Version:

Last Modified: February 27, 2020

Calculates the number of zeros of a real polynomial in a real interval. This node does not determine the values of the zeros.

Real polynomial. The first element of the array relates to the constant coefficient of the polynomial.

Leftmost point of the real interval. **start** must be less than or equal to **end**.

**Default: **0

Rightmost point of the real interval.

**Default: **0

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.

**Default: **No error

Number of zeros of the real polynomial in the interval.

Error information.

The node produces this output according to standard error behavior.

Standard Error Behavior

**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.

This node uses the Sturm algorithm to calculate the number of zeros. Let *p*(*x*) be a real polynomial in *x* and let *p*'(*x*) be the derivative of *p*(*x*). If *d*(*x*) denotes the greatest common divisor (GCD) of *p*(*x*) and *p*'(*x*), such that

*d*(*x*) = gcd(*p*(*x*), *p*'(*x*))

Then *p*(*x*) has multiple zeros, if *d*(*x*) is a nonconstant polynomial. In other words, the polynomial *p*(*x*)/*d*(*x*) has only single zeros.

Repeating this idea, you can combine *p*(*x*) as the product of simple polynomials, each of which has single real zeros. The number of zeros of *p*(*x*) is equal to the sum of all zeros of the defined simple polynomials that have only single zeros.

To determine the number of zeros of a real polynomial *p*(*x*) with the single zero property, use the following Euclidean algorithm:

$p\left(x\right)={q}_{1}\left(x\right){p}_{1}\left(x\right)-{p}_{2}\left(x\right)\phantom{\rule{0ex}{0ex}}{p}_{1}\left(x\right)={q}_{2}\left(x\right){p}_{2}\left(x\right)-{p}_{3}\left(x\right)\phantom{\rule{0ex}{0ex}}\vdots \phantom{\rule{0ex}{0ex}}{p}_{r-2}\left(x\right)={q}_{r-1}\left(x\right){p}_{r-1}\left(x\right)-{p}_{r}\left(x\right)$

The Sturm's chain (*p*(*x*), *p*_{1}(*x*), ..., *p*_{r}(*x*)) determines the two values *W*(**start**) and *W*(**end**), where *W*(*x*) is the number of sign changes of the Sturm's chain.

The number of all zeros of *p*(*x*) is exactly equal to *W*(**end**) - *W*(**start**).

To determine all real zeroes of a real polynomial, choose **start** and **end** that satisfies the following conditions:

$\text{start}=-\mathrm{max}\{\frac{\left(\left|{a}_{0}\right|+\dots +\left|{a}_{n-1}\right|\right)}{\left|{a}_{n}\right|},1\}$

$end=\mathrm{max}\{\frac{\left(\left|{a}_{0}\right|+\dots +\left|{a}_{n-1}\right|\right)}{\left|{a}_{n}\right|},1\}$

where *a*_{0}, *a*_{1}, ..., *a*_{n} are the elements of the polynomial.

**Where This Node Can Run: **

Desktop OS: Windows

FPGA: Not supported

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