# Documentation

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# polyval

Polynomial evaluation

## Syntax

```y = polyval(p,x)[y,delta] = polyval(p,x,S)y = polyval(p,x,[],mu)[y,delta] = polyval(p,x,S,mu)```

## Description

`y = polyval(p,x)` returns the value of a polynomial of degree `n` evaluated at `x`. The input argument `p` is a vector of length `n+1` whose elements are the coefficients in descending powers of the polynomial to be evaluated.

y = p1xn + p2xn–1 + … + pnx + pn+1

`x` can be a matrix or a vector. In either case, `polyval` evaluates `p` at each element of `x`.

`[y,delta] = polyval(p,x,S)` uses the optional output structure `S` generated by `polyfit` to generate error estimates `delta`. `delta` is an estimate of the standard deviation of the error in predicting a future observation at `x` by `p(x)`. If the coefficients in `p` are least squares estimates computed by `polyfit`, and the errors in the data input to `polyfit` are independent, normal, and have constant variance, then `y`±`delta` contains at least 50% of the predictions of future observations at `x`.

`y = polyval(p,x,[],mu)` or ```[y,delta] = polyval(p,x,S,mu)``` use $\stackrel{^}{x}=\left(x-{\mu }_{1}\right)/{\mu }_{2}$ in place of `x`. In this equation, ${\mu }_{1}=\text{mean}\left(x\right)$ and ${\mu }_{2}=\text{std}\left(x\right)$. The centering and scaling parameters `mu` = [μ1,μ2] are optional output computed by `polyfit`.

## Examples

The polynomial $p\left(x\right)=3{x}^{2}+2x+1$ is evaluated at x = 5, 7, and 9 with

```p = [3 2 1]; polyval(p,[5 7 9])```

which results in

```ans = 86 162 262```

For another example, see `polyfit`.

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### Tips

The `polyvalm(p,x)` function, with `x` a matrix, evaluates the polynomial in a matrix sense. See `polyvalm` for more information.