<s>
PROSE	O
was	O
the	O
mathematical	O
4GL	B-Language
virtual	B-Architecture
machine	I-Architecture
that	O
established	O
the	O
holistic	O
modeling	O
paradigm	O
known	O
as	O
Synthetic	O
Calculus	O
(	O
AKA	O
MetaCalculus	O
)	O
.	O
</s>
<s>
It	O
was	O
the	O
first	O
commercial	O
language	O
to	O
employ	O
automatic	B-Algorithm
differentiation	I-Algorithm
(	O
AD	O
)	O
,	O
which	O
was	O
optimized	O
to	O
loop	O
in	O
the	O
instruction-stack	O
of	O
the	O
CDC	O
6600	O
CPU	O
.	O
</s>
<s>
During	O
some	O
or	O
maybe	O
all	O
of	O
the	O
iterative	O
model-subroutine	O
calls	O
the	O
engine	O
would	O
invoke	O
automatic	B-Algorithm
differentiation	I-Algorithm
of	O
the	O
formulas	O
in	O
the	O
model	O
holarchy	O
with	O
respect	O
to	O
the	O
model	O
's	O
input-unknowns	O
(	O
arguments	O
)	O
defined	O
in	O
the	O
calling	O
template	O
.	O
</s>
<s>
The	O
extended	O
arithmetic	O
of	O
automatic	B-Algorithm
differentiation	I-Algorithm
and	O
its	O
ability	O
to	O
dynamically	O
differentiate	O
numerical	O
integration	O
gave	O
rise	O
to	O
the	O
unique	O
mode	O
of	O
holarchy	O
modeling	O
illustrated	O
in	O
Figure	O
1	O
.	O
</s>
<s>
This	O
report	O
,	O
also	O
published	O
as	O
a	O
textbook	O
,	O
described	O
Quasilinearization	O
,	O
an	O
alternative	O
to	O
"	O
dynamic	B-Algorithm
programming	I-Algorithm
"	O
,	O
invented	O
by	O
the	O
same	O
author	O
,	O
Richard	B-Algorithm
Bellman	I-Algorithm
.	O
</s>
<s>
And	O
these	O
engines	O
require	O
partial	O
derivatives	O
of	O
all	O
downstream	O
variables	O
dependent	O
upon	O
those	O
unknowns	O
,	O
which	O
are	O
evaluated	O
by	O
automatic	B-Algorithm
differentiation	I-Algorithm
arithmetic	O
.	O
</s>
<s>
As	O
these	O
subprograms	O
include	O
the	O
numerical	O
integration	O
of	O
the	O
system	O
dynamics	O
(	O
differential	O
equations	O
)	O
model	O
,	O
the	O
automatic	B-Algorithm
differentiation	I-Algorithm
arithmetic	O
includes	O
differentiation	O
of	O
the	O
integration	O
algorithm	O
of	O
the	O
simulation	O
engine	O
(	O
and	O
the	O
quadrature	O
solver	O
)	O
,	O
to	O
evaluate	O
derivatives	O
of	O
the	O
boundary	O
(	O
end-point	O
of	O
the	O
integrated	O
curves	O
)	O
conditions	O
with	O
respect	O
to	O
the	O
initial	O
conditions	O
.	O
</s>
<s>
Only	O
automatic	B-Algorithm
differentiation	I-Algorithm
,	O
with	O
its	O
exact	O
chain-rule	O
propagation	O
is	O
feasible	O
.	O
</s>
<s>
Some	O
simulation	O
solvers	O
(	O
JANUS	O
,	O
MERCURY	O
,	O
MINERVA	O
,	O
MERLIN	O
and	O
PEGASUS	O
)	O
could	O
not	O
be	O
nested	O
in	O
automatic	B-Algorithm
differentiation	I-Algorithm
contexts	O
of	O
correlation	O
and	O
optimization	O
because	O
they	O
were	O
not	O
overloaded	O
for	O
automatic-differentiation	O
arithmetic	O
.	O
</s>
<s>
Thus	O
hybrid	O
versions	O
,	O
JANISIS	O
(	O
ISIS	O
or	O
JANUS	O
)	O
and	O
GEMINI	O
(	O
MERLIN	O
or	O
NEPTUNE	O
)	O
were	O
introduced	O
,	O
which	O
would	O
work	O
efficiently	O
in	O
automatic	B-Algorithm
differentiation	I-Algorithm
mode	O
or	O
ordinary	O
arithmetic	O
mode	O
(	O
differentiation	O
internally	O
turned	O
off	O
)	O
.	O
</s>
