoints',[1+1i,1-1i])

q = -0.0000 - 3.1416i

Vector-Valued Function

Create the vector-valued function $$f(x)=[\sin x,\sin 2x,\sin 3x,\sin 4x,\sin 5x]$$ and integrate from x=0 to x=1. Specify 'ArrayValued',true to evaluate the integral of an array-valued or vector-valued function.

fun = @(x)sin((1:5)*x);
q = integral(fun,0,1,'ArrayValued',true)

q = 1×5

    0.4597    0.7081    0.6633    0.4134    0.1433

Improper Integral of Oscillatory Function

Create the function $$f(x)=x^5{e}^{-x}\sin x$$.

fun = @(x)x.^5.*exp(-x).*sin(x);

Evaluate the integral from x=0 to x=Inf , adjusting the absolute and relative tolerances.

format long
q = integral(fun,0,Inf,'RelTol',1e-8,'AbsTol',1e-13)

q = 
 -14.999999999998360

Input Arguments

fun — Integrand

function handle

Integrand, specified as a function handle, which defines the function to be integrated from xmin to xmax.

For scalar-valued problems, the function y = fun(x) must accept a vector argument, x, and return a vector result, y. This generally means that fun must use array operators instead of matrix operators. For example, use .* (times) rather than * (mtimes). If you set the 'ArrayValued' option to true, then fun must accept a scalar and return an array of fixed size.

xmin — Lower limit of x

real number | complex number

Lower limit of x, specified as a real (finite or infinite) scalar value or a complex (finite) scalar value. If either xmin or xmax are complex, then integral approximates the path integral from xmin to xmax over a straight line path.

Data Types: double | single
Complex Number Support: Yes

xmax — Upper limit of x

real number | complex number

Upper limit of x, specified as a real number (finite or infinite) or a complex number (finite). If either xmin or xmax are complex, integral approximates the path integral from xmin to xmax over a straight line path.

Data Types: double | single
Complex Number Support: Yes

Name-Value Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: integral(fun,a,b,'AbsTol',1e-12) sets the absolute error tolerance to approximately 12 decimal places of accuracy.

AbsTol — Absolute error tolerance

1e-10 (default) | nonnegative real number

Absolute error tolerance, specified as the comma-separated pair consisting of 'AbsTol' and a nonnegative real number. integral uses the absolute error tolerance to limit an estimate of the absolute error, |q – Q|, where q is the computed value of the integral and Q is the (unknown) exact value. integral might provide more decimal places of precision if you decrease the absolute error tolerance.

Note

AbsTol and RelTol work together. integral might satisfy the absolute error tolerance or the relative error tolerance, but not necessarily both. For more information on using these tolerances, see the Tips section.

Example: integral(fun,a,b,'AbsTol',1e-12) sets the absolute error tolerance to approximately 12 decimal places of accuracy.

Data Types: single | double

RelTol — Relative error tolerance

1e-6 (default) | nonnegative real number

Relative error tolerance, specified as the comma-separated pair consisting of 'RelTol' and a nonnegative real number. integral uses the relative error tolerance to limit an estimate of the relative error, |q – Q|/|Q|, where q is the computed value of the integral and Q is the (unknown) exact value. integral might provide more significant digits of precision if you decrease the relative error tolerance.

Note

RelTol and AbsTol work together. integral might satisfy the relative error tolerance or the absolute error tolerance, but not necessarily both. For more information on using these tolerances, see the Tips section.

Example: integral(fun,a,b,'RelTol',1e-9) sets the relative error tolerance to approximately 9 significant digits.

Data Types: single | double

ArrayValued — Array-valued function flag

false or 0 (default) | true or 1

Array-valued function flag, specified as the comma-separated pair consisting of 'ArrayValued' and a numeric or logical 1 (true) or 0 (false). Set this flag to true or 1 to indicate that fun is a function that accepts a scalar input and returns a vector, matrix, or N-D array output.

The default value of false indicates that fun is a function that accepts a vector input and returns a vector output.

Example: integral(fun,a,b,'ArrayValued',true) indicates that the integrand is an array-valued function.

Waypoints — Integration waypoints

vector

Integration waypoints, specified as the comma-separated pair consisting of 'Waypoints' and a vector of real or complex numbers. Use waypoints to indicate points in the integration interval that you would like the integrator to use in the initial mesh:

• Add more evaluation points near interesting features of the function, such as a local extrema.

• Integrate efficiently across discontinuities of the integrand by specifying the locations of the discontinuities.

• Perform complex contour integrations by specifying complex numbers as waypoints. If xmin, xmax, or any entry of the waypoints vector is complex, then the integration is performed over a sequence of straight line paths in the complex plane. In this case, all of the integration limits and waypoints must be finite.

Do not use waypoints to specify singularities. Instead, split the interval and add the results of separate integrations with the singularities at the endpoints.

Example: integral(fun,a,b,'Waypoints',[1+1i,1-1i]) specifies two complex waypoints along the interval of integration.

Data Types: single | double
Complex Number Support: Yes

Tips

• The integral function attempts to satisfy:

  abs(q - Q) <= max(AbsTol,RelTol*abs(q))

  where q is the computed value of the integral and Q is the (unknown) exact value. The absolute and relative tolerances provide a way of trading off accuracy and computation time. Usually, the relative tolerance determines the accuracy of the integration. However if abs(q) is sufficiently small, the absolute tolerance determines the accuracy of the integration. You should generally specify both absolute and relative tolerances together.

• If you are specifying single-precision limits of integration, or if fun returns single-precision results, you might need to specify larger absolute and relative error tolerances.

References

[1] L.F. Shampine “Vectorized Adaptive Quadrature in MATLAB^®,” Journal of Computational and Applied Mathematics, 211, 2008, pp.131–140.

Extended Capabilities

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.