.\"
.de Id
..
.de Sp
.if n .sp
.if t .sp 0.4
..
.TH integrate 4rheolef "rheolef-6.5" "rheolef-6.5" "rheolef-6.5"
.\" label: /*Class:integrate
.SH NAME
\fBintegrate\fP - integrate a function or an expression
.\" skip: @findex  integrate
.SH DESCRIPTION

Integrate an expression over a domain by using a quadrature formulae.
There are three main usages of the integrate function, depending upon the
type of the expression.
(i) When the expression is a numerical one, it leads to a numerical value.
(ii) When the expression involves a symbolic test-function see test(2),
the result is a linear form, represented by the \fBfield\fP class.
(iii) When the expression involves both  symbolic trial- and test-functions see test(2),
the result is a bilinear form, represented by the \fBfield\fP class.
.SH SYNOPSYS

.\" begin_example
.Sp
.nf
 template <class T, class M, class Expr>
 T integrate (const geo_basic<T,M>& omega, Expr expr,
   quadrature_option_type qopt)
.PP
 template <class T, class M, class Expr>
 field integrate (const geo_basic<T,M>& omega, VFExpr expr,
   quadrature_option_type qopt)
.PP
 template <class T, class M, class Expr>
 form integrate (const geo_basic<T,M>& omega, VFExpr expr,
   form_option_type fopt)
.Sp
.fi
.\" end_example
.SH EXAMPLE

.\" begin_example
.Sp
.nf
  Float f (const point& x);
  ...
  quadrature_option_type qopt;
  Float value = integrate (omega, f, qopt);
  field lh = integrate (omega, f*v, qopt);
.Sp
.fi
.\" end_example
The last argument specifies the quadrature formulae used
for the computation of the integral.
The expression can be any function, classs-function or
any linear or nonlinear field expression see field(2).
.SH DEFAULT ARGUMENTS

In the case of a linear form, the domain is optional: by default it is
the full domain definition of the test function.
.\" begin_example
.Sp
.nf
  field l1h = integrate (f*v, qopt);
.Sp
.fi
.\" end_example
When the integration is perfomed on a subdomain, this subdomain
simply replace the first argument and a domain name could also be used:
.\" begin_example
.Sp
.nf
  field l2h = integrate (omega["boundary"], f*v, qopt);
  field l3h = integrate ("boundary", f*v, qopt);
.Sp
.fi
.\" end_example
The quadrature formulae is required, except when a test and/or trial
function is provided in the expression to integrate.
In that case, the quadrature formulae is deduced from the space
containing the test (or trial) function.
When a test function is suppied, let k be its polynomial degree.
Then the default quadrature is
choosen to be exact at least for 2*k+1 polynoms.
When both a test and trial functions are suppied, let k1 and k2 be their polynomial degrees.
Then the default quadrature is
choosen to be exact at least for k1+k2+1 polynoms.
Also, when the expression is a constant, the quadrature function is optional:
in that case, the constant is also optional and the following call:
.\" begin_example
.Sp
.nf
  Float meas = integrate (omega);
.Sp
.fi
.\" end_example
is valid and returns the measure of the domain.
.\" END

.\" LENGTH = 3
.SH SEE ALSO
test(2), test(2), field(2)