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polyval in MATLAB evaluates a polynomial at the points you give it: y = polyval(coeffs,x) returns the value of the polynomial whose coefficients are in coeffs (highest power first) at every element of x. Used together with polyfit, it turns a fitted curve into predicted values, and its optional second output delta gives an error estimate for those predictions. This guide covers the syntax, a worked example, the fitted-model workflow and how to read delta correctly. It reflects the MathWorks documentation read on 2026-11-07.
What polyval is used for in research
Researchers use polyval to evaluate a polynomial: a calibration curve at new readings, a trend line at future time points, or a fitted model on a fine grid for plotting. The MathWorks Polynomials page describes a polynomial in MATLAB as a vector of coefficients in descending order, so [1 -4 4] represents x² − 4x + 4. polyval evaluates it, polyvalm evaluates a matrix polynomial, and polyfit finds the coefficients from data.
The polyval formula
For coeffs = [c1 c2 ... cn+1], polyval(coeffs,x) computes c1*x^n + c2*x^(n-1) + ... + cn*x + cn+1. The coefficient vector has length n+1 for a polynomial of degree n. It may be double or single, and complex values are allowed.
Install or access
polyval is a MATLAB function documented in the official MathWorks help at mathworks.com/help/matlab/ref/polyval.html. Get MATLAB from MathWorks; your institution’s site-licence administrator or the MathWorks website can tell you which access route applies to you. To confirm the function is available in your session, type help polyval or doc polyval at the command prompt.
Step-by-step: evaluate a polynomial and a fitted model
1. Evaluate a polynomial you already know
- Write the coefficients in descending powers:
coeffs = [3 2 1];represents 3x² + 2x + 1. - Choose the query points:
x = [5 7 9]; - Call the function:
y = polyval(coeffs,x)
The MathWorks documentation shows this result: y = 86 162 262. You can check the first value by hand: 3·25 + 2·5 + 1 = 86. The output y has the same size as x.
2. Fit with polyfit, then evaluate with polyval
- Fit a degree-n polynomial:
coeffs = polyfit(x,y,n); - Build the points where you want predictions, for example
xq = linspace(min(x),max(x),200); - Evaluate:
yfit = polyval(coeffs,xq); - Plot the data and the curve together:
plot(x,y,'o',xq,yfit,'-')
A simple check on a made-up data set: with x = [0 1 2 3] and y = [1 3 9 19] (exactly 2x² + 1), polyfit(x,y,2) returns coefficients close to [2 0 1] up to floating-point rounding, and polyval at x = 4 returns about 33.
3. Fit with normalisation (centerScale)
polyfit can return a third output, centerScale, when you request it. The documentation defines it as [mean(x) std(x)] when Normalize=true and as 0 and 1 when Normalize=false; requesting the third output makes Normalize=true the default. The fit is then done on the centred and scaled x, so you must pass centerScale back to polyval. The documented pattern is:
[coeffsNorm,~,centerScale] = polyfit(x,y,5,Normalize=true);
yfit = polyval(coeffsNorm,x,[],centerScale);The empty [] is a placeholder for the fitStats argument, which sits in third position in the polyval call.
Syntax summary
| Call | Returns |
|---|---|
y = polyval(coeffs,x) |
Polynomial values at x |
[y,delta] = polyval(coeffs,x,fitStats) |
Values plus a standard-error estimate; fitStats comes from polyfit |
y = polyval(coeffs,x,[],centerScale) |
Values when the fit used normalisation |
[y,delta] = polyval(coeffs,x,fitStats,centerScale) |
Both of the above |
Output and how to read it
y has the same size as x. delta, when requested, is the standard error for predicting a future observation and also has the same size as x. It is a prediction-interval quantity, not a confidence interval for the fitted line, so it is wider than the uncertainty of the curve itself. To get delta, call polyfit with the second output ([coeffs,fitStats] = polyfit(x,y,n)); fitStats is a structure with the fields R, df, normr and rsquared.
The documentation gives these readings:
- For large samples,
y ± Δcontains approximately 68% of future observations andy ± 2Δapproximately 95%. - If the coefficients come from a
polyfitleast-squares fit and the errors are independent, normally distributed and of constant variance, theny ± Δcontains at least 50% of future observations.
When you report results, state which of these conditions you checked. See assumptions of simple regression for residual checks and linear regression scatter plots for plotting the fitted line. For the predicted value symbol, see y hat (ŷ).
Troubleshooting
Wrong values: coefficient order
MATLAB expects descending powers. Passing [1 2 3] means x² + 2x + 3, not 1 + 2x + 3x². The order is the most common silent error when moving between tools; see the cross-walk in the table below.
Unstable or odd fits: ill-conditioning
The polyfit documentation states that an ill-conditioned fit can result from repeated or nearly repeated query points, or widely spaced query points, and advises adding distinct query points. It also notes that high-order fits have a large Vandermonde condition number and that Normalize=true can improve the numerical properties. If you normalise, remember to pass centerScale to polyval.
Matrix versus element-wise evaluation
polyval works element by element on x. To evaluate a polynomial with a square matrix as the argument, use polyvalm.
Rounding at high degree
Evaluation is subject to floating-point rounding; the NumPy documentation notes rounding error for high-degree polynomials in its Horner-scheme implementation. See IEEE 754 floating-point for background on why such errors arise.
Free alternatives and coefficient-order cross-walk
| Tool | Function | Coefficient order | Notes (from official docs) |
|---|---|---|---|
| MATLAB | polyval(coeffs,x) |
Highest power first | Optional delta output with fitStats; centerScale for normalised fits |
| GNU Octave | polyval(p,x) |
Highest power first | [y,dy] = polyval(p,x,s) with s from polyfit; y ± dy covers at least 50% of future predictions; polyval(p,x,[],mu) evaluates at (x − mu(1))/mu(2) |
| NumPy (legacy API) | numpy.polyval(p,x) |
Highest degree first | Documented as legacy; uses Horner’s scheme |
NumPy (numpy.polynomial) |
numpy.polynomial package |
Lowest power first | Preferred by NumPy for new code; mixing orders gives wrong results without an error |
GNU Octave is free software; NumPy is open-source. Check each project’s own documentation for current installation routes.
Frequently asked questions
What does polyval do in MATLAB?
It evaluates a polynomial, given as a vector of coefficients in descending order, at each element of the input x and returns an array of the same size.
Is matlab polyval the same as polyfit?
No. polyfit(x,y,n) finds the coefficients of a degree-n least-squares polynomial; polyval uses coefficients to compute values. They are normally used as a pair.
What is a polyval example?
polyval([3 2 1],[5 7 9]) returns 86 162 262, the values of 3x² + 2x + 1 at 5, 7 and 9.
What is delta in polyval?
The standard error for predicting a future observation at each point of x. Roughly, y ± Δ is a 68% prediction interval for large samples and y ± 2Δ is 95%; with a polyfit fit and independent, normal, constant-variance errors, y ± Δ is at least a 50% interval.
Why is the third argument empty when I pass centerScale?
The third position is fitStats. Passing [] skips it while centerScale goes in fourth position.
Sources
- MathWorks, polyval, read 2026-11-07.
- MathWorks, polyfit, read 2026-11-07.
- MathWorks, Polynomials, read 2026-11-07.
- GNU Octave manual, Evaluating Polynomials, read 2026-11-07.
- NumPy documentation, numpy.polyval, read 2026-11-07.
Related reading on this site: MATLAB mod function and non-linear relationships.
