matlab中stc,Scattered translates collocation matrix

stcol

Scattered translates collocation matrix

Syntax

colmat = stcol(centers,x,type)

colmat = stcol(...,'tr')

Description

colmat = stcol(centers,x,type) is the

matrix whose (i,j)th entry is

ψj(x(:,i)),i=1:size(x,2),j=1:n

with the bivariate functions ψj and the

number n depending on the centers and the

character vector type, as detailed in the description of

stmak.

centers and x must be matrices with the same

number of rows.

The default for type is the character vector

'tp', and for this default, n equals

size(centers,2), and the functions

ψj are given by

ψj(x)=ψ(x−centers(:,j)),j=1:n

with ψ the thin-plate spline basis function

ψ(x)=|x|2log|x|2

and with |x| denoting the Euclidean norm of the vector

x.

Note

See stmak for a description of

other possible values for type.

The matrix colmat is the coefficient matrix in the linear

system

∑jajψj(x(:,i))=yi,i=1:size(x,2)

that the coefficients aj of the function

f =

Σjajψj

must satisfy in order that f interpolate the value

yi at the site

x(:,i), all i.

colmat = stcol(...,'tr') returns the

transpose of the matrix returned by stcol(...).

Examples

Example 1. The following evaluates and plots the

function

f(x)=ψ(x−c1)+ψ(x−c2)+ψ(x−c3)−3.5ψ(x)

on a regular mesh, with ψ the above thin-plate basis function,

and with c1,

c2,

c3 three points on the unit circle;

see the figure below.

a = [0,2/3*pi,4/3*pi]; centers = [cos(a), 0; sin(a), 0];

[xx,yy] = ndgrid(linspace(-2,2,45));

xy = [xx(:) yy(:)].';

coefs = [1 1 1 -3.5];

zz = reshape( coefs*stcol(centers,xy,'tr') , size(xx));

surf(xx,yy,zz), view([240,15]), axis off

Example 2. The following also evaluates, on the

same mesh, and plots the length of the gradient of the function in Example 1.

zz = reshape( sqrt(...

([coefs,0]*stcol(centers,xy,'tp10','tr')).^2 + ...

([coefs,0]*stcol(centers,xy,'tr','tp01')).^2),

size(xx));

figure, surf(xx,yy,zz), view([220,-15]), axis off