02/06

announcements
+ hand in E3
+ AEW still open until 1pm thursday
+ new sections
        Sec 15 #647222 T 1220pm UP 205
        Sec 16 #658508 W 1220pm PH 403
reminders: bonus: online evals, Qs in lec (write name, cuid, netid, Q)
topics (vector concat, conditionals)
+ E3 (for notes and much more: see solutions to E3)
+ Pascal's triangle
+ subarrays (indexing)
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print rows 0..N of pascal's triangle

notes:
+ by "row", we mean "the non-zero elements of a row"
+ print 1+N rows: 0, 1, 2, ..., N
+ for now, without nice formatting (bacteria: also no formatting!)
+ pay attention to vector concat

triangle                matlab representation of row    i
0   0   1   0   0       [ 1 ]                           0
 \ / \ / \ / \ /          |\                            
  0   1   1   0         [ 1 1 ]                         1
 / \ / \ / \ / \          |\|\                          
0   1   2   1   0       [ 1 2 1 ]                       2
 \ / \ / \ / \ /          |\|\|\                        
  1   3   3   1         [ 1 3 3 1 ]                     3
 / \ / \ / \ / \          |\|\|\|\                      
1   4   6   4   1       [ 1 4 6 4 1 ]                   4

solution 1:

    N = 4;
    row = [ 1 ];
    disp(row);
    for i = 1:N
        new = zeros(1,i+1);
        for j = 1:i
            new(j) = new(j) + row(j);
            new(j+1) = new(j+1) + row(j);
        end
        row = new;
        disp(row);
    end

observation:

[ 1 3 3 1 ] makes 2 copies: [ 1 3 3 1 ] and [ 1 3 3 1 ]
  |\|\|\|\                    | | | |          \ \ \ \ 
                              1 3 3 1           1 3 3 1
 
so the result is the sum [ 1 3 3 1 ]
                           |\|\|\|\ 
                           1 3 3 1 *    = [ row 0 ]
                           * 1 3 3 1    = [ 0 row ]
                         [ 1 4 6 4 1 ]  

solution 2:

    N = 4;
    row = [ 1 ];
    disp(row);
    for i = 1:N
        row = [row 0] + [0 row];
        disp(row);
    end

note similarities with P2.4 (bacteria):
+ both take an old row/population and produce a new row/population
+ both process each item twice (add left + add right versus age + replicate)
+ both can be solved with nested loops or with just one loop
  (might also need conditionals -- only mature bacteria replicate)
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subarrays

+ we have already seen $x(i)$ for $i$ an integer 1..$length(x)$
  means "i-th element of x"
+ Matlab uses 1-relative indexing     1   2   3
                                    +---+---+---+
                                    | 7 | 8 | 5 |
                                    +---+---+---+
  Java uses 0-relative indexing       0   1   2   3
+ matlab allows one to access *subarrays*: multiple elements from an array
+ ex: every other element of x (1st, 3rd, 5th, ...): $x(1:2:end)$
+ ex: 5th, 1st, 1st, 8th: $x([5 1 1 8])$
+ ex: first five elements: $x(1:5)$
+ ex: in reverse order: $x(end:-1:1)$

two different behaviors
+ read: positions must be in range 1..length(x)
+ write: positions >= 1; if fall off right end, array is auto-extended with 0s

review: update rule for $lhs = rhs$
+ evaluate $lhs$ to get locations
+ evaluate $rhs$ to get values
+ declare $lhs$ if non-existent
+ put values into locations (cross out old values if any)

question: why does order of 2nd and 3rd step matter?
+ start from scratch: $x(5) = x(5)+1$

assignment: 
+ generally, size of $lhs$ must match size of $rhs$
    ok: x([1 2]) = x([11 10])  (assuming length(x)>=11)
   bad: x([1 2 3]) = x([11 10])
+ exception 1: auto-replicate scalar $rhs$
    ok: x([1 2]) = x(11)
+ exception 2: assigning $[]$ deletes
    ok: x([1 2]) = []
+ question: have deletion, how do insertion?  answer: use concat
    ex: insert 3 between x(5), x(6): x = [x(1:5) 3 x(6:end)]

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questions in lecture:

Q: can you access matrix elements in any order?
A: yes.

Q: in E3.3 solution, is $median(x)$ instead of $sum(x)-max(x)-min(x)$ ok?
A: yes.

Q: can you insert a new spot into an array without giving it a value?
A1: no, you need to provide a value, e.g. 0.
A2: depends on what you meant by "value" -- $nan$ (Not A Number) can be used.

Q: (re: solution-2 for pascal's triangle)  
Q: why isn't loop variable $i$ used inside the loop?  is that ok?
A1: the loop is just to repeat something $N$ times.
A2: as seen in E2.3, it is ok to not use the loop variable in the loop body.

Q: can you use $fliplr([row 0])$ instead of $[0 row]$?
A: yes, but it is faster and simpler to use $[0 row]$.