# Routine lsapprox

Polynomial approximation of tabulated data using linear least-squares.

### Description

This routine uses Linear Least-Squares to compute a multivariate polynomial approximation $f:\mathbb{R}^n\rightarrow\mathbb{R}^{n_1\times n_2}$ of a set of samples $\left\{y_k\in\mathbb{R}^{n_1\times n_2}, k\in [1, N]\right\}$ obtained for different values $\left\{x_k\in\mathbb{R}^n,k \in [1, N]\right\}$ of some explanatory variables $x\in\mathbb{R}^n$. The maximum degree of the polynomial function is set by the user. No maximum admissible error can be specified and a full polynomial expression is obtained, which minimizes the global relative error and the root-mean-square error between $\left\{f(x_k), k\in [1, N]\right\}$ and $\left\{y_k, k\in [1, N]\right\}$ for each entry (see errapprox for precise definitions).

### Syntax

[fdata,flfr]=lsapprox(X,Y,names,maxdeg{,options})

### Input arguments

The first four input arguments are mandatory:

 X Values $\left\{x_k\in\mathbb{R}^n,k \in [1, N]\right\}$ of the explanatory variables $x$ ($n\times N$ array, where X(:,k) corresponds to $x_k$). Y Samples $\left\{y_k\in\mathbb{R}^{n_1\times n_2}, k\in [1, N]\right\}$ to be approximated ($n_1\times n_2\times N$ array where Y(:,:,k) corresponds to $y_k$ in the general case, or possibly $1\times N$ array where Y(k) corresponds to $y_k$ if $n_1=n_2=1$). names Names of the explanatory variables $x$ ($1\times n$ cell array of strings). maxdeg Maximum degree of the approximating polynomial function $f$.

The fifth input argument options is an optional structured variable with fields:

 maxexp Maximum exponent of each explanatory variable in the approximating polynomial function $f$ ($1\times n$ array). The default value is options.maxexp=maxdeg*ones(1,n). XeqYeq Constraint to be exactly met (no least squares). options.Yeq is a $n_1\times n_2$ array and options.Xeq is a $n\times 1$ vector with associated values of the explanatory variables. The default values are options.Xeq=[] and options.Yeq=[]. trace Trace of execution (0=no, 1=text, 2=text+figures). The default value is options.trace=1. viewpoint This option is applicable only if 3-D graphs are to be displayed (options.trace=2 and $n=2$). It represents the deviation with respect to the default viewpoint (see plotapprox). The default value is options.viewpoint=[0 0].

### Output arguments

 fdata Values $\left\{f(x_k)\in\mathbb{R}^{n_1\times n_2},k \in [1, N]\right\}$ of the approximating function $f$ (same size as Y). flfr Linear fractional representation of the approximating function $f$ (GSS object if the GSS library is installed, LFR object otherwise if the LFR toolbox is installed).

### Caution

If some values in X are much larger than 1, they must be scaled or normalized.

### Example

Drag coefficient of a generic fighter aircraft model:
load data_cx
Approximation on a rough grid:
maxdeg=12; options.trace=2; options.viewpoint=[-100 0]; [fdata,flfr]=lsapprox(X1,Y1,names,maxdeg,options); Validation on a fine grid:
plotapprox(Xv,Yv,names,flfr,options); olsapprox
qpapprox
tracker
koala
errapprox
plotapprox