# auxiliary.maple -- Maple code to compute ADM auxiliary quantities
# $Header$

#
# auxiliary - compute and optionally generate C code for auxiliary quantities
## inverse_metric - metric determinant and inverse metric
## extrinsic_curvature_trace_raise - compute trace+contravariant K_ij
#

###############################################################################

#
# This function computes various auxiliary variables which are
# algebraic functions of the basic ADM fields {g_dd,K_dd}, and optionally
# also generates C code for them (the auxiliar variables).
#
# Code files are written to the directory ../gr.cg/ .
#
# Inputs (Maple):
#	N
#	g_dd
#	K_dd
#
# Outputs (Maple + code file "inverse_metric.c"):
#	g_uu = g_uu__fnd(g_dd)
#
# Outputs (Maple + code file "extrinsic_curvature_trace_raise.c"):
#	K = K__fnd(g_uu, K_dd)
#	K_uu = K_uu__fnd(g_uu, K_dd)
#
# Arguments:
# cg_flag = (in) false ==> Compute Maple expressions only.
#		 true ==> Compute Maple expressions and generate C code.
# 
auxiliary :=
proc(cg_flag::boolean)

inverse_metric(cg_flag);
extrinsic_curvature_trace_raise(cg_flag);

NULL;
end;

###############################################################################

#
# This function computes the inverse metric, and optionally also
# generates C code for this.
#
# Inputs (Maple):
#	N
#	g_dd
#
# Outputs (Maple + code file "inverse_metric.c"):
#	g_uu = g_uu__fnd(g_dd)
#
# Arguments:
# cg_flag = (in) false ==> Compute Maple expressions only.
#		 true ==> Compute Maple expressions and generate C code.
#
inverse_metric :=
proc(cg_flag::boolean)
global
  @include "../maple/coords.minc",
  @include "../maple/gfa.minc",
  @include "../gr/gr_gfas.minc";

printf("%a...\n", procname);

assert_fnd_exists(g_dd);
assert_fnd_exists(g_uu, fnd);

g_uu__fnd := linalg[inverse](g_dd);

if (cg_flag)
   then codegen2(g_uu__fnd, 'g_uu', "../gr.cg/inverse_metric.c");
fi;

NULL;
end;

###############################################################################

#
# This function computes the trace of the extrinsic curvature, K,
# and the contravariant extrinsic curvature, K_uu, and optionally also
# generates C code for this.
#
# Inputs (as global variables):
#	N
#	K_dd
#	g_uu
#
# Outputs (Maple + code file "extrinsic_curvature_trace_raise.c"):
#	K = K__fnd(g_uu, K_dd)
#	K_uu = K_uu__fnd(g_uu, K_dd)
#
# Arguments:
# cg_flag = (in) false ==> Set up Maple variables only.
#		 true ==> Set up Maple variables and generate C code.
#
extrinsic_curvature_trace_raise :=
proc(cg_flag::boolean)
global
  @include "../maple/coords.minc",
  @include "../maple/gfa.minc",
  @include "../gr/gr_gfas.minc";
local i, j, m, n;

printf("%a...\n", procname);

assert_fnd_exists(g_uu);
assert_fnd_exists(K_dd);
assert_fnd_exists(K, fnd);
assert_fnd_exists(K_uu, fnd);

K__fnd := simplify(
	    msum('g_uu[i,j] * K_dd[i,j]', 'i'=1..N, 'j'=1..N)
		  );

	for i from 1 to N
	do
	for j from i to N		# upper triangle only
	do
	K_uu__fnd[i,j] := simplify(
			    msum('g_uu[i,m] * g_uu[j,n] * K_dd[m,n]',
				 'm'=1..N, 'n'=1..N)
				  );
	end do;
	end do;

if (cg_flag)
   then codegen2([K__fnd, K_uu__fnd],
		 ['K', 'K_uu'],
		 "../gr.cg/extrinsic_curvature_trace_raise.c");
fi;

NULL;
end;