## Contents

## Demo script rewrite from gridfit demo

This script file is designed to be used in cell mode from the matlab editor, or best of all, use the publish to HTML feature from the matlab editor. Older versions of matlab can copy and paste entire blocks of code into the Matlab command window.

```
clc; clear; close all;
```

## Topographic data

load bluff_data; x=bluff_data(:,1); y=bluff_data(:,2); z=bluff_data(:,3); % Two ravines on a hillside. Scanned from a % topographic map of an area in upstate New York. plot3(x,y,z,'.')

% Turn the scanned point data into a surface gx=0:4:264; gy=0:4:400; smoothness = 0.00001; g=regularizeNd([x,y] ,z, {gx,gy}, smoothness); % equivalent calls % g=regularizeNd([x,y] ,z, {gx,gy}, smoothness, 'linear', 'normal'); % g=regularizeNd([x,y] ,z, {gx,gy}, smoothness, [], []); % alternative calls % g=regularizeNd([x,y] ,z, {gx,gy}, smoothness, [], '\'); % g=regularizeNd([x,y] ,z, {gx,gy}, smoothness, 'cubic', 'normal'); % g=regularizeNd([x,y] ,z, {gx,gy}, smoothness, 'cubic', '\'); figure colormap(hot(256)); surf(gx,gy,g'); camlight right; lighting phong; shading interp line(x,y,z,'marker','.','markersize',4,'linestyle','none'); title 'Use topographic contours to recreate a surface'

## Fitting a trigonometric surface

clear % make the random number sequence repeatable rng(20170401); n1 = 15; n2 = 15; theta = rand(n1,1)*pi/2; r = rand(1,n2); x = cos(theta)*r; y = sin(theta)*r; x=x(:); y=y(:); x = [[0 0 1 1]';x;x;1-x;1-x]; y = [[0 1 0 1]';y;1-y;y;1-y]; figure plot(x,y,'.') title 'Data locations in the x-y plane'

z = sin(4*x+5*y).*cos(7*(x-y))+exp(x+y); xi = linspace(0,1,51); [xg,yg]=meshgrid(xi,xi); zgd = griddata(x,y,z,xg,yg); figure surf(xi,xi,zgd) colormap(hot(256)) camlight right lighting phong title 'Griddata on trig surface' % Note the wing-like artifacts along the edges, due % to the use of a Delaunay triangulation in griddata.

xGrid = {xi,xi}; smoothness = 1e-6; zGrid = regularizeNd([x,y], z, xGrid, smoothness); figure surf(xi,xi,zGrid') colormap(hot(256)) camlight right lighting phong title(sprintf('regularizeNd to trig surface, smoothness=%g', smoothness))

## The trig surface with highly different scalings on the x and y axes

xs = x/100; xis = xi/100; ys = y*100; yis = xi*100; % griddata has problems with badly scaled data [xg,yg]=meshgrid(xis,yis); zgd = griddata(xs,ys,z,xg,yg); figure surf(xg,yg,zgd) colormap(hot(256)) camlight right lighting phong title 'Serious problems for griddata on badly scaled trig surface' % No need for scaling with regularizeNd zgrids = regularizeNd([xs,ys],z,{xis,yis}, smoothness); % plot the result figure surf(xis,yis,zgrids) colormap(hot(256)) camlight right lighting phong title(sprintf('regularizeNd on badly scaled trig surface, smoothness=%g', smoothness))

## Fitting the "peaks" surface

clear n = 100; x = (rand(n,1)-.5)*6; y = (rand(n,1)-.5)*6; z = peaks(x,y); xi = linspace(-3,3,101); smoothness = 0.0001; zpgf = regularizeNd([x,y],z,{xi,xi}, smoothness, 'cubic'); [xg,yg]=meshgrid(xi,xi); zpgd = griddata(x,y,z,xg,yg,'cubic'); figure surf(xi,xi,zpgd) colormap(jet(256)) camlight right lighting phong title 'Griddata (method == cubic) on peaks surface' figure surf(xi,xi,zpgf') colormap(hsv(256)) camlight right lighting phong title('regularizeNd with cubic interp method to peaks surface')

## No need for tiles with regularizeNd

gridfit had a option called 'tiles'. It was made for handling large problems. However, this option is not in regularizeNd. The case used in gridfit took several minutes to calculate. However, with the advances in hardware and software, the problem solves in a few seconds. I have solved problems with as many as 3000 grid points (2e6 unknowns) in under a minute.

n = 100000; x = rand(n,1); y = rand(n,1); z = x+y+sin((x.^2+y.^2)*10); xnodes = 0:.00125:1; ynodes = xnodes; smoothness = 0.001; tic; zg = regularizeNd([x,y], z, {xnodes,ynodes}, smoothness); toc; figure; surf(xnodes, ynodes, zg') shading interp colormap(jet(256)) camlight right lighting phong title 'No need for tiles with regularizeNd'

Elapsed time is 6.051789 seconds.