2/20

note: contains potentially useful information not presented during lecture --
      such lines are marked with !s.

announcements
+ catch-up review session this sun, 3pm, kimball b11
+ E5 was due online at 8am; solutions already posted
+ course & prelim online evaluation form -- hopefully available thu or fri

reminders
+ P5 due online 8am thu
+ online lecture evaluation
+ turn in questions in lecture for bonus
+ there are lots of unused office hours

topics
+ arrays as lookups
+ chars as array indices
+ demo of loop invariant for max example in 2/15 lecture
+ setting matlab Path
+ matlab scripts
+ matlab functions
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array as lookup table

+ suppose given $mm$ in range 1..12,
  we want to print 3 letter abbreviation of the month

+ could use big conditional (if-elseif-elseif-....else end)

+ (could use $switch$)

+ easier to just look it up in a table:

    monthname = 'JanFebMarAprMayJunJulAugSepOctNovDec';
    i = 3*mm;
    monthname(i-2:i)

chars as array indices

+ recall: a char is internally represented as a small integer

+ therefore, should be able to use a char as an index into an array

+ matlab unhappy: doesn't understand using a char as an index

+ so, convert from char to integer using $double$ or $0+$,
  e.g. $double('a')$ or $0+'a'$

example:

given array $y$ of digits 1..9 (e.g. y = [3 1 4 1 5 9 2 6 5 3 5],
print the string where each digit is subsituted with letters as follows:

     1 2 3 4 5 6 7 8 9
     f e e d f a c e s

+ can do with a loop and conditionals

+ better way: 

    map = 'feedfaces';
    map(y)
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notes about the online loop invariant demo

!should work under netscape, might need plug-in for internet explorer

!code at top
!
!+ refer to code from 2/15 lecture for full code, e.g. here
!  + $prompt$ initialization not shown
!  + $stopping = 0$ not shown
!+ $maxsofar$ is a conceptual, mathematical quantity, not a program variable
!+ black arrow: points to line of code that was just executed
!  + cyan (light blue) line of code: guard of loop or conditional is true
!  + magenta (purple) line of code: guard of loop or conditional is false
!  + blue line of code: "regular" line of code (not loop, not conditional)
!+ green box, green "maxsofar", green "curmax": loop invariant is true
!+ red box, red "maxsofar", red "curmax": loop invariant is false
!
!
!"maxsofar", "num", "input", "curmax", numbers at bottom of gray rectangle:
!
!+ blue line above numbers: input values preceding num (excluding num)
!+ "maxsofar" above a number: 
!  $maxsofar$ is above the maximum input value preceding num (excluding num)
!
!+ "num", yellow line above a number: above the latest number read from input
!
!+ "input": input values typed in one by one by user
!
!+ "curmax" below a number, blue line below numbers:
!  $curmax$ is under the maximum input value of the underlined numbers,
!  which are the input values preceding $num$ and also possibly including $num$

_______________________________________________________________________________

motivation for scripts

consider this code:

    a =   5; x = linspace(-pi,pi,397); y = rem(x.^2,sin(a*x)); plot(x,y)
    ...other code...
    a =  10; x = linspace(-pi,pi,397); y = rem(x.^2,sin(a*x)); plot(x,y)
    ...other code...
    a =  50; x = linspace(-pi,pi,397); y = rem(x.^2,sin(a*x)); plot(x,y)
    ...other code...
    a = 100; x = linspace(-pi,pi,397); y = rem(x.^2,sin(a*x)); plot(x,y)
    ...other code...
    a = 500; x = linspace(-pi,pi,397); y = rem(x.^2,sin(a*x)); plot(x,y)

+ because of the "other code", cannot use a loop to reduce redundancy.

instead, we can write a script $myplot.m$:

    $x = linspace(-pi,pi,397); y = rem(x.^2,sin(a*x)); plot(x,y)$

and modify the original code to be:

    a =   5; 
    ...other code...
    a =  10; myplot
    ...other code...
    a =  50; myplot
    ...other code...
    a = 100; myplot
    ...other code...
    a = 500; myplot

observe: to *call* or *invoke$ the script, simply type in script name

caveat: need to let matlab know where to look for scripts and functions

+ use Set Path in File menu or $path$ command
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motivation for functions

suppose now we want to have code like this:

    x = 17; y = 6*6;
    fred = rem(x^3, y);
    % plot with a <-- 17 * fred
    x = fred; bob = sqrt(x) - y;
    % plot with a <-- bob
    x = x + y;
    % plot with a <-- x-1

that is, like the earlier code, except

+ "other code" is complicated code that computes various constants,
  some of which we wish to use as $a$, but $a$ is not set by other code

+ $x$, $y$ are now variables used by "other code"

+ script $myplot$ is no good:

  + $a$ is not set, and even if it were, still have the problem that

  + script $myplot$ would clobber values of $x$, $y$ used by "other code"

solution: use functions

+ *arguments* (values) are *passed in* to functions and 
  are automatically assigned to the function's *parameters* (variables)

+ functions have variables in a different *scope*
  + a scope is sort of like a room (not perfect analogy)
  + idea: code is run in a scope (room), using the variables there
  + therefore, variables in different scopes are different, but can have
    the same name and be used without confusion

! + functions can also *return* values back to caller,
!   e.g. probably a good idea to return $x$ and $y$ from $myplot$.

modify myplot.m to now be a function:
 _________________________________________________________________
|                                                                 |
| function [x,y] = myplot(a)                                      |
|     x = linspace(-pi,pi,397); y = rem(x.^2,sin(a*x)); plot(x,y) |
|_________________________________________________________________| 

+ calling code is now:

    x = 17; y = 6*6;
    fred = rem(x^3, y);
    myplot(17 * fred); % plot with a <-- 17 * fred
    x = fred; bob = sqrt(x) - y;
    myplot(bob); % plot with a <-- bob
    x = x + y;
    myplot(x-1); % plot with a <-- x-1
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preview of next lecture:

! returning versus printing a value

! tracing with function calls
! 
! to find an identifier (variable or function name):
! + look inside current scope 
!   (but do not look inside scopes within current scope)
! + if found, great
! + else, move to immediately enclosing scope.  repeat.
! + if there is no enclosing scope, then identifier is not known/visible
! 
! box scope diagrams for function call
! 1. evaluate arguments
! 2. draw activation record --function name, box-- 
!    at same scope the function is found
! 3. inside activation record: write in parameters, fill in with argument values
!    (1st parameter <-- 1st value, 2nd <-- 2nd, 3rd <-- 3rd, etc.)
! 4. inside activation record: run function body
! 5. copy out return values
! 6. tear down activation record: cross out

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answers to questions asked in lecture -- important!

! Q: why do you multiply $mm$ by 3 in the name of month example?
! A: each abbreviation is 3 letters long.  here is what we want:
! A: mm = 1: monthname(1:3) = 'Jan'
! A: mm = 2: monthname(4:6) = 'Feb'
! A: mm = 3: monthname(7:9) = 'Mar'
! A: mm = 4: monthname(10:12) = 'Apr'
! A: etc.

Q: how does $monthname((3*mm-2):3*mm)$ know the value of $mm$?
A: the problem statement says "given $mm$", 
A: which means "assume somehow has already somehow given $mm$ a value".

Q: are there spaces in $monthname$?
A: no

Q: can you lookup an entire array of letters or just one letter at a time?
A: yes, you can lookup an entire array.

! Q: what is the advantage of using a lookup table?
! A: it is less verbose/redundant than using lots of conditionals;
! A: it can also be much faster than computing the answer

! Q: how do you initialize an array to store $'t'$ at $x('a')$?
! A: $x(0+'a') = 't';$  
! A: 
! A: note: you don't have to pre-allocate an array before storing
! A: values into it.  for example, if you start up matlab fresh, you
! A: can immediately run
! A: 
! A:     >> x(5) = 27;
! A: 
! A: and matlab will store 27 at position 5 in $x$; when you assign to part
! A: of a vector, matlab will automatically extend the vector (with 0s) to
! A: be large enough to hold that element.
! A:
! A: note: remember the rules for assignment $lhs <-- rhs$:
! A: 1. evaluate $lhs$ to get a (potential) location
! A: 2. evaluate $rhs$ to get a value
! A: 3. if location $lhs$ does not exist, then *declare* (create) it
! A: 4. put value into location, erasing the old value
! A:
! A: since step 2 occurs before step 3, if $x$ does not exist,
! A: then $x(2) = x(2)+1;$ is an error: 
! A: 
! A: step 1: the location "position 2 of x"
! A: step 2: "add 1 to value at position 2 of x", oops!  no such location yet!

Q: does a loop invariant have to say what the loop is doing, 
Q: or just something that is "unchanging" 
Q: (true every time the loop guard is checked)?
A: technically: only something true every time the loop guard is checked.
A: pragmatically, to be useful: what the loop is doing

Q: is the stopping value always 0?
A: no, which is why good programming style requires us to define a
A: constant (e.g. $stopping$) to hold the value, so that we can easily
A: change the stopping value if need be.
! A: ideally, we would compute $prompt$ from $stopping$, e.g.
! A: $prompt = ['value (' num2str(stopping) ' to stop)? '];$
! A: this way, when $stopping$ is changed, $prompt$ is automatically
! A: changed, without any further programming effort.

Q: what is $maxsofar$?  what is its point?
A: it is a conceptual variable, a mathematical definition, used to
A: give a definition to $curmax$ for use in the loop invariant

Q: why do use a $while$ loop in the max example, not a $for$ loop?
A: a $for$ loop is used when you know all the values in advance.
A: here, we are reading input that the user types in, so we have
A: no idea or control over what the values are; we must keep looping
A: until we see the stopping value

Q: why exclude current input value $num$ from the definition of $maxsofar$?
A: because the stopping value is not part of the input sequence, so
A: including $num$ could improperly affect the answer.  for example,
A: suppose all the input numbers were negative.  then at the end of the
A: program, $maxsofar$ should be the largest negative value seen.  however,
A: if $num$ is included, then $maxsofar$ would be set to 0, which is wrong.

! Q: stylewise, should you indent the body of a function?
! A, revised: yes.  (not required for P3, but required from now on)
! A, revised: this is because the body is a substructure of the function.