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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])

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