.\"
.de Id
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.de Sp
.if n .sp
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.TH integrate 4rheolef "rheolef-7.0" "rheolef-7.0" "rheolef-7.0"
.\" 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 SYNOPSIS

.\" begin_example
.Sp
.nf
 Float integrate (geo domain);
 Float integrate (geo domain, quadrature_option qopt);
 Value integrate (geo domain, Expression, quadrature_option qopt);
.PP
 field integrate (Expression);
 field integrate (Expression, quadrature_option qopt);
 field integrate (geo domain, Expression);
 field integrate (geo domain, Expression, quadrature_option qopt);
.PP
 form integrate (Expression);
 form integrate (Expression, integrate_option qopt);
 form integrate (geo domain, Expression);
 form integrate (geo domain, Expression, integrate_option qopt);
.Sp
.fi
.\" end_example
.SH EXAMPLE

For computing the measure of a domain:
.\" begin_example
.Sp
.nf
  Float meas_omega = integrate (omega);
.Sp
.fi
.\" end_example
For computing the integral of a function:
.\" begin_example
.Sp
.nf
  Float f (const point& x);
  ...
  quadrature_option qopt;
  qopt.set_order (3);
  Float int_f = integrate (omega, f, qopt);
.Sp
.fi
.\" end_example
The last argument specifies the quadrature formulae
(see quadrature_option(2))
used for the computation of the integral.
The function can be replaced by any field-valued expression (see field(2)).
For computing a right-hand-side of a variational formulation
with the previous function \fBf\fP:
.\" begin_example
.Sp
.nf
  space Xh (omega, "P1");
  test v (Xh);
  field lh = integrate (f*v);
.Sp
.fi
.\" end_example
For computing a bilinear form:
.\" begin_example
.Sp
.nf
  trial u (Xh);
  test v (Xh);
  form m = integrate (u*v);
.Sp
.fi
.\" end_example
The expression \fBu*v\fP can be replaced by any bilinear expression (see field(2)).
.SH DEFAULT ARGUMENTS

In the case of a linear or bilinear form, the domain is optional: by default it is
the full domain definition of the test function.
Also, the quadrature formulae is optional: by default, its order
is \fB2*k+1\fP where \fBk\fP is the polynomial degree of the
\fBXh\fP space associated to the test function \fBv\fP.
When both a test \fBu\fP and trial \fBv\fP functions are supplied, let k1 and k2 be their polynomial degrees.
Then the default quadrature is chosen to be exact at least for k1+k2+1 polynoms.
When the integration is performed 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);
  field l3h = integrate ("boundary", f*v);
.Sp
.fi
.\" end_example
For convenience, only the domain name can be supplied.
.\" END

.\" LENGTH = 5
.SH SEE ALSO
test(2), test(2), quadrature_option(2), field(2), field(2)
.SH COPYRIGHT
Copyright (C) 2000-2018 Pierre Saramito <Pierre.Saramito@imag.fr> GPLv3+: GNU GPL version 3 or later <http://gnu.org/licenses/gpl.html>.
This is free software: you are free to change and redistribute it.  There is NO WARRANTY, to the extent permitted by law.