demo_interpolation.html

Demo of Interpolation in MATLAB

hold <span class="string">on</span>;
plot(u,p,<span class="string">':g'</span>,<span class="string">'linewidth'</span>,1.5)
plot(t,v,<span class="string">'-r'</span>,<span class="string">'linewidth'</span>,1.5)
hold <span class="string">off</span>;
figure(h2)
</pre><img vspace="5" hspace="5" src="demo_interpolation_06.png"> <h2>Piecewise Linear Interpolation<a name="7"></a></h2>
         <p>The Matlab plotting routines use piecewise linear interpolation</p><pre class="codeinput">h2=figure(2);
plot(x,y,<span class="string">'-o'</span>,<span class="string">'markerfacecolor'</span>,<span class="string">'b'</span>,<span class="string">'markersize'</span>,9,<span class="string">'linewidth'</span>,1.5)
axis([0 7 10 25])
title(<span class="string">'Piecewise Linear Interpolation'</span>,<span class="string">'fontsize'</span>,20)
xlabel(<span class="string">'x'</span>,<span class="string">'fontsize'</span>,18')
ylabel(<span class="string">'y'</span>,<span class="string">'fontsize'</span>,18')
grid <span class="string">on</span>
</pre><img vspace="5" hspace="5" src="demo_interpolation_07.png"> <h2>Cubic Spline Interpolation using pchip<a name="8"></a></h2>
         <p>pchip is a Matlab function that implements "Piecewise Cubic Hermite Interpolating Polynomial" as described in NCM, Chapter
            2.
         </p><pre class="codeinput">w = pchip(x,y,t);
h2=figure(2);
plot(x,y,<span class="string">'o'</span>,<span class="string">'markerfacecolor'</span>,<span class="string">'b'</span>,<span class="string">'markersize'</span>,9)
axis([0 7 0 25])
title(<span class="string">'Cubic Interpolation with pchip'</span>,<span class="string">'fontsize'</span>,20)
xlabel(<span class="string">'x'</span>,<span class="string">'fontsize'</span>,18')
ylabel(<span class="string">'y'</span>,<span class="string">'fontsize'</span>,18')
grid <span class="string">on</span>
hold <span class="string">on</span>;
plot(u,p,<span class="string">':g'</span>,<span class="string">'linewidth'</span>,1.5)
plot(t,v,<span class="string">':r'</span>,<span class="string">'linewidth'</span>,1.5)
plot(t,w,<span class="string">'-m'</span>,<span class="string">'linewidth'</span>,1.5)
hold <span class="string">off</span>;
figure(h2);
</pre><img vspace="5" hspace="5" src="demo_interpolation_08.png"> <h2>Cubic Spline Interpolation using spline<a name="9"></a></h2>
         <p>spline is a Matlab function that implements "Piecewise Cubic Hermite Interpolating Polynomial" as described in NCM, Chapter
            2. It uses a different condition to estimate slopes at the "knots".
         </p><pre class="codeinput">z = spline(x,y,t);
h2=figure(2);
plot(x,y,<span class="string">'o'</span>,<span class="string">'markerfacecolor'</span>,<span class="string">'b'</span>,<span class="string">'markersize'</span>,9)
axis([0 7 0 25])
title(<span class="string">'Cubic Interpolation with spline'</span>,<span class="string">'fontsize'</span>,20)
xlabel(<span class="string">'x'</span>,<span class="string">'fontsize'</span>,18')
ylabel(<span class="string">'y'</span>,<span class="string">'fontsize'</span>,18')
grid <span class="string">on</span>
hold <span class="string">on</span>;
plot(u,p,<span class="string">':g'</span>,<span class="string">'linewidth'</span>,1.5)
plot(t,v,<span class="string">':r'</span>,<span class="string">'linewidth'</span>,1.5)
plot(t,w,<span class="string">':m'</span>,<span class="string">'linewidth'</span>,1.5)
plot(t,z,<span class="string">'-k'</span>,<span class="string">'linewidth'</span>,1.5);
hold <span class="string">off</span>;
</pre><img vspace="5" hspace="5" src="demo_interpolation_09.png"> <p class="footer"><br>
            Published with MATLAB&reg; 7.6<br></p>
      </div>
      <!--
##### SOURCE BEGIN #####
%% Demonstrations of Polynomial Interpolation
% Polynomial interpolation is a method for computing intermediate values
% between given data points. All interpolation methods make assumptions
% about the behavior of the function to compute values at intermediate
% points. The different models produce different interpolation curves. This
% demonstration will illustrate linear interpolation, full polynomial
% interpolation that constructs a polynomial with the number of
% coefficients equal to the number of points, and two methods of cubic
% spline interpolation. The development follows that in the book Numerical
% Computing with Matlab" by Cleve Moler, Chapter 3.

%% Example 1: A small number of data points

% Set up some data points
x = (0:3)';
y = [-5; -6; -1; 16];
disp('Data Points')
disp([x';y'])

h1 = figure(1);
plot(x,y,'o','markerfacecolor','b','markersize',9)
title('Data Points','fontsize',20)
xlabel('x','fontsize',18')
ylabel('y','fontsize',18')
grid on

%% Linear Interpolation
% The standard Matlab plot routine draws straight lines between the data
% points.

hold on;
plot(x,y,'-o','linewidth',1.5)
title('Linear Fit','fontsize',20)
hold off;

%% Cubic Polynomial Fit
% Here we will calculate a cubic polynomial to pass through the data
% points. We will do the calculation by setting up a matrix of powers of x
% and then inverting the X*c=y form to get the coefficients. When four data
% points are given, this is a full polynomial fit. The problem with this
% method is that X is poorly conditioned. This method is not used in
% practice.

V = [x.^3 x.^2 x ones(length(x),1)]
c = V\y

% Specify a vector of independent values
u = (0:.1:3)';

% Construct an array of the powers of u
U = [u.^3 u.^2 u ones(length(u),1)];

% Calculate the polynomial values and plot
p = U*c;
plot(x,y,'REPLACE_WITH_DASH_DASHo','linewidth',1.5)
title('Full Cubic Fit - Matrix Method','fontsize',20)
xlabel('x','fontsize',18')
ylabel('y','fontsize',18')
grid on
hold on;
plot(u,p,'r','linewidth',1.5)
hold off

% Condition number of the matrix X
disp('Condition Number of V')
disp(cond(V,2));

%% A Second Example
x = (1:6)';
y = [16; 18; 21; 17; 15; 12];
h2 = figure(2);
plot(x,y,'o','markerfacecolor','b','markersize',9)
axis([0 7 0 25])
title('Data Points','fontsize',20)
xlabel('x','fontsize',18')
ylabel('y','fontsize',18')
grid on

%% Solve for the coefficients using linear equations
V = [x.^3 x.^2 x ones(length(x),1)]
c = V\y

% Specify a vector of independent values
u = (0.75:.05:6.25)';

% Construct an array of the powers of u
U = [u.^3 u.^2 u ones(length(u),1)];

% Calculate the polynomial values and plot
p = U*c;
hold on;
plot(u,p,'g','linewidth',1.5)
title('Cubic Polynomial Fit REPLACE_WITH_DASH_DASH Matrix Method','fontsize',20)
hold off

% Condition number of the matrix X
disp('Condition Number of V')
disp(cond(V,2));

%% Lagrange form of polynomial interpolation
% The function polyinterp.m does a full Lagrange form interpolation.
t = (0.75:.05:6.25)';
v = polyinterp(x,y,t);
h2=figure(2);
plot(x,y,'o','markerfacecolor','b','markersize',9)
axis([0 7 0 25])
title('Full Interpolation with Lagrange Form','fontsize',20)
xlabel('x','fontsize',18')
ylabel('y','fontsize',18')
grid on
hold on;
plot(u,p,':g','linewidth',1.5)
plot(t,v,'-r','linewidth',1.5)
hold off;
figure(h2)

%% Piecewise Linear Interpolation
% The Matlab plotting routines use piecewise linear interpolation
h2=figure(2);
plot(x,y,'-o','markerfacecolor','b','markersize',9,'linewidth',1.5)
axis([0 7 10 25])
title('Piecewise Linear Interpolation','fontsize',20)
xlabel('x','fontsize',18')
ylabel('y','fontsize',18')
grid on

%% Cubic Spline Interpolation using pchip
% pchip is a Matlab function that implements "Piecewise Cubic 
% Hermite Interpolating Polynomial" as described in NCM, Chapter 2.

w = pchip(x,y,t);
h2=figure(2);
plot(x,y,'o','markerfacecolor','b','markersize',9)
axis([0 7 0 25])
title('Cubic Interpolation with pchip','fontsize',20)
xlabel('x','fontsize',18')
ylabel('y','fontsize',18')
grid on
hold on;
plot(u,p,':g','linewidth',1.5)
plot(t,v,':r','linewidth',1.5)
plot(t,w,'-m','linewidth',1.5)
hold off;
figure(h2);

%% Cubic Spline Interpolation using spline
% spline is a Matlab function that implements "Piecewise Cubic 
% Hermite Interpolating Polynomial" as described in NCM, Chapter 2. It uses
% a different condition to estimate slopes at the "knots".

z = spline(x,y,t);
h2=figure(2);
plot(x,y,'o','markerfacecolor','b','markersize',9)
axis([0 7 0 25])
title('Cubic Interpolation with spline','fontsize',20)
xlabel('x','fontsize',18')
ylabel('y','fontsize',18')
grid on
hold on;
plot(u,p,':g','linewidth',1.5)
plot(t,v,':r','linewidth',1.5)
plot(t,w,':m','linewidth',1.5)
plot(t,z,'-k','linewidth',1.5); 
hold off;


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