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- Linear Equations
- ODE
- Integration
- Handle Graphics.
- Exercises on this days topics

Lecture 5

- Linear equations
- Using left division
- MATLAB uses Gaussian elemination

- Has the solution in MATLAB
- X=A\Y

- Example:
Solve the system of linear equation:

Lecture 5

- MATLAB can solve system of first order ordinary differential equations with known initial condition.
- MATLAB offers a number of different solvers
- ode45
- This is the typical first solver to try on a new problem.

- ode15s
- This is typical to try if ode45 fails or are too inefficient. Solves stiff problems.
- Stiff problem are described as problem in which the time constants vary a lot.

- ode45

Lecture 5

- General form
- [time,x]=solver(fh,t,x0)
- time: Time values
- x: contains the solution for each time value
- solver: one of MATLAB’s ode solvers
- fh: A function handle to the function that describes the differential equations
- t: is the time span T0 to TFINAL
- x0: Initial condition

- [time,x]=solver(fh,t,x0)

Lecture 5

Example:

Solve the differential equation:

Lecture 5

Lecture 5

- Higher order differential equations must be rewritten into system of first order differential equations.

y

b

m

F

k

Lecture 5

- Trapezoidal numerical integration.
- z = trapz(x,y)
- computes the integral of y with respect to x using the trapezoidal method.

- z = trapz(x,y)
- The trapezoidal technique is used when we only know the integrand in a number of specific points.
- With a smaller delta-x the numerical integration becomes more accurate

Lecture 5

- A set of low level functions that control the characteristic of a graphic object.
- Changing grids, line color etc. that is not supported by the standard LinSpec option in the plot command.
- MATLAB graphics system is based on a hierarchical system of Graphical Objects

Lecture 5

- Each graphical object are known by a unique name: Handle
- Each graphical object has special data: properties
- A handle is automatically returned by any command that creates a graphic object.
- Hndl=figure

Lecture 5

Lecture 5

- When an object is created all of its properties are initialized to default values.
- plot(y) uses the default line color, line style, line width etc.

- All properties can be examined using:
- get(Handle,’PropertyName’);

- All properties can be changed using:
- set(Handle,’PropertyName’,Value1’,….);
- If value are left out MATLAB display a list of possible property values for that actual property name

Lecture 5

- To view all properties:
- x=[0:0.1:2];
- y=x.^2;
- Hndl=plot(x,y)%Handle to the line.
- result=get(Hndl)
- result will be a structure containing all properties to the line

Lecture 5

Lecture 5

- To change the line width from default 0.5 to 5:
- set(Hndl,’LineWidth’,5)

Lecture 5

- Functions that return handles
- gcf
- Get current figure.

- gca
- Get current axes in the current figure.

- gco
- Get current object in the current figure.

- findobj
- Finds a graphics object with a specific property value

- gcf
- The current object is defined as the last object clicked on with the mouse.

Lecture 5

- Position of figure Objects
- [left bottom width height]

- Units can be: pixels, inches, cm, points and normalized coordinates.
- Normalized coordinates are between 0-1.
- (0,0) = Lower left corner
- (1,1) = Upper right corner

- Normalized coordinates are recommended to use if possible

Lecture 5

- Position of axes and uicontrol Objects
- Also a 4-element vector.
- [left bottom width height]

- Position is specified relative to the figure that contains the object.
- Default units:
- Normalized coordinates within the figure.

Lecture 5

- Position of text Objects
- The position of a text object refers to the actual axes.
- x,y in 2D
- x,y,z in 3D

- The position of the text object relative to a specified point is controlled by:
- HorizontalAlignment
- {Left},Center,Right

- VerticalAlignment
- {Middle}, Top, Cap, Baseline, Bottom

- HorizontalAlignment

Lecture 5

Example:

-Construct a dartboard with 10 circles.

-Add text in each circle what the point would be if the dart should hit within that circle.

-Construct a frame around the dartboard.

-Simulate that a person throw 10 darts at the

board with the standard deviation 4 in x and y direction.

Lecture 5

Lecture 5

Lecture 5