matlab多项式拟合

polyfit函数

Descriptionp = polyfit(x,y,n) finds the coefficients
of a polynomial p(x) of degree n that
fits the data, p(x(i)) to y(i), in a
least squares sense. The result p is a row vector of length n+1 containing
the polynomial coefficients in descending powers
[p,S] = polyfit(x,y,n) returns
the polynomial coefficients p and a structure S for
use with polyval to obtain error estimates or predictions.
Structure S contains fields R, df,
and normr, for the triangular factor from a QR decomposition
of the Vandermonde matrix of X, the degrees of freedom,
and the norm of the residuals, respectively. If the data Y are
random, 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 y are independent normal with constant
variance, polyval produces error bounds that contain
at least 50% of the predictions.[p,S,mu] = polyfit(x,y,n) finds
the coefficients of a polynomial in
where
 and
. mu is
the two-element vector
. This centering
and scaling transformation improves the numerical properties of both the polynomial
and the fitting algorithm.ExamplesThis example involves fitting the error function, erf(x),
by a polynomial in x. This is a risky project because erf(x) is
a bounded function, while polynomials are unbounded, so the fit might not
be very good. First generate a vector of x points, equally spaced
in the interval
; then evaluate erf(x) at
those points. x = (0: 0.1: 2.5)';
y = erf(x); The coefficients in the approximating polynomial of degree 6 are p = polyfit(x,y,6)

p =

 0.0084  -0.0983   0.4217   -0.7435  0.1471   1.1064  0.0004There are seven coefficients and the polynomial is
To see how good the fit is, evaluate the polynomial at the data points
withf = polyval(p,x); A table showing the data, fit, and error is  table = [x y f y-f]
 
 table =

    0          0          0.0004    -0.0004
    0.1000     0.1125     0.1119     0.0006
    0.2000     0.2227     0.2223     0.0004
    0.3000     0.3286     0.3287    -0.0001
    0.4000     0.4284     0.4288    -0.0004
    ...
    2.1000     0.9970     0.9969     0.0001
    2.2000     0.9981     0.9982    -0.0001
    2.3000     0.9989     0.9991    -0.0003
    2.4000     0.9993     0.9995    -0.0002
    2.5000     0.9996     0.9994     0.0002So, on this interval, the fit is good to between three and four digits.
Beyond this interval the graph shows that the polynomial behavior takes over
and the approximation quickly deteriorates. x = (0: 0.1: 5)';
y = erf(x);
f = polyval(p,x);
plot(x,y,'o',x,f,'-')
axis([0  5  0  2])
AlgorithmThe polyfit M-file forms the Vandermonde matrix,
, whose elements are powers
of
.
It then uses the backslash operator, \,
to solve the least squares problem
You can modify the M-file to use other functions of
 as the basis functions.