# polyfit

Polynomial curve fitting

## Syntax

• p = polyfit(x,y,n) example
• [p,S] = polyfit(x,y,n)
• [p,S,mu] = polyfit(x,y,n) example

## Description

example

p = polyfit(x,y,n) returns the coefficients for a polynomial p(x) of degree n that is a best fit (in a least-squares sense) for the data in y. The coefficients in p are in descending powers, and the length of p is n+1

$p\left(x\right)={p}_{1}{x}^{n}+{p}_{2}{x}^{n-1}+...+{p}_{n}x+{p}_{n+1}.$

[p,S] = polyfit(x,y,n) also returns a structure S that can be used as an input to polyval to obtain error estimates.

example

[p,S,mu] = polyfit(x,y,n) also returns mu, which is a two-element vector with centering and scaling values. mu(1) is mean(x), and mu(2) is std(x). Using these values, polyfit centers x at zero and scales it to have unit standard deviation

$\stackrel{^}{x}=\frac{x-\overline{x}}{{\sigma }_{x}}\text{\hspace{0.17em}}.$

This centering and scaling transformation improves the numerical properties of both the polynomial and the fitting algorithm.

## Examples

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### Fit Polynomial to Trigonometric Function

Generate 10 points equally spaced along a sine curve in the interval [0,4*pi].

x = linspace(0,4*pi,10);
y = sin(x);

Use polyfit to fit a 7th-degree polynomial to the points.

p = polyfit(x,y,7);

Evaluate the polynomial on a finer grid and plot the results.

x1 = linspace(0,4*pi);
y1 = polyval(p,x1);
figure
plot(x,y,'o')
hold on
plot(x1,y1)
hold off

### Fit Polynomial to Set of Points

Create a vector of 5 equally spaced points in the interval [0,1], and evaluate at those points.

x = linspace(0,1,5);
y = 1./(1+x);

Fit a polynomial of degree 4 to the 5 points. In general, for n points, you can fit a polynomial of degree n-1 to exactly pass through the points.

p = polyfit(x,y,4);

Evaluate the original function and the polynomial fit on a finer grid of points between 0 and 2.

x1 = linspace(0,2);
y1 = 1./(1+x1);
f1 = polyval(p,x1);

Plot the function values and the polynomial fit in the wider interval [0,2], with the points used to obtain the polynomial fit highlighted as circles. The polynomial fit is good in the original [0,1] interval, but quickly diverges from the fitted function outside of that interval.

figure
plot(x,y,'o')
hold on
plot(x1,y1)
plot(x1,f1,'r--')
legend('y','y1','f1')

### Fit Polynomial to Error Function

First generate a vector of x points, equally spaced in the interval [0,2.5], and then evaluate erf(x) at those points.

x = (0:0.1:2.5)';
y = erf(x);

Determine the coefficients of the approximating polynomial of degree 6.

p = polyfit(x,y,6)
p =

0.0084   -0.0983    0.4217   -0.7435    0.1471    1.1064    0.0004

To see how good the fit is, evaluate the polynomial at the data points and generate a table showing the data, fit, and error.

f = polyval(p,x);
T = table(x,y,f,y-f,'VariableNames',{'X','Y','Fit','FitError'})
T =

X        Y          Fit         FitError
___    _______    __________    ___________

0          0    0.00044117    -0.00044117
0.1    0.11246       0.11185     0.00060836
0.2     0.2227       0.22231     0.00039189
0.3    0.32863       0.32872    -9.7429e-05
0.4    0.42839        0.4288    -0.00040661
0.5     0.5205       0.52093    -0.00042568
0.6    0.60386       0.60408    -0.00022824
0.7     0.6778       0.67775     4.6383e-05
0.8     0.7421       0.74183     0.00026992
0.9    0.79691       0.79654     0.00036515
1     0.8427       0.84238      0.0003164
1.1    0.88021       0.88005     0.00015948
1.2    0.91031       0.91035    -3.9919e-05
1.3    0.93401       0.93422      -0.000211
1.4    0.95229       0.95258    -0.00029933
1.5    0.96611       0.96639    -0.00028097
1.6    0.97635       0.97652    -0.00016704
1.7    0.98379       0.98379     8.3306e-07
1.8    0.98909       0.98893     0.00016278
1.9    0.99279       0.99253     0.00025791
2    0.99532       0.99508     0.00024347
2.1    0.99702       0.99691      0.0001131
2.2    0.99814       0.99823    -8.8548e-05
2.3    0.99886       0.99911    -0.00025673
2.4    0.99931       0.99954    -0.00022451
2.5    0.99959       0.99936     0.00023151

In this interval, the interpolated values and the actual values agree fairly closely. Create a plot to show how outside this interval, the extrapolated values quickly diverge from the actual data.

x1 = (0:0.1:5)';
y1 = erf(x1);
f1 = polyval(p,x1);
figure
plot(x,y,'o')
hold on
plot(x1,y1,'-')
plot(x1,f1,'r--')
axis([0  5  0  2])
hold off

### Use Centering and Scaling to Improve Numerical Properties

Create a table of population data for the years 1750 - 2000 and plot the data points.

year = (1750:25:2000)';
pop = 1e6*[791 856 978 1050 1262 1544 1650 2532 6122 8170 11560]';
T = table(year, pop)
plot(year,pop,'o')
T =

year       pop
____    _________

1750     7.91e+08
1775     8.56e+08
1800     9.78e+08
1825     1.05e+09
1850    1.262e+09
1875    1.544e+09
1900     1.65e+09
1925    2.532e+09
1950    6.122e+09
1975     8.17e+09
2000    1.156e+10

Use polyfit with three outputs to fit a 5th-degree polynomial using centering and scaling, which improves the numerical properties of the problem. polyfit centers the data in year at 0 and scales it to have a standard deviation of 1, which avoids an ill-conditioned Vandermonde matrix in the fit calculation.

[p,~,mu] = polyfit(T.year, T.pop, 5);

Use polyval with four inputs to evaluate p with the scaled years, (year-mu(1))/mu(2). Plot the results against the original years.

f = polyval(p,year,[],mu);
hold on
plot(year,f)
hold off

## Input Arguments

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### x — Query pointsvector

Query points, specified as a vector. The points in x correspond to the fitted function values contained in y.

Warning messages result when x has repeated (or nearly repeated) points or if x might need centering and scaling.

Data Types: single | double
Complex Number Support: Yes

### y — Fitted values at query pointsvector

Fitted values at query points, specified as a vector. The values in y correspond to the query points contained in x.

Data Types: single | double
Complex Number Support: Yes

### n — Degree of polynomial fitpositive integer scalar

Degree of polynomial fit, specified as a positive integer scalar. n specifies the polynomial power of the left-most coefficient in p.

A warning message results if n is greater than or equal to length(x).

## Output Arguments

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### p — Least-squares fit polynomial coefficientsvector

Least-squares fit polynomial coefficients, returned as a vector. p has length n+1 and contains the polynomial coefficients in descending powers with the highest power being n.

Use polyval to evaluate p at query points.

### S — Error estimation structurestructure

Error estimation structure. This optional output structure is primarily used as an input to the polyval function to obtain error estimates. S contains the following fields:

FieldDescription
RTriangular factor from a QR decomposition of the Vandermonde matrix of x
dfDegrees of freedom
normrNorm of the residuals

If the data in y is random, then an estimate of the covariance matrix of p is (Rinv*Rinv')*normr^2/df, where Rinv is the inverse of R.

If the errors in the data in y are independent and normal with constant variance, then [y,delta] = polyval(...) produces error bounds that contain at least 50% of the predictions. That is, y ± delta contains at least 50% of the predictions of future observations at x.

### mu — Centering and scaling valuestwo element vector

Centering and scaling values, returned as a two element vector. mu(1) is mean(x), and mu(2) is std(x). These values center the query points in x at zero with unit standard deviation.

Use mu as the fourth input to polyval to evaluate p at the scaled points, (x - mu(1))/mu(2).

## Limitations

• In problems with many points, increasing the degree of the polynomial fit using polyfit does not always result in a better fit. High-order polynomials can be oscillatory between the data points, leading to a poorer fit to the data. In those cases, you might use a low-order polynomial fit (which tends to be smoother between points) or a different technique, depending on the problem.

• Polynomials are unbounded, oscillatory functions by nature. Therefore, they are not well-suited to extrapolating bounded data or monotonic (increasing or decreasing) data.

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### Algorithms

polyfit uses x to form Vandermonde matrix V with n+1 columns, resulting in the linear system

$\left(\begin{array}{cccc}{x}_{1}^{n+1}& {x}_{1}^{n}& \cdots & 1\\ {x}_{2}^{n+1}& {x}_{2}^{n}& \cdots & 1\\ ⋮& ⋮& \ddots & ⋮\\ {x}_{n}^{n+1}& {x}_{n}^{n}& \cdots & 1\end{array}\right)\left(\begin{array}{c}{p}_{1}\\ {p}_{2}\\ ⋮\\ {p}_{n}\end{array}\right)=\left(\begin{array}{c}{y}_{1}\\ {y}_{2}\\ ⋮\\ {y}_{n}\end{array}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}},$

which polyfit solves with p = V\y. Since the columns in the Vandermonde matrix are powers of the vector x, the condition number of V is often large for high-order fits, resulting in a singular coefficient matrix. In those cases centering and scaling can improve the numerical properties of the system to produce a more reliable fit.