A short overiew of files needed to proof periodic orbis of the Mackey-Glass equation.
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Scripts (non-intlab):
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MG_orbit_guess.m
	Provides a first guess for the numerical solutions in the case where MG has a long
	stable periodic solution.

MG_orbit_newton.m
	Refine the solution found in MG_orbit_guess.m and computes the auxiliary variables needed 
	for automatic differentiation scheme.
		
MG_continue.m
	Continues and subsequently refines the solution found in MG_orbit_newton.m to other 
	possible parameters, like the ones used to find the solutions in the chaotic regime.



Scripts (intlab; these assume intlab has been initiated)
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proof_orbit_11.m
	Used to construct the proof of the short length solution for the first set of parameters
	(Solution 1 in Table 1).
	
proof_orbit_12.m
	Used to construct the proof of the medium length solution for the first set of paramters
	(Solution 3 in Table 1).

proof_orbit_22.m
	Used to construct the proof of the short length solution for the chaotic regime
	(Solution 2 in Table 1).
	
proof_orbit_21.m
	Used to construct the proof of the medium length solution for the chaotic regime
	(Solution 4 in Table 1).


Functions (non-intlab)
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dde_integ.m
	A numerical forward integrator. (A simplified wrapper for dde23)
		
F_0.m
	Numerically approximates F from for Wright's equation

DF_0.m
	Numerically calculates the n \times n approximate derivative of F_0.
	

plot_solution.m
	Plots solutions.
	
x2c.m
	Used to split x = (\eta_0,\eta_1,\eta_2,\eta_3,c_0,c_1,c_2,c_3) into its individual components.


Functions (intlab)
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A_extend.m
	Used to extend the approximate derivative in order to calculate the Y-bound.
	
DcG_IA.m
	Computes the middle 2n+1 rows of the derivative of G, where n is derived from the length 
	of non-zero components of c and the number of padded zeroes (n_pad). It also outputs the 
	absolute value of the 	components of each psuedo-convolution operator, which will be used 
	in the Z1 bounds.
	
DetaG_IA.m
	Computes the derivative of G w.r.t. \eta.

DG_IA.m
	Combines the values of DcG_IA and DetaG_IA in a single matrix.
	
F_IA.m
	Computes F using interval arithmetic.
	
norm_X.m
	Computes the norm for a given nu.
	
norm_X_op.m
	Computes the operator norm for a given nu.
	
symm_mat.m
	Symmetrises a given matrix as remarked in Section 4.5.
	
symm_vec.m
	Symmetrises a given vector as remarked in Section 4.5.

x2c.m
	Used to split x = (\eta_0,\eta_1,\eta_2,\eta_3,c_0,c_1,c_2,c_3) into its individual components.
	(intlab version)
	
rig_conv_2sided.m	
	Computes rigorous enclosures for the two-sided convolutions based on fft. 
		
Y_bound.m
	Computes the Y-bound.
	
Z0_bound.m
	Computes the Z_0 bound.
	
Z1_bound.m
	Computes the Z_1 bound from the Z^1_finite (Z1L_IA) and Z^1_infinite (Z1R_IA) bounds.
	
Z1_matrix.m
	Computes the matrix needed for the Z^1_finite (Z1L_IA) bound.
	
Z1L_bound.m
	Computes the Z^1_finite Bound.

Z1R_bound.m
	Computes the Z^1_infinite Bound.

Z2_bound_toep.m
	Computes the Z^2 bound.
	
Z3_bound_toep.m
	Computes the Z^3 bound.





