Page 310 - Numerical Analysis Using MATLAB and Excel

Chapter 7 Finite Differences and Interpolation

⁄
MATLAB interpolates the impedance at ω = 792 rad s and displays the following message:

Magnitude of Z at w=792 rad/s is 217.034 Ohms

Two−dimensional plots were briefly discussed in Chapter 1. For convenience, we will review the
following commands which can be used for two−dimensional interpolation.
1. mesh(Z) − Plots the values in the matrix Z as height values above a rectangular grid, and con-
nects adjacent points to form a mesh surface.

2. [X,Y]=meshgrid(x,y) − Generates interpolation arrays which contain all combinations of the
x and y points which we specify. X and Y comprise a pair of matrices representing a rectangular
,
(=
grid of points in the x – y plane. Using these points, we can form a function z f x y ) where
z is a matrix.

Example 7.13

Generate the plot of the function
sin R
Z = ----------- (7.61)
R
⁄
xy
in three dimensions , , and . This function is the equivalent of the function y = sin x x in
z
two dimensions. Here, is a matrix that contains the distances from the origin to each point in
R
the pair of XY,[ ] matrices that form a rectangular grid of points in the x – y plane.
Solution:

The matrix that contains the distances from the origin to each point in the pair of XY,[ ]
R
matrices, is
2
R = X + Y 2 (7.62)
We let the origin be at x y,( 0 0 ) ( = 00 ) , and the plot in the intervals 2π– ≤≤ 2π and
,
x
– 2π ≤≤ 2π . Then, we write and execute the following MATLAB script.
y

% This is the script for Example_7_13
x=−2*pi: pi/24: 2*pi; % Define interval in increments of pi/24
y=x; % y must have same number of points as x
[X,Y]=meshgrid(x,y); % Create X and Y matrices
R=sqrt(X.^ 2 + Y.^ 2); % Compute distances from origin (0,0) to x−y points
Z=sin(R)./ (R+eps); % eps prevents division by zero
mesh(X,Y,Z); % Generate mesh plot for Z=sin(R)/R
xlabel('x'); ylabel('y'); zlabel('z');

7−32 Numerical Analysis Using MATLAB® and Excel®, Third Edition
Copyright © Orchard Publications