<s>
QUADPACK	B-Algorithm
is	O
a	O
FORTRAN	O
77	O
library	B-Library
for	O
numerical	B-Algorithm
integration	I-Algorithm
of	O
one-dimensional	O
functions	O
.	O
</s>
<s>
It	O
was	O
included	O
in	O
the	O
SLATEC	B-Application
Common	I-Application
Mathematical	I-Application
Library	I-Application
and	O
is	O
therefore	O
in	O
the	O
public	O
domain	O
.	O
</s>
<s>
The	O
individual	O
subprograms	O
are	O
also	O
available	O
on	O
netlib	B-Application
.	O
</s>
<s>
The	O
GNU	B-Application
Scientific	I-Application
Library	I-Application
reimplemented	O
the	O
QUADPACK	B-Algorithm
routines	O
in	O
C	B-Language
.	O
SciPy	B-Application
provides	O
a	O
Python	B-Language
interface	O
to	O
part	O
of	O
QUADPACK	B-Algorithm
.	O
</s>
<s>
The	O
main	O
focus	O
of	O
QUADPACK	B-Algorithm
is	O
on	O
automatic	O
integration	O
routines	O
in	O
which	O
the	O
user	O
inputs	O
the	O
problem	O
and	O
an	O
absolute	B-Algorithm
or	I-Algorithm
relative	I-Algorithm
error	I-Algorithm
tolerance	O
and	O
the	O
routine	O
attempts	O
to	O
perform	O
the	O
integration	O
with	O
an	O
error	O
no	O
larger	O
than	O
that	O
requested	O
.	O
</s>
<s>
There	O
are	O
nine	O
such	O
automatic	O
routines	O
in	O
QUADPACK	B-Algorithm
,	O
in	O
addition	O
to	O
a	O
number	O
of	O
non-automatic	O
routines	O
.	O
</s>
<s>
All	O
but	O
one	O
of	O
the	O
automatic	O
routines	O
use	O
adaptive	B-Algorithm
quadrature	I-Algorithm
.	O
</s>
<s>
Each	O
of	O
the	O
adaptive	B-Algorithm
routines	O
also	O
have	O
versions	O
suffixed	O
by	O
E	O
that	O
have	O
an	O
extended	O
parameter	O
list	O
that	O
provides	O
more	O
information	O
and	O
allows	O
more	O
control	O
.	O
</s>
<s>
The	O
two	O
general-purpose	O
routines	O
most	O
suitable	O
for	O
use	O
without	O
further	O
analysis	O
of	O
the	O
integrand	O
are	O
QAGS	O
for	O
integration	O
over	O
a	O
finite	O
interval	O
and	O
QAGI	B-Algorithm
for	O
integration	O
over	O
an	O
infinite	O
interval	O
.	O
</s>
<s>
These	O
two	O
routines	O
are	O
used	O
in	O
GNU	B-Language
Octave	I-Language
(	O
the	O
quad	O
command	O
)	O
and	O
R	B-Language
(	O
the	O
integrate	O
function	O
)	O
.	O
</s>
<s>
QAGS	O
uses	O
global	O
adaptive	B-Algorithm
quadrature	I-Algorithm
based	O
on	O
21-point	O
Gauss	B-Algorithm
–	I-Algorithm
Kronrod	I-Algorithm
quadrature	I-Algorithm
within	O
each	O
subinterval	O
,	O
with	O
acceleration	B-Algorithm
by	O
Peter	O
Wynn	O
's	O
epsilon	O
algorithm	O
.	O
</s>
<s>
QAGI	B-Algorithm
is	O
the	O
only	O
general-purpose	O
routine	O
for	O
infinite	O
intervals	O
,	O
and	O
maps	O
the	O
infinite	O
interval	O
onto	O
the	O
semi-open	O
interval	O
(	O
0	O
,	O
1 ]	O
using	O
a	O
transformation	O
then	O
uses	O
the	O
same	O
approach	O
as	O
QAGS	O
,	O
except	O
with	O
15-point	O
rather	O
than	O
21-point	O
Gauss	B-Algorithm
–	I-Algorithm
Kronrod	I-Algorithm
quadrature	I-Algorithm
.	O
</s>
<s>
For	O
an	O
integral	O
over	O
the	O
whole	O
real	O
line	O
,	O
the	O
transformation	O
used	O
is	O
:	O
This	O
is	O
not	O
the	O
best	O
approach	O
for	O
all	O
integrands	O
:	O
another	O
transformation	O
may	O
be	O
appropriate	O
,	O
or	O
one	O
might	O
prefer	O
to	O
break	O
up	O
the	O
original	O
interval	O
and	O
use	O
QAGI	B-Algorithm
only	O
on	O
the	O
infinite	O
part	O
.	O
</s>
