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Fig. 3.6
Goniometricus's MP graph related to the MP grammar given in Table 3.10
The least square evaluation of the following system of equations with unknowns
c
1
,...
c
6
will provide the regulators of the grammar we are searching for:
⎧
⎨
c
1
+
c
2
C
[
0
]
−
c
3
S
[
0
]
−
c
4
C
[
0
]
=
Δ
C
[
0
]
c
3
S
]
.......................................
=
......
c
1
+
[
0
]+
c
4
C
[
0
]
−
c
5
−
c
6
S
[
0
]
=
Δ
S
[
0
(3.15)
⎩
c
2
C
[
n
]
−
c
3
S
[
0
]
−
c
4
C
[
n
]
=
Δ
C
[
n
]
c
3
S
[
n
]+
c
4
C
[
n
]
−
c
5
−
c
6
S
[
0
]
=
Δ
S
[
n
]
.
The least square evaluation of system (3.15) provides sine and cosine with an ap-
proximation error of 10
−
14
. The initial values of
C
and
S
are 4
,
3 respectively (for
10
−
3
keeping fluxes positive),
τ
=
×
2
π
(the period is 2
π
), and
ν
=
μ
=
1. The
values of the constants are reported in Table 3.11.
Ta b l e 3 . 1 1
Goniometricus's MP grammar. Parameters
C
−
3and
S
−
3 shift the values of
C
and
S
in the interval
[
−
1
,
1
]
.
r
1
:0
→
C
ϕ
1
=
0
.
0030015
+
0
.
0009995
C
r
2
:
C
→
S
ϕ
2
=
0
.
001
C
+
0
.
001
S
r
3
:
S
→
0
ϕ
3
=
0
.
0029985
+
0
.
0010005
S
3.3.2
Generalization of the Goniometricus Model
The approximation power of MP grammars may increase by considering “memo-
ries”. Given a substance
A
, we denote by
A
−
m
(
m
∈
N
)the
memory
of
A
of level
m
.
It is a substance obtained from
A
such that
A
−
m
[
i
]=
A
[
i
−
m
]
. Therefore its time
series
T
A
−
m
is obtained from the time series
T
A
=[
t
A
,
1
,
t
A
,
2
,...,
t
A
,
n
]
of
A
by the
