The following MATLAB program determines the first and second derivatives of the data given in the problem applying the finite difference schemes and developing
a custom user defined function firstsecondderivatives(x,y).
For second derivative:
The finite difference schemes used are the following:
For first derivative:
- At first point (three point forward)
- At last point (three point backward)
- At all other points (two point central)
For second derivative:
- At first point (four point forward)
- At last point (four point backward)
- At all other points (three point central)
DATA SET
| x | -1 | -0.5 | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 |
| f(x) | -3.632 | -0.3935 | 1 | 0.6487 | -1.282 | -4.518 | -8.611 | -12.82 | -15.91 | -15.88 | -9.402 | 9.017 |
function [dy, ddy] = firstsecondderivatives(x,y)
% The function calculates the first & second derivative of a function that is given by a set
% of
points. The first derivatives at the first and last points are calculated by
% the 3
point forward and 3 point backward finite difference scheme respectively.
% The
first derivatives at all the other points are calculated by the 2 point
% central
approach.
% The
second derivatives at the first and last points are calculated by
% the 4
point forward and 4 point backward finite difference scheme respectively.
% The
second derivatives at all the other points are calculated by the 3 point
% central
approach.
n = length (x);
dy = zeros;
ddy = zeros;
% Input
variables:
% x:
vector with the x the data points.
% y:
vector with the f(x) data points.
% Output
variable:
% dy:
Vector with first derivative at each point.
% ddy:
Vector with second derivative at each point.
dy(1) = (-3*y(1)
+ 4*y(2) - y(3)) / 2*(x(2) - x(1)); % First derivative
ddy(1) = (2*y(1)
- 5*y(2) + 4*y(3) - y(4)) / (x(2) - x(1))^2; % Second
derivative
for i =
2:n-1
dy(i) = (y(i+1)
- y(i-1)) / 2*(x(i+1) - x(i-1));
ddy(i) = (y(i-1)
- 2*y(i) + y(i+1)) / (x(i-1) - x(i))^2;
end
dy(n) = (y(n-2)
- 4*y(n-1) + 3*y(n)) / 2*(x(n) - x(n-1));
ddy(n) =
(-y(n-3) + 4*y(n-2) - 5*y(n-1) + 2*y(n)) / (x(n) - x(n-1))^2;
figure(1)
subplot (3,1,1)
plot (x, y, 'linewidth', 2, 'color', 'k'),
grid
title('x
VS f(x)')
ylabel ('f(x)')
subplot (3,1,2)
plot (x, dy, 'linewidth', 2, 'color', 'b'),
grid
title('x
VS dy')
ylabel ('dy:
First Derivative')
subplot (3,1,3)
plot (x, ddy, 'linewidth', 2, 'color', 'r'),
grid
title('x
VS dy')
ylabel ('ddy:
Second Derivative'), xlabel('x')
figure(2)
plot (x, y, 'linewidth', 2, 'color', 'k');
hold on;
plot (x, dy, 'linewidth', 2, 'color', 'b');
hold on;
plot (x, ddy, 'linewidth', 2, 'color', 'r'),
legend
ylabel ('y,
dy, ddy'), xlabel('x')
end
The following
operations are done in MATLAB command window to run the above function.
Program Outputs:
>> x = -1:0.5:4.5;
>> y=[-3.632 -0.3935 1
0.6487 -1.282 -4.518 -8.611 -12.82 -15.91 -15.88 -9.402 9.017];
>>
firstsecondderivatives(x,y)
ans =
Columns 1 through 11
2.0805
2.3160 0.5211 -1.1410
-2.5833 -3.6645 -4.1510
-3.6495 -1.5300 3.2540
12.4485
Column 12
12.1947
We can also plot the derivatives, below is the figure for the function, first and second order derivatives respectively.
 |
| Plotting the first and second order numerical derivatives |
