# Special Polynomials (Chebyshev) (G Dataflow)

Last Modified: June 25, 2019

Evaluates the Chebyshev polynomial of a given order.  ## x

Real number.

Default: 0 ## order

Nonnegative order of the Chebyshev polynomial.

Default: 0 ## 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 ## Chebyshev polynomial

Value of the n th Chebyshev polynomial at x. ## 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 Chebyshev Polynomial

The following equation defines the Chebyshev polynomial:

${T}_{n}\left(x\right)=\mathrm{cos}\left(n\text{}\text{}\text{}{\mathrm{cos}}^{-1}\left(x\right)\right)$

where

• x is any real number.
• n is the nonnegative order of the Chebyshev polynomial. Note

The result of this definition does not look like a polynomial at first glance, but you can use trigonometric rules to show that T n is a polynomial of degree n in the variable x.

T n (x) forms the base of the Chebyshev approximation. For ij, the following equation gives the Chebyshev approximation.

${\int }_{-1}^{1}\frac{{T}_{i}\left(x\right){T}_{j}\left(x\right)}{\sqrt{1-{x}^{2}}}dx=0$

All T n (x) form an orthogonal system over the weight function

$\frac{1}{\sqrt{1-{x}^{2}}}$

The following illustration shows the four Chebyshev polynomials of order n = 0, 1, 2, 3. Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported

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