Numerical integration - MATLAB integral- MathWorks Deutschland (2024)

Numerical integration

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Syntax

q = integral(fun,xmin,xmax)

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

Description

example

q = integral(fun,xmin,xmax) numericallyintegrates function fun from xmin to xmax usingglobal adaptive quadrature and default error tolerances.

example

q = integral(fun,xmin,xmax,Name,Value) specifiesadditional options with one or more Name,Value pairarguments. For example, specify 'WayPoints' followedby a vector of real or complex numbers to indicate specific pointsfor the integrator to use.

Examples

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Improper Integral

Open Live 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

Parameterized Function

Open Live Script

Create the function $f (x) = 1 / (x^{3} - 2 x - c)$ 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

Open Live 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 longq1 = 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

Open Live Script

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

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

Open Live Script

Create the vector-valued function $f (x) = [\sin x, \sin 2 x, \sin 3 x, \sin 4 x, \sin 5 x]$ 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

Open Live Script

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

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

Input Arguments

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`fun` — Integrand
function handle

Integrand, specified as a function handle, which defines thefunction 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 (finiteor infinite) scalar value or a complex (finite) scalar value. If either xmin or xmax arecomplex, then integral approximates the pathintegral from xmin to xmax overa 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 arecomplex, integral approximates the path integralfrom xmin to xmax over a straightline path.

Data Types: double | single
Complex Number Support: Yes

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

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 worktogether. integral might satisfy the absoluteerror tolerance or the relative error tolerance, but not necessarilyboth. 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 worktogether. integral might satisfy the relativeerror tolerance or the absolute error tolerance, but not necessarilyboth. 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 “VectorizedAdaptive Quadrature in MATLAB^®,” Journalof Computational and Applied Mathematics, 211, 2008, pp.131–140.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

You must enable support for variable-size arrays.
The integral function does not support function handles that return sparse matrix output.

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.

Version History

Introduced in R2012a

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Numerical integration - MATLAB integral - MathWorks Deutschland (2024)

Syntax

Description

Examples

Improper Integral

Parameterized Function

Singularity at Lower Limit

Complex Contour Integration Using Waypoints

Vector-Valued Function

Improper Integral of Oscillatory Function

Input Arguments

`fun` — Integrand
function handle

`xmin` — Lower limit of x
real number | complex number

`xmax` — Upper limit of x
real number | complex number

Name-Value Arguments

`AbsTol` — Absolute error tolerance
`1e-10` (default) | nonnegative real number

`RelTol` — Relative error tolerance
`1e-6` (default) | nonnegative real number

`ArrayValued` — Array-valued function flag
`false` or `0` (default) | `true` or `1`

`Waypoints` — Integration waypoints
vector

Tips

References

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Thread-Based Environment
Run code in the background using MATLAB® `backgroundPool` or accelerate code with Parallel Computing Toolbox™ `ThreadPool`.

Version History

See Also

Topics

MATLAB-Befehl

Americas

Europe

Asia Pacific

References

Numerical integration - MATLAB integral - MathWorks Deutschland (2024)

Syntax

Description

Examples

Improper Integral

Parameterized Function

Singularity at Lower Limit

Complex Contour Integration Using Waypoints

Vector-Valued Function

Improper Integral of Oscillatory Function

Input Arguments

fun — Integrand function handle

xmin — Lower limit of x real number | complex number

xmax — Upper limit of x real number | complex number

Name-Value Arguments

AbsTol — Absolute error tolerance 1e-10 (default) | nonnegative real number

RelTol — Relative error tolerance 1e-6 (default) | nonnegative real number

ArrayValued — Array-valued function flag false or 0 (default) | true or 1

Waypoints — Integration waypoints vector

Tips

References

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

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

Version History

See Also

Topics

MATLAB-Befehl

Americas

Europe

Asia Pacific

References

`fun` — Integrand
function handle

`xmin` — Lower limit of x
real number | complex number

`xmax` — Upper limit of x
real number | complex number

`AbsTol` — Absolute error tolerance
`1e-10` (default) | nonnegative real number

`RelTol` — Relative error tolerance
`1e-6` (default) | nonnegative real number

`ArrayValued` — Array-valued function flag
`false` or `0` (default) | `true` or `1`

`Waypoints` — Integration waypoints
vector

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Thread-Based Environment
Run code in the background using MATLAB® `backgroundPool` or accelerate code with Parallel Computing Toolbox™ `ThreadPool`.