Math 171 Spring 2005, Lab 5

Due Tuesday, February 8, 11:59 P.M.

1. Overview
2. Practicing Matlab Commands
3. Assignment

Overview

Unlike Maple, Matlab's design focuses on providing a language for implementing numerical algorithms and working with matrices. While Maple provides a wider range of functionality, using Matlab in its area of specialization is generally more efficient and convenient. This lab will introduce you to working with matrices interactively; Matlab programming will be covered later.

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Practicing Matlab Commands

As with Maple, in order to run Matlab you may need to make some changes to your environment variables. Make sure your PATH variable contains the /usr/matlab/6.1/bin directory. Now either start a new shell or type

source ~/.zprofile

The examples that follow will introduce you to all the Matlab commands you need to complete the assignment (with one exception that you must lookup yourself). Enter each of the following Matlab commands before continuing to the assignment section.

• Matlab has the ability to record all input and output into a file via the diary command. Use the command as follows:

  
  diary practice
  
  

  to record all your practice commands into the textfile named practice in the current working directory.
• By default, Matlab's working directory will be the same as your shell's when Matlab was started. Matlab emulates several UNIX shell commands, including pwd and cd. This allows you to change the working directory without restarting.

  
  diary off
  pwd
  ls
  mkdir lab5
  cd lab5
  pwd
  diary practice
  
  

• Matrices are entered in Matlab by surrounding a list containing elements of the matrix with square brackets. Rows are separated by semicolons; entries within a row may be separated by spaces or commas:

  
  [1,2,3;4 5,6;7 8 9]
  
  

• The ans variable always contains the result of the last command you entered into Matlab. Additional variables can be assigned values using the = operator.

  
  M = ans
  
  

• You can look up commands for a specified purpose using the lookfor command, and read help on commands using help:

  
  lookfor determinant
  help det
  det(M)
  lookfor echelon
  help rref
  rref(M)
  
  

• For convenience, Matlab provides several functions for generating standard matrices:

  
  eye(5)
  ones(5)
  triu(ans)
  tril(ones(5))
  ones(3,5)
  zeros(5)
  zeros(5,3)
  diag([2,4,8])
  rand
  rand(3,2)
  rand(3)
  diag(ans)
  diag(ans)
  
  

• Vectors with values in a specified range may also be generated using the : operator:

  
  t = 1:10
  u = 10:1
  u = 10:-1:1
  v = 0:5:100
  w = 100:-0.25:90
  
  

• Performing arithmetic between a matrix and a scalar results in application of the arithmetic to each element of the matrix:

  
  M+5
  x=3
  M*x
  M/x
  ans/3
  
  

• Matrix elements and submatrices can be accessed by appending (row number, column number) to the matrix name. A colon separated range of row or column numbers (or just a colon to indicate all rows/columns) may also be used:

  
  M(1,1)
  M(:,3)
  M(3,:)
  M(3,2:3)
  
  

• To form an augmented matrix, define the matrix as usual, but use the name of a matrix variables instead of scalars:

  
  scol = [2;2;2]
  dcol = [3 4; 5 6; 7 8]
  [scol dcol]
  [dcol; scol scol]
  
  

• The usual matrix arithmetic operations are provided by the +,-,*, and ^ operators. The the ' postfix operator takes the transpose of a matrix:

  
  A=[2,4,8,16;3,9,27,81;5,25,125,625]
  B=[A(3,:);A(2,:);A(1,:)]
  A-B
  A+B
  A*B
  A'
  A*B'
  C=ans
  C*C*C
  C^3
  C^-1
  C*ans
  inv(C)
  
  

• In addition to ordinary matrix arithmetic, Matlab can perform componentwise arithmetic on the elements of two matrices with identical dimensions. The componentwise form of each operator is accessed by prepending a dot (.) to the operator. Additionally, Matlab provides componentwise comparison of matrices with the == operator.

  
  A.*B
  A.^2
  A./B
  C^-1==inv(C)
  
  

• Matlab provides several commands to control output formatting:

  
  v = 1.5*10^9;
  V = [v,2*v,3*v]
  v
  V
  format long
  V
  format long g
  V
  format short
  V
  1.5
  rats(1.5)
  I = eye(75);
  I
  more on
  I
  more off
  
  

  Note that Matlab is case sensitive, i.e. v and V are two separate variables. Terminating a Matlab command with a semicolon suppresses the output for that command.
• Matlab provides a few commands for memory management:

  
  bigmat=eye(5000);
  who
  whos
  size(bigmat)
  clear bigmat
  whos
  size(bigmat)
  clear
  whos
  
  

  Be careful about using the clear command without any arguments, this will erase all your currently saved data without prompting for confirmation.
• In addition to providing a convenient tool for implementing your own numerical algorithms, Matlab comes pre-packaged with programs for several standard numerical algorithms. For example, you can use the quad command to perform numerical integration with Simpson's rule, and the diff command calculates finite differences, which can be used to approximate derivatives:

  
  quad('X.^2',-1,1)
  quad('1./(2*sqrt(X))',4,16)
  quad('cos(X)',0,pi/2)
  X=-1:0.1:1
  Y=X.^2
  dY=diff(Y)
  dX=diff(X)
  yp=dY./dX
  
  

  Note that the function passed as the first argument to the quad command is actually a matrix formula, so you should use componentwise operators in the function definition unless performing arithmetic between a matrix and a scalar.
• Matlab provides plotting of one vector versus another with the plot command:

  
  plot(X,Y)
  plot(X,yp)
  length(X)
  length(yp)
  X2=X(2:length(X));
  length(X2)
  plot(X2,yp)
  plot(X,Y,X2,yp)
  
  

• Finally, Matlab provides limited symbolic algebraic capability via the syms command, which can be used to create and list symbolic variables:

  
  syms
  syms x
  int(x^2)
  int(x^2,-1,1)
  diff(x^2)
  syms
  clear
  syms
  x=0:10
  diff(x)
  
  

  Note that the behavior of the diff and int commands behave differently when the variable x is declared symbolically. It turns out that there are several separate Matlab programs called diff; Matlab decides which one to use depending on whether x is symbolic. This is called function overloading; the Matlab help files will usually tell you when a function is overloaded. You can read help on the symbolic form of diff by typing help sym/diff.
• When you have finished practicing the commands above, you can exit Matlab using either the quit or exit command:

  
  quit
  
  

  Alternately, just type clear before moving on to the assignment.

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Assignment

1. Make sure you know which command to use for each of the tasks below. Then move into your lab directory and type

   
   clear
   diary off
   diary assignment
   
   

   and type the commands for each of the remaining tasks.
2. Use exponential notation to express the inverse of the 10x10 upper diagonal matrix with 5's on the main diagonal and 3's above the main diagonal.
3. Express the answer to the previous step using fractions.
4. Find the determinant of the 25x25 tridiagonal (i.e. 3-banded) matrix with 1's above the main diagonal, 2's on the main diagonal, and 3's below the main diagonal.
5. Use augmented matrices to enter and determine the fourth power of the matrix

   1	1	1	0	0	0	0	0	0
   1	1	1	0	0	0	0	0	0
   1	1	1	0	0	0	0	0	0
   0	0	0	2	2	2	0	0	0
   0	0	0	2	2	2	0	0	0
   0	0	0	2	2	2	0	0	0
   0	0	0	0	0	0	3	3	3
   0	0	0	0	0	0	3	3	3
   0	0	0	0	0	0	3	3	3

6. Use matrix arithmetic to solve the simultaneous system of equations from lab 4:

   5x1	+	3x2	+	2x3	=	0
   8x1	+	4x2	+	2x3	=	-3
   3x1	+	5x2	+	7x3	=	6

7. Solve the system of equations above using row reduction.
8. Approximate the area of a semicircle of radius 2 using numerical integration.
9. Find the antiderivative and derivative of y=2xe^-x2.
10. Approximate the first and second derivatives of the function y=x^3 numerically. Plot all the functions on the same graph.
11. Determine the sum of all prime numbers less than or equal to 1000.
12. When you have finished all of the tasks above, quit Matlab and submit your lab using the ~gbutler/bin/m171_submit command.

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