% Class 7 
% Coded by NFR on 2.5.19
% Piecewise Interpolation
%
clc
close all
t = [0 20 40 56 68 80 84 96 104 110];
v = [0 20 20 38 80 80 100 100 125 125];
tvec = linspace(0,110,1000);
% Linear Interpolation
subplot(2,2,1)
plot(t,v,'o')
hold on
v_vec = interp1(t,v,tvec,'linear');
plot(tvec,v_vec)
% Nearest Interpolation
subplot(2,2,2)
plot(t,v,'o')
hold on
v_vec = interp1(t,v,tvec,'nearest');
plot(tvec,v_vec)
% Spline Interpolation
subplot(2,2,3)
plot(t,v,'o')
hold on
v_vec = interp1(t,v,tvec,'spline');
plot(tvec,v_vec)
% Pchip Interpolation
subplot(2,2,4)
plot(t,v,'o')
hold on
v_vec = interp1(t,v,tvec,'pchip');
plot(tvec,v_vec)
% Quick example of 2 dimensional interpolation
Time = [1 2 3 4]; % Hours
Sugar = [2.2 3.4 5 7]; % Cups of sugar
Modulus = [3 5 8 9;4 7 10 12; 5 9 13 15;...
    6 10 14 18]; % measure of hardness
% What is the modulus of a cake that is 2.5 hr and 4.2 cups sugar?
Mod_Solve = interp2(Time, Sugar, Modulus, 2.5, 4.2,'cubic')
% SPLINES
% Let's use the t and v data sets again
% What is the velocity at t = 87 using splines
V_ans = spline(t,v,87)
% Plot on the pchip plot
plot(87,V_ans,'k+')
% Example 18.12 from the book
Func_1 = @(t) sin(t).^2;
xvec = linspace(0,2*pi,8);
yvec = Func_1(xvec);
figure
fplot(Func_1,[0 2*pi])
hold on
plot(xvec,yvec,'o')
% Part (a) cubic spline with not-a-knot conditions
xq = linspace(0, 2*pi,1000);
yq = spline(xvec,yvec,xq);
plot(xq,yq,'k-')
% Part (b) cubic spline + clamped end conditions
% Exact slope at beginning and end are zeros
d1 = @(t) 2*sin(t)*cos(t);
slopestart = d1(0);
slopeend = d1(2*pi);
yvec2 = [slopestart yvec slopeend];
yq_2 = spline(xvec,yvec2,xq);
plot(xq,yq_2,'g-')
% Examples of contour and surface plots
F2 = @(x,y) sin(x).*tan(y)./x.^2; % '.' are used for element wise notation
xvec = linspace(1,2,200);
yvec = linspace(.5,1,100);
[xmat, ymat] = meshgrid(xvec,yvec);
Z = F2(xmat,ymat);
figure
contourf(xvec,yvec,Z)
% Add a label
textstring = '\leftarrow this is an arrow';
text(1.5,.8,textstring)
% Output the plot as an image
saveas(gcf,'Output.png')
% Draw lines on a plot
line([1.2 1.2] ,[0.5 1])
figure
surf(xvec,yvec,Z)
%
% Here is a copy of 18.12 solution from 2017 class
% Practice with 18.12 from the book
%
F1 = @(x) sin(x).^2;
xp = linspace(0,2*pi,8);
yp = F1(xp);
% Part a, use the default 'not-a-knot' condition for the spline
xq = linspace(0,2*pi,1000);
yq = spline(xp,yp,xq);
plot(xp,yp,'o',xq,yq,':')
hold on
fplot(F1,[0 2*pi])
% Part b where we 'clamp' the ends
d1 = @(t) 2*sin(t)*cos(t);
slopestart = d1(0);
slopeend = d1(2*pi);
yp_clamp = [slopestart yp slopeend];
yq_clamp = spline(xp,yp_clamp,xq);
figure
plot(xp,yp,'o',xq,yq_clamp,':')
hold on
fplot(F1,[0 2*pi])
% Part c - use pchip for the interpolation
yq_pchip = pchip(xp,yp,xq);
figure
plot(xp,yp,'o',xq,yq_pchip,':')
hold on
fplot(F1,[0 2*pi])