GNU Octave Manual Version 3 by John W. Eaton, David Bateman, Søren Hauberg Paperback (6"x9"), 568 pages ISBN 095461206X RRP £24.95 ($39.95) 
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 loworder polynomial
to data. This usually means that it is necessary to fit the polynomial
in a leastsquares 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 piecewise 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 piecewise 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 isd1, d1, ..., dk, length (xi)]
.See also mkpp, unmkpp, spline
 Function File: pp = mkpp (x, p)
 Function File: pp = mkpp (x, p, d)

Construct a piecewise polynomial structure from sample points x and coefficients p. The ith row of p,
p (i,:)
, contains the coefficients for the polynomial over the ith interval, ordered from highest to lowest. There must be one row for each interval in x, sorows (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 piecewise 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. Ifd > 1
,p (r, i, :)
contains the coefficients for the rth polynomial defined on interval i. However, this is stored as a 2D array such thatc = reshape (p (:, j), n, d)
givesc (i, r)
is the jth coefficient of the rth polynomial over the ith 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
ISBN 095461206X  GNU Octave Manual Version 3  See the print edition 