integral

Numerical integration

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Syntax

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

Description

example

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

example

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

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Open Script

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
Open Script

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
Open Script

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, specifying 12 decimal places of accuracy.

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

q2 =

  -1.000000000000010
Open Script

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
Open Script

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 =

    0.4597    0.7081    0.6633    0.4134    0.1433
Open Script

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 =

 -14.999999999998364

Related Examples

Input Arguments

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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.

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

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 Pair 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 single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'AbsTol',1e-12 sets the absolute error tolerance to approximately 12 decimal places of accuracy.

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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. The default value is 1e-10.

    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: 'AbsTol',1e-12 sets the absolute error tolerance to approximately 12 decimal places of accuracy.

Data Types: single | double

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. The default value is 1e-6.

    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: 'RelTol',1e-9 sets the relative error tolerance to approximately 9 significant digits.

Data Types: single | double

Array-valued function flag, specified as the comma-separated pair consisting of 'ArrayValued' and either false, true, 0, or 1. Set this flag to true 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: 'ArrayValued',true indicates that the integrand is an array-valued function.

Integration waypoints, specified as the comma-separated pair consisting of 'Waypoints' and a vector of real or complex numbers. Use waypoints to indicate any points in the integration interval that you would like the integrator to use. You can use waypoints to integrate efficiently across discontinuities of the integrand. Specify the locations of the discontinuities in the vector you supply.

You can specify waypoints when you want to perform complex contour integration. If xmin, xmax, or any entry of the waypoints vector is complex, the integration is performed over a sequence of straight line paths in the complex plane.

Example: 'Waypoints',[1+1i,1-1i] specifies two complex waypoints along the interval of integration.

Data Types: single | double
Complex Number Support: Yes

More About

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Tips

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

  • 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 specify a complex value for xmin, xmax, or any waypoint, all of your limits and waypoints must be finite.

  • 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.

See Also

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Introduced in R2012a

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