Numerically evaluate double integral — tiled method

specifies additional options with one or more `q`

= quad2d(`fun`

,`a,b`

,`c,d`

,`Name,Value`

)`Name,Value`

pair
arguments. For example, you can specify `'AbsTol'`

and
`'RelTol'`

to adjust the error thresholds that the algorithm
must satisfy.

Integrate

$$y\mathrm{sin}(x)+x\mathrm{cos}(y)$$

over $$-\pi \le x\le 2\pi $$ and $$0\le y\le \pi $$.

fun = @(x,y) y.*sin(x)+x.*cos(y); Q = quad2d(fun,pi,2*pi,0,pi)

Q = -9.8696

Compare the result to the true value of the integral, $$-{\pi}^{2}$$.

-pi^2

ans = -9.8696

Integrate the function

$${\left[{(x+y)}^{1/2}{(1+x+y)}^{2}\right]}^{-1}$$

over the region $$0\le x\le 1$$ and $$0\le y\le 1-x$$. This integrand is infinite at the origin (0,0), which lies on the boundary of the integration region.

fun = @(x,y) 1./(sqrt(x + y) .* (1 + x + y).^2 ); ymax = @(x) 1 - x; Q = quad2d(fun,0,1,0,ymax)

Q = 0.2854

The true value of the integral is $$\pi /4-1/2$$.

pi/4 - 0.5

ans = 0.2854

`quad2d`

`quad2d`

begins by mapping the region of integration to a rectangle. Consequently, it may have trouble integrating over a region that does not have four sides or has a side that cannot be mapped smoothly to a straight line. If the integration is unsuccessful, some helpful tactics are leaving `Singular`

set to its default value of `true`

, changing between Cartesian and polar coordinates, or breaking the region of integration into pieces and adding the results of integration over the pieces.

For instance:

fun = @(x,y)abs(x.^2 + y.^2 - 0.25); c = @(x)-sqrt(1 - x.^2); d = @(x)sqrt(1 - x.^2); quad2d(fun,-1,1,c,d,'AbsTol',1e-8,... 'FailurePlot',true,'Singular',false);

Warning: Reached the maximum number of function evaluations (2000). The result fails the global error test.

The failure plot shows two areas of difficulty, near the points `(-1,0)`

and `(1,0)`

and near the circle $${x}^{2}+{y}^{2}=0.25$$.

Changing the value of `Singular`

to `true`

will cope with the geometric singularities at `(-1,0)`

and `(1,0)`

. The larger shaded areas may need refinement but are probably not areas of difficulty.

Q = quad2d(fun,-1,1,c,d,'AbsTol',1e-8, ... 'FailurePlot',true,'Singular',true);

Warning: Reached the maximum number of function evaluations (2000). The result passes the global error test.

From here you can take advantage of symmetry:

Q = 4*quad2d(fun,0,1,0,d,'Abstol',1e-8,... 'Singular',true,'FailurePlot',true)

Q = 0.9817

However, the code is still working very hard near the singularity. It may not be able to provide higher accuracy:

Q = 4*quad2d(fun,0,1,0,d,'Abstol',1e-10,... 'Singular',true,'FailurePlot',true);

Warning: Reached the maximum number of function evaluations (2000). The result passes the global error test.

At higher accuracy, a change in coordinates may work better.

```
polarfun = @(theta,r) fun(r.*cos(theta),r.*sin(theta)).*r;
Q = 4*quad2d(polarfun,0,pi/2,0,1,'AbsTol',1e-10);
```

It is best to put the singularity on the boundary by splitting the region of integration into two parts:

Q1 = 4*quad2d(polarfun,0,pi/2,0,0.5,'AbsTol',5e-11); Q2 = 4*quad2d(polarfun,0,pi/2,0.5,1,'AbsTol',5e-11); Q = Q1 + Q2;

`fun`

— Function to integratefunction handle

Function to integrate, specified as a function handle. The function
`Z = fun(X,Y)`

must accept 2-D matrices
`X`

and `Y`

of the same size and
return a matrix `Z`

of corresponding values. Therefore, the
function must be vectorized (that is, you must use elementwise operators
such as `.^`

instead of matrix operators such as
`^`

). The inputs and outputs of the function must be
either single or double precision.

**Example: **`@(x,y) x.^2 - y.^2`

**Data Types: **`function_handle`

`a,b`

— scalars

*x* limits of integration, specified as scalars.

**Data Types: **`single`

| `double`

**Complex Number Support: **Yes

`c,d`

— scalars | function handles

*y* limits of integration, specified as scalars or
function handles. Each limit can be specified as a scalar or a function
handle. If the limits are specified as function handles, then they are
functions of the *x* limit of integration ```
ymin =
@x c(x)
```

and `ymax = @(x) d(x)`

. The function
handles `ymin`

and `ymax`

must accept
matrices and return matrices of the same size with the corresponding values.
The inputs and outputs of the functions must be either single or double
precision.

**Data Types: **`single`

| `double`

| `function_handle`

**Complex Number Support: **Yes

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`

.

`quad2d(@(x,y) x.*y.^2, 0, 1, 0, 2, 'AbsTol',1e-3)`

specifies the absolute tolerance for the integration as
`1e-3`

.`'AbsTol'`

— Absolute error tolerance`1e-5`

(default) | scalarAbsolute error tolerance, specified as the comma-separated pair
consisting of `'AbsTol'`

and a scalar.

`quad2d`

attempts to satisfy ```
ERRBND <=
max(AbsTol,RelTol*|Q|)
```

. This is absolute error control
when `|Q|`

is sufficiently small and relative error
control when `|Q|`

is larger. A default tolerance value
is used when a tolerance is not specified. The default value of
`AbsTol`

is 1e-5. The default value of
`RelTol`

is `100*eps(class(Q))`

.
This is also the minimum value of `RelTol`

. Smaller
`RelTol`

values are automatically increased to the
default value.

`'RelTol'`

— Relative error tolerance`100*eps(class(q))`

(default) | scalarRelative error tolerance, specified as the comma-separated pair
consisting of `'RelTol'`

and a scalar.

`quad2d`

attempts to satisfy ```
ERRBND <=
max(AbsTol,RelTol*|Q|)
```

. This is absolute error control
when `|Q|`

is sufficiently small and relative error
control when `|Q|`

is larger. A default tolerance value
is used when a tolerance is not specified. The default value of
`AbsTol`

is 1e-5. The default value of
`RelTol`

is `100*eps(class(Q))`

.
This is also the minimum value of `RelTol`

. Smaller
`RelTol`

values are automatically increased to the
default value.

`'MaxFunEvals'`

— Maximum number of evaluations of `fun`

`2000`

(default) | scalarMaximum number of evaluations of `fun`

, specified as
the comma-separated pair consisting of `'MaxFunEvals'`

and a scalar. Use this option to limit the number of times
`quad2d`

evaluates the function
`fun`

.

`'FailurePlot'`

— Toggle to generate failure plot`false`

or
`0`

(default) | `true`

or `1`

Toggle to generate failure plot, specified as the comma-separated pair
consisting of `'FailurePlot'`

and a numeric or logical
`1`

(`true`

) or
`0`

(`false`

). Set
`FailurePlot`

to `true`

or
`1`

to generate a graphical representation of the
regions needing further refinement when `MaxFunEvals`

is reached. No plot is generated if the integration succeeds before
reaching `MaxFunEvals`

. The failure plot contains
(generally) 4-sided regions that are mapped to rectangles internally.
Clusters of small regions indicate the areas of difficulty in the
integration.

`'Singular'`

— Toggle to transform boundary singularities`true`

or
`1`

(default) | `false`

or `0`

Toggle to transform boundary singularities, specified as the
comma-separated pair consisting of `'Singular'`

and a
numeric or logical `1`

(`true`

) or
`0`

(`false`

). By default,
`quad2d`

employs transformations to weaken
boundary singularities for better performance. Set
`'Singular'`

to `false`

or
`0`

to turn these transformations off, which can
provide a performance benefit on some smooth problems.

`q`

— Calculated integralscalar

Calculated integral, returned as a scalar.

`E`

— Error boundscalar

Error bound, returned as a scalar. The error bound provides an upper bound
on the error between the calculated integral *q* and the
exact value of the integral *I* such that *E* = | *q* -
*I* |.

[1] L.F. Shampine, "Matlab Program for Quadrature in 2D."
*Applied Mathematics and Computation.* Vol. 202, Issue 1, 2008,
pp. 266–274.

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

Generated code issues a warning if the size of the internal storage arrays is not large enough. If a warning occurs, as a workaround, you can try to divide the region of integration into pieces and sum the integrals over each piece.

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