ENGG 1801 Engineering Computing

1
ENGG 1801Engineering Computing

• MATLAB Lecture 5 Tutorial (Weeks 9-10 )
• Matrices and Graphing in MATLAB

2
Outline of Lecture

• Scalars, vectors and matrices, generating them.
• Scalar and array operations, programming advice.
• Plotting and formatting 2D graphs.
• Functions of two variables 3D graphs.
• Creating 3D graphs, different types.

3
Scalars, Vectors and Matrices

• Scalar 2.4
• Vector 1.5 3.1
• Or 2
• 3
• 1
• Matrix

3 1 2 5 6 8 7 1 7
4
Generating matrices

• X 10.53 5,4,3,2,1 2210
• gives 1 1.5 2 2.5 3
• 5 4 3 2 1
• 2 4 6 8 10
• Y X(,34) Z X(23,23)
• gives 2 2.5 gives 4 3
• 3 2 4 6
• 6 8

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Generating Matrices (cont)

• Other options
• X X' gives transpose (rows to columns, columns
  to rows)
• Zeros and ones
• A zeros (3) gives a 3x3 matrix of 0
• B ones (3,2) gives a 3x2 matrix of 1

6
Scalar and Array Operations

• Operations on scalars
• Normal mathematics
• Operations on arrays
• Element-by-Element Operations
• Specified by putting a . directly after
  variable name
• Required for multiplication, division and
  exponentiation
• NOT required for addition and subtraction
• NOT required for (scalar nonscalar) or
  (nonscalar / scalar)

7
Miscellaneous Advice for Programming, Including
Matrices

• Colon may be used to enter matrices
• Enter starting point, step, end point
• e.g., H 2210 gives 2 4 6 8 10
• Type the semi-colon to prevent results of
  operations being shown in the command window
• 3 Ellipses can be used if you wish to continue
  typing on the next line
• Edit matrices by double clicking on the matrix in
  the Workspace window
• Sign to enter comments to your code

8
Class Exercise

• Calculate the following

4 1 1 8 7 6 3 3 2
3 1 2 5 6 8 7 1 7
?


4 1 8 7 3 3
3 1 2 5 6 8 7 1 7
?


.
4 1 1 8 7 6 3 3 2
3 1 2 5 6 8 7 1 7
?

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Class Exercise Answers
4 1 1 8 7 6 3 3 2
26 16 13 92 71 57 57 35 27
3 1 2 5 6 8 7 1 7


4 1 8 7 3 3
26 16 92 71 57 35
3 1 2 5 6 8 7 1 7


.
4 1 1 8 7 6 3 3 2
12 1 2 40 42 48 21 3 14
3 1 2 5 6 8 7 1 7

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Summary Note on size of matrices

• To do a matrix multiplication AB, the number of
  columns in A must be equal to the number of rows
  in B
• To do an array multiplication both matrices must
  have the same number of rows AND columns

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Plotting Graphs

• plot command
• Used to graph data
• Can specify
• Grids
• Markers for data points eg
• plot(x,T,x,T,o)
• Line style
• Many plots on one window (subplots)

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Formatting Graphs

• Lines
• Default solid blue line
• Dashed line --
• Dotted line
• Dash-dot -.
• Markers
• Use ., , , o, x, d, v, lt, gt
• Colours
• b, g, y, m, r, k, c

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• Problem plot the expression
• Solution
• Define x values
• x 10.15
• y ((2x.2).exp(x))./(sqrt(x)-100)
• Plot results
• plot(x,y),...
• title('Computing Sample Question'),...
• xlabel('x value'),...
• ylabel('y value'),...
• grid

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• Problem subplots
• This example illustrates subplots
• x 10.15
• y1 2x.2
• y2 exp(x)
• y3 sqrt(x)-100
• y (y1.y2)./y3
• Plot results
• subplot(2,2,1), plot(x,y1,'o'),...
• title('y1 as a function of x'),...
• xlabel('x value'),...
• ylabel('y1 value'),grid
• subplot(2,2,2), plot(x,y2,''),...
• title('y2 as a function of x'),...
• xlabel('x value'),...
• ylabel('y2 value'),grid

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Functions of Two Variables

• Wish to plot in 3 dimensions (i.e., 2 variables
  against function value)
• First generate a two dimensional grid in the x-y
  plane
• Then evaluate function value at the grid points
  to develop a 3 dimensional surface

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Functions of Two variables (continued)

• The 2-D grid consists of 2 matrices
• X-coordinates at all points in the grid (m) and
• Y- coordinates at all points in the grid (n)
• New matrices are size m x n
• e.g., if x goes from 2 to 2 in increments of 1
  and y goes from 1 to 2 in increments of 1 then

-1 -1 -1 -1 -1 0 0 0 0 0 1 1 1
1 1 2 2 2 2 2
-2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1
2 -2 -1 0 1 2
y_grid
x_grid
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Creating 3-D grids

• Use the meshgrid command
• x_grid, y_grid meshgrid(x,y)
• Creates 2 new matrices as shown on previous slide
  from vectors x and y
• Then calculate the function value for each of the
  values in vectors x and y, ie
• If z 1/(1 x2 y2)
• Then code is
• z 1./(1 x_grid.2 y_grid.2)
• Result is an m x n matrix

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Types of 3-D plots

• Mesh(x_grid,y_grid,z) gives an open mesh plot

• Number of points in original vector determines
  closeness of mesh

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Types of 3-D plots

• surf(x_grid,y_grid,z) gives a shaded mesh plot

• Number of points in original vector determines
  closeness of mesh

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Types of 3-D plots

• contour(x_grid,y_grid,z) gives a contour plot

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Types of 3-D plots

• meshc(x_grid,y_grid,z) gives a contour plot
  combined with a mesh plot

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Conclusions

• MATLAB is focussed on manipulating matrices.
• Scalars lead to vectors, lead to matrices.
• MATLAB is good for plotting graphs
• subplots are useful.
• 2D 3D graphing available.