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int3d

definite 3D integral by quadrature and cubature method

Syntax

[result,err]=int3d(X,Y,Z,f [,nf[,params]])

Arguments

X

a 4 by NUMTET array containing the abscissae of the vertices of the NUMTET tetrahedrons.

Y

a 4 by NUMTET array containing the ordinates of the vertices of the NUMTET tetrahedrons.

Z

a 4 by NUMTET array containing the third coordinates of the vertices of the NUMTET tetrahedrons.

f

external (function or list or string) defining the integrand f(xyz,nf), where xyz is the vector of a point coordinates and nf is the number functions

nf

the number of functions to integrate (default is 1)

params

a real vector [minpts, maxpts, epsabs, epsrel]. The default value is [0, 1000, 0.0, 1.d-5].

epsabs

Desired bound on the absolute error.

epsrel

Desired bound on the relative error.

minpts

Minimum number of function evaluations.

maxpts

Maximum number of function evaluations. The number of function evaluations over each subregion is 43

result

the integral value or vector of the integral values.

err

estimates of absolute errors.

Description

The function calculates an approximation to a given vector of definite integrals

I  I  I (F ,F ,...,F )      dx(3)dx(2)dx(1),
1  2      numfun

where the region of integration is a collection of NUMTET tetrahedrons and where

F = F (X(1),X(2),X(3)), J = 1,2,...,NUMFUN.
J   J

A globally adaptive strategy is applied in order to compute approximations result(k) hopefully satisfying, for each component of I, the following claim for accuracy: abs(I(k)-result(k))<=max(epsabs,epsrel*abs(I(k)))

int3d repeatedly subdivides the tetrahedrons with greatest estimated errors and estimates the integrals and the errors over the new subtetrahedrons until the error request is met or maxpts function evaluations have been used.

A 43 point integration rule with all evaluation points inside the tetrahedron is applied. The rule has polynomial degree 8.

If the values of the input parameters epsabs or epsrel are selected great enough, an integration rule is applied over each tetrahedron and the results are added up to give the approximations result(k). No further subdivision of the tetrahedrons will then be applied.

When int3d computes estimates to a vector of integrals, all components of the vector are given the same treatment. That is, I(Fj) and I(Fk) for j not equal to k, are estimated with the same subdivision of the region of integration. For integrals with enough similarity, we may save time by applying int3d to all integrands in one call. For integrals that varies continuously as functions of some parameter, the estimates produced by int3d will also vary continuously when the same subdivision is applied to all components. This will generally not be the case when the different components are given separate treatment.

On the other hand this feature should be used with caution when the different components of the integrals require clearly different subdivisions.

References

Fortran routine dcutet.f

Examples

X=[0;1;0;0];
Y=[0;0;1;0];
Z=[0;0;0;1];
[RESULT,ERROR]=int3d(X,Y,Z,'int3dex')
// computes the integrand exp(x*x+y*y+z*z) over the
//tetrahedron (0.,0.,0.),(1.,0.,0.),(0.,1.,0.),(0.,0.,1.)

//integration over a cube  -1<=x<=1;-1<=y<=1;-1<=z<=1

//  bottom  -top-     right    -left-   front   -rear-
X=[ 0, 0,    0, 0,    0, 0,    0, 0,    0, 0,    0, 0;
-1,-1,   -1,-1,    1, 1,   -1,-1,   -1,-1,   -1,-1;
1,-1,    1,-1,    1, 1,   -1,-1,    1,-1,    1,-1;
1, 1,    1, 1,    1, 1,   -1,-1,    1, 1,    1, 1];
Y=[ 0, 0,    0, 0,    0, 0,    0, 0,    0, 0,    0, 0;
-1,-1,   -1,-1,   -1, 1,   -1, 1,   -1,-1,    1, 1;
-1, 1,   -1, 1,    1, 1,    1, 1,   -1,-1,    1, 1;
1, 1,    1, 1,   -1,-1,   -1,-1,   -1,-1,    1, 1];
Z=[ 0, 0,    0, 0,    0, 0,    0, 0,    0, 0,    0, 0;
-1,-1,    1, 1,   -1, 1,   -1, 1,   -1,-1,   -1,-1;
-1,-1,    1, 1,   -1,-1,   -1,-1,   -1, 1,   -1, 1;
-1,-1,    1, 1,    1, 1,    1, 1,    1, 1,    1, 1];

function v=f(xyz, numfun),v=exp(xyz'*xyz),endfunction
[result,err]=int3d(X,Y,Z,f,1,[0,100000,1.d-5,1.d-7])

function v=f(xyz, numfun),v=1,endfunction
[result,err]=int3d(X,Y,Z,f,1,[0,100000,1.d-5,1.d-7])

• intc — Cauchy integral
• intl — Cauchy integral
• int2d — definite 2D integral by quadrature and cubature method 