Astronomical Applications Department, U.S. Naval Observatory Sensitivity Integral Page 27
=
(
)
F
,
1
2
1
2
k
( )
A
k
1
k
k
( )
B
k
1
k
is purely a function of the frequencies (and coefficients
a
i
) and is well-behaved. Hence,
the frequency function
(
)
F
,
1
2
need be calculated only once per integration over the sky
, making evaluation of
very inexpensive.
6.3.3. Fortran Subroutines
Now we create optimized fortran subroutines to calculate the fourth-order expansion,
defint_x4
(
)
S
4
, , , ,
,
r s g T
1
1
(
)
+
1
1
2520
s
2
r
2
1
5
(
)
- +
1
g
2
=
280 a
4
(
)
+
+
2
3
3 (
)
+
+
+
+
+
+
6
6
5
15
4
21
3
18
2
9
3
1
4
315 a
3
(
)
+
2 (
)
+
+
2
2
2 (
)
+
+
+
+
4
4
3
6
2
4
2
1
3
+
360 a
2
(
)
+
+
+
+
+
+
6
7
5
21
4
35
3
35
2
21
7
1
2
+
420 a
1
(
)
+
2 (
)
+
+
2
3
3 (
)
+ +
2
1
1
504 a
0
(
)
+
+
+
+
5
10
10
2
5
3
4
+
+
and the corresponding error function,
. First we process the series approximation.
Create a Maple procedure to evaluate the series approximation expression.
:=
tmp
(
)
subs
,
,
(
)
seq
,
=
a
k
.
a k
=
k
..
0
4
=
1
1
(
)
rhs defint_x4
F :=
(
)
optimize/makeproc
,
[
]
(
)
optimize
,
tmp tryhard
=
parameters
[
]
,
,
, , ,
1
(
)
seq
,
.
a i
=
i
..
0
4
s r g
The cost of the optimized expression sequence is
(
)
cost
(
)
optimize
,
tmp tryhard
+
+
32 additions
52 multiplications
10 assignments
Create a fortran subroutine from the Maple procedure.
fortran F optimized
=
mode
double
=
precision
double
,
,
,
,
(
=
filename
"d:/FAMEStuff/PhotonSensitivity/expansion4.f" )
Next, we split the error function into geometrical and frequency parts, so that the
frequency part can be evaluated independently. Create a Maple procedure to evaluate the
frequency function:
(
)
remove
,
,
has
(
)
rhs error_expr
s
Page 27
