# Quadrature (3D » Formula) (G Dataflow)

Performs 3D numerical integration using the adaptive Lobatto quadrature approach. You define the integrand using a formula.  ## integrand

Expression to integrate. The first, second, and third integral variables must be x, y, and z, respectively. ## upper limits

Upper limits of the integral. ### x upper limit

Upper limit of the first integral variable x.

Default: 1 ### y upper limit

Upper limit of the second integral variable y.

Default: 1 ### z upper limit

Upper limit of the third integral variable z.

Default: 1 ## lower limits

Lower limits of the integral. ### x lower limit

Lower limit of the first integral variable x.

Default: 0 ### y lower limit

Lower limit of the second integral variable y.

Default: 0 ### z lower limit

Lower limit of the third integral variable z.

Default: 0 ## tolerance

Accuracy of the quadrature. A smaller tolerance leads to a more accurate result but requires more computation time.

How Does Tolerance Affect the Accuracy of the Quadrature?

This node compares the difference between the 4-points and 7-points Lobatto quadratures on the interval and uses tolerance to terminate the calculation iteration. If the difference is less than tolerance, this node stops the calculation iteration and moves on to the next interval.

Default: 1E-05 ## error in

Error conditions that occur before this node runs.

The node responds to this input according to standard error behavior.

Standard Error Behavior

Many nodes provide an error in input and an error out output so that the node can respond to and communicate errors that occur while code is running. The value of error in specifies whether an error occurred before the node runs. Most nodes respond to values of error in in a standard, predictable way.

error in does not contain an error error in contains an error  If no error occurred before the node runs, the node begins execution normally.

If no error occurs while the node runs, it returns no error. If an error does occur while the node runs, it returns that error information as error out.

If an error occurred before the node runs, the node does not execute. Instead, it returns the error in value as error out.

Default: No error ## result

Integral result. ## error out

Error information.

The node produces this output according to standard error behavior.

Standard Error Behavior

Many nodes provide an error in input and an error out output so that the node can respond to and communicate errors that occur while code is running. The value of error in specifies whether an error occurred before the node runs. Most nodes respond to values of error in in a standard, predictable way.

error in does not contain an error error in contains an error  If no error occurred before the node runs, the node begins execution normally.

If no error occurs while the node runs, it returns no error. If an error does occur while the node runs, it returns that error information as error out.

If an error occurred before the node runs, the node does not execute. Instead, it returns the error in value as error out.

## Algorithm for Evaluating the Integral

This node evaluates the following integral:

$∫ z 0 z 1 ∫ y 0 y 1 ∫ x 0 x 1 f ( x , y , z ) d x d y d z$

where

• x1 is x upper limit
• x0 is x lower limit
• y1 is y upper limit
• y0 is y lower limit
• z1 is z upper limit
• z0 is z lower limit

To obtain high accuracy, this node divides an interval cube into sub-cubes when the integrand f(x, y, z) varies sharply.

Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported

Web Server: Not supported in VIs that run in a web application