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diff --git a/Lagrange_interpolation.m b/Lagrange_interpolation.m
new file mode 100644 (file)
index 0000000..e53a219
--- /dev/null
+++ b/Lagrange_interpolation.m
@@ -0,0 +1,32 @@
+function [pnx] = Lagrange_interpolation(z,)
+%%
+%  Implementation to evaluate an interpolating polynomial p_n(x) 
+%  at the point x = z. The polynomial uses the standard Lagrange
+%  basis functions.
+%
+%  INPUT:
+%
+%     z    - 1x1 value to evaluate
+%     ???
+%
+%  OUTPUT:
+%
+%     pnx - value of the polynomial interpolant at x = z
+%
+
+%%
+%  compute the polynomial interpolation sum evaluated at x = z
+   pnx =
+   for i =
+
+
+
+      for j =
+
+
+
+      end
+
+      pnx =
+   end
+end

diff --git a/diff.asv b/diff.asv
new file mode 100644 (file)
index 0000000..41e56bc
--- /dev/null
+++ b/diff.asv
@@ -0,0 +1,55 @@
+%% Center
+
+D_x_center = diag(0.5*ones(5,1),1) - diag(0.5*ones(5,1),-1);
+D_x_center(1,1) = 1;
+D_x_center(6,6) = 1;
+interval = [0:1/5:1];
+h = 1/5;
+D_x_center
+
+f_x_1 = [1;1;1;1;1;1];
+f_x_2 = interval.';
+f_x_3 = (interval.^2).';
+
+diff1 = 1/h * D_x_center * f_x_1
+diff2 = 1/h * D_x_center * f_x_2
+diff3 = 1/h * D_x_center * f_x_3
+
+%% Forward
+
+D_x_forward = diag(ones(5,1),1) - diag(ones(6,1), 0);
+D_x_forward(6,5) = -1;
+D_x_forward(6,6) = 1;
+
+interval = [0:1/5:1];
+h = 1/5;
+
+f_x_1 = [1;1;1;1;1;1];
+f_x_2 = interval.';
+f_x_3 = (interval.^2).';
+
+diff1 = 1/h * D_x_forward * f_x_1
+diff2 = 1/h * D_x_forward * f_x_2
+diff3 = 1/h * D_x_forward * f_x_3
+
+%% Backwards
+
+D_x_back = D_x_forward;
+
+interval = [0:1/5:1];
+h = 1/5;
+
+f_x_1 = [1;1;1;1;1;1];
+f_x_2 = interval.';
+f_x_3 = (interval.^2).';
+
+diff1 = 1/h * D_x_back * f_x_1
+diff2 = 1/h * D_x_back * f_x_2
+diff3 = 1/h * D_x_back * f_x_3
+
+
+
+%% Bias
+
+bound_low = ((-3*f_x_1(1)) + (4*f_x_1(2)) - (f_x_1(3)))/(2*h)
+bound_up = ((f_x_1(4)) - (4*f_x_1(5)) + (3*f_x_1(6)))/(2*h)
\ No newline at end of file

diff --git a/diff.m b/diff.m
new file mode 100644 (file)
index 0000000..64eff1a
--- /dev/null
+++ b/diff.m
@@ -0,0 +1,86 @@
+%% Center
+
+D_x_center = diag(0.5*ones(5,1),1) - diag(0.5*ones(5,1),-1);
+D_x_center(1,1) = 1;
+D_x_center(6,6) = 1;
+interval = [0:1/5:1];
+h = 1/5;
+D_x_center
+
+f_x_1 = [1;1;1;1;1;1];
+f_x_2 = interval.';
+f_x_3 = (interval.^2).';
+
+diff1 = 1/h * D_x_center * f_x_1
+diff2 = 1/h * D_x_center * f_x_2
+diff3 = 1/h * D_x_center * f_x_3
+
+%% Forward
+
+D_x_forward = diag(ones(5,1),1) - diag(ones(6,1), 0);
+D_x_forward(6,5) = -1;
+D_x_forward(6,6) = 1;
+
+interval = [0:1/5:1];
+h = 1/5;
+
+f_x_1 = [1;1;1;1;1;1];
+f_x_2 = interval.';
+f_x_3 = (interval.^2).';
+
+diff1 = 1/h * D_x_forward * f_x_1
+diff2 = 1/h * D_x_forward * f_x_2
+diff3 = 1/h * D_x_forward * f_x_3
+
+%% Backwards
+
+D_x_back = D_x_forward;
+
+interval = [0:1/5:1];
+h = 1/5;
+
+f_x_1 = [1;1;1;1;1;1];
+f_x_2 = interval.';
+f_x_3 = (interval.^2).';
+
+diff1 = 1/h * D_x_back * f_x_1
+diff2 = 1/h * D_x_back * f_x_2
+diff3 = 1/h * D_x_back * f_x_3
+
+
+
+%% Bias
+
+bound_low = ((-3*f_x_3(1)) + (4*f_x_3(2)) - (f_x_3(3)))/(2*h)
+bound_up = ((f_x_3(4)) - (4*f_x_3(5)) + (3*f_x_3(6)))/(2*h)
+
+
+
+%% EXP
+
+D_x_center = diag(0.5*ones(5,1),1) - diag(0.5*ones(5,1),-1);
+D_x_center(1,1) = 1;
+D_x_center(6,6) = 1;
+
+N = 5;
+interval = [0:1/N:1];
+h = 1/N;
+
+f_prim = zeros(length(interval), 1);
+for i=1:length(interval)
+    f_prim(i) = 4*exp(sin(4*interval(i)))*cos(4*interval(i));
+end
+f_prim
+
+
+
+
+
+
+
+
+
+
+
+
+

diff --git a/joukowsky_airfoil_data.m b/joukowsky_airfoil_data.m
new file mode 100644 (file)
index 0000000..63bb3a1
--- /dev/null
+++ b/joukowsky_airfoil_data.m
@@ -0,0 +1,2 @@
+x_joukow = [-1.7550, -1.7400, -1.6960, -1.6010, -1.4830, -1.2950, -1.0430, -0.8354, -0.5833, -0.2872, 0.1997, 0.8372, 1.3570,  1.69500];
+y_joukow = [ 0.0893,  0.1461,  0.2196,  0.3199,  0.4088,  0.5115,  0.6042,  0.6529,  0.6852,  0.6903, 0.6297, 0.4284, 0.1643, -0.04687];
\ No newline at end of file

diff --git a/joukowsky_airfoil_image.m b/joukowsky_airfoil_image.m
new file mode 100644 (file)
index 0000000..e120675
--- /dev/null
+++ b/joukowsky_airfoil_image.m
@@ -0,0 +1,30 @@
+function joukowsky_airfoil_image
+%%
+%  This perfoms the conformal mapping to create a particular instance of a Joukowsky airfoil
+%
+  theta = linspace(0,2*pi,300); % angles
+  r     = 1.0;                  % radius
+  s     = 0.175 + 1i*0.215;     % circle origin
+  zeta  = s + r*exp(1i*theta);  % original circle values in complex space (written in polar form)
+  k     = r - s;                % transformation parameter
+  z     = zeta + k^2./zeta;     % conformal mapping of airfoil surface in complex space
+  z     = z*exp(1i*pi/11);      % rotate airfoil location
+  
+  plot(-real(z),imag(z),'-k','LineWidth',1.5) % plot the transformed airfoil
+
+% plot the top and bottom curves
+%   plot(topx,topy,'-k',botx,boty,'-k','LineWidth',1.5)
+   hold on
+% set limits and size of figure
+   set(gcf,'Position',[10, 10, 1500, 400])
+   set(gca,'FontSize',16);
+   xlim([-2.0 2.0])
+   ylim([-0.2 0.9])
+% fill with gray color for better visualization
+   h1 = fill(-real(z),imag(z),'k','LineStyle','none');
+   set(h1,'facealpha',0.05);
+%
+%    for i = 1:length(z)
+%       plot(-real(z(i)),imag(z(i)),'ro','MarkerSize',10)
+%    end
+end
\ No newline at end of file