Polynomial Interpolation - Untitled
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### 27.5 Polynomial Interpolation

Octave comes with good support for various kinds of interpolation, most of which are described in 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 are 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 ith 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
>   Samples 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 coeffients 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.