Contoh polyval…
Examples
This 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 0.0004
There are seven coefficients and the polynomial is
To see how good the fit is, evaluate the polynomial at the data points with
f = 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 0.0002
So, 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])
Algorithm
The 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.
