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26.5 Polynomial Interpolation

Octave comes with good support for various kinds of interpolation, most of which are described in section 27 Interpolation. One simple alternative to the functions described in the aforementioned chapter, is to fit a single polynomial to some given data points. To avoid a highly fluctuating polynomial, one most often wants to fit a low-order polynomial to data. This usually means that it is necessary to fit the polynomial in a least-squares sense, which is what the polyfit function does.

Function File: [p, s] = polyfit (x, y, n)

Return the coefficients of a polynomial p(x) of degree n that minimizes sumsq (p(x(i)) - y(i)),

to best fit the data in the least squares sense.

The polynomial coefficients are returned in a row vector.

If two output arguments are requested, the second is a structure containing the following fields:

R

The Cholesky factor of the Vandermonde matrix used to compute the polynomial coefficients.

X

The Vandermonde matrix used to compute the polynomial coefficients.

df

The degrees of freedom.

normr

The norm of the residuals.

yf

The values of the polynomial for each value of x.

In situations where a single polynomial isn't good enough, a solution is to use several polynomials pieced together. The function mkpp creates a piece-wise polynomial, ppval evaluates the function created by mkpp, and unmkpp returns detailed information about the function.

The following example shows how to combine two linear functions and a quadratic into one function. Each of these functions is expressed on adjoined intervals.

x = [-2, -1, 1, 2];
p = [ 0,  1, 0;
      1, -2, 1;
      0, -1, 1 ];
pp = mkpp(x, p);
xi = linspace(-2, 2, 50);
yi = ppval(pp, xi);
plot(xi, yi);

Function File: yi = ppval (pp, xi)

Evaluate piece-wise polynomial pp at the points xi. If pp.d is a scalar greater than 1, or an array, then the returned value yi will be an array that is d1, d1, ..., dk, length (xi)].

See also mkpp, unmkpp, spline

Function File: pp = mkpp (x, p)

Function File: pp = mkpp (x, p, d)

Construct a piece-wise polynomial structure from sample points x and coefficients p. The i-th row of p, p (i,:), contains the coefficients for the polynomial over the i-th interval, ordered from highest to lowest. There must be one row for each interval in x, so rows (p) == length (x) - 1.

You can concatenate multiple polynomials of the same order over the same set of intervals using p = [ p1; p2; ...; pd ]. In this case, rows (p) == d * (length (x) - 1).

d specifies the shape of the matrix p for all except the last dimension. If d is not specified it will be computed as round (rows (p) / (length (x) - 1)) instead.

See also unmkpp, ppval, spline

Function File: [x, p, n, k, d] = unmkpp (pp)

Extract the components of a piece-wise polynomial structure pp. These are as follows:

x

Sample points.

p

Polynomial coefficients for points in sample interval. p (i, :) contains the coefficients for the polynomial over interval i ordered from highest to lowest. If d > 1, p (r, i, :) contains the coefficients for the r-th polynomial defined on interval i. However, this is stored as a 2-D array such that c = reshape (p (:, j), n, d) gives c (i, r) is the j-th coefficient of the r-th polynomial over the i-th interval.

n

Number of polynomial pieces.

k

Order of the polynomial plus 1.

d

Number of polynomials defined for each interval.

See also mkpp, ppval, spline