# POLYFIT2W

Compute a weighted 2D polynomial fit.

## Contents

## Syntax

[p, S] = polyfit2w(x,y,z,w,n,m)

## Description

Compute the polynomial coefficients of a function of 2 variables x and y of degrees n and m, respectively, that fit the data in z in a least-squares sense, using weights w. x,y,z and w can be vectors or matrices of the same size. S is a structure containing three elements: the Cholesky factor of the Vandermonde matrix, the degrees of freedom and the norm of the residuals.

The weighted regression problem is formulated in matrix format as:

A'*W*z = A'*W*A*p where the matrix A contains the x,y data

if n = 3 and m = 1, A = [y.*x.^3 y.*x.^2 y.*x y x.^3 x.^2 x ones(length(x),1)] Note that the various xy products are column vectors of length(x).

The polynomial coeffiecients are ordered as [p31 p21 p11 p01 p30 p20 p10 p00]' for the computation.

The coefficents of the output p matrix are arranged as shown:

p31 p30 p21 p20 p11 p10 p01 p00

The indices on the elements of p correspond to the order of x and y associated with that element.

For a solution to exist, the number of ordered triples [x,y,z] must equal or exceed (n+1)*(m+1). Note that m or n may be zero.

To evaluate the resulting polynominal function, use polyval2d.