Computer Science 121. Scientific Computing Winter 2016 Chapter 8 Loops

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1 Computer Science 121 Scientific Computing Winter 2016 Chapter 8 Loops

2 Loops: Motivation We've already seen algorithms (findsmallest, Fibonacci, factorial) that repeat the same steps over and over Without an explicit repeat instruction, this becomes tedious and error-prone. Example: computing a square root

3 Square Root Algorithm: Basic Idea The square root of x must lie somewhere between 0 and x : i.e., 0 is its lower bound, and x is its upper bound. Somewhere between is vague, so we'll define it as halfway between : add the lower bound and upper bound and divide by 2, to approximate x. If our approximation is too big, lower the upper bound; if too small, raise the lower bound. The more we repeat this, the closer we get to x.

4 Square Root Algorithm: Pseudocode 1) Lower = 0; Upper = x 2) Repeat until happy: 3) S = (Upper + Lower) / 2 4) If S 2 > x, let Upper = S (was too big!) 5) Otherwise Lower = S (was too small!) 6) Report S

5 Square Root Algorithm: Example: Compute 2 Lower = 0; Upper = 2 S = (0+1)/2 = = 1, so too small: Lower = 1 Lower = 1; Upper = 2 S = (1+2)/2 = = 2.25, so too big: Upper = 1.5 Lower = 1; Upper = 1.5 S = (1+1.5)/2 = = , so too small: Lower = 1.25 Lower = 1.25; Upper = 1.5 S = ( )/2 = = 1.891, so too small: Lower = etc. Does this work for any positive number n?

6 Square Root Algorithm: Upgrade for 0 < x < 1 1) If x < 1: Lower = x; Upper = 1 2) Otherwise: Lower = 0; Upper = x 3) Repeat until happy: 4) S = (Upper + Lower) / 2 5) If S 2 > x, let Upper = S (was too big!) 6) Otherwise Lower = S (was too small!) 7) Report S

7 def mysqrt(x): if x < 1: else: lower = x upper = 1 lower = 0 upper = x Square Root Algorithm: Python at Last! for k in range(1000): # 1000 reps get us pretty close! s = (lower+upper)/2 if s**2 > x: else: return s upper = s lower = s

8 for loops: general syntax List of values is usually range(n), where n = some large number Variable name is sometimes a "dummy": (not used) range is like NumPy arange

9 for loops: non-numerical example def grumpycat(names): for name in names: print("i hate " + name) >>> grumpycat(["joe", "Sally", "Fred"]) I hate Joe I hate Sally I hate Fred

10 8.2 Accumulators Of ourse, we'd use the built-in sum function in a real program But understanding the general pattern of accumulators is important

11 Accumulators Anything that reduces an array of values to a single value is an accumulator So: sum, prod, min, max, mean, stdev,...

12 8.3 Nested Loops Any code we like can be inside a loop: including another loop! Example: Let's write a function that computes the probability of rolling a certain sum in pair of dice:

13 8.3 Nested Loops Any code we like can be inside a loop: including another loop! Example: Let's write a function that computes the probability of rolling a certain sum in pair of dice: Can we use our NumPy Fu to do this without loops?

14 8.4 Nested Loops: Optimal Job Matching Four people, four tasks One person per task at a given time Each person is better at some tasks, worse at others Represent an assignment of tasks as an array, e.g.: [2,1,4,3] = Alison on 2, Basil on 1, Clyde on 4, Daisy on 3 Do an exhaustive search of all assignments to find the best outcome

15 How about this?

16 Exclude Impossible Assignments, via set

17 Can we do it in a more general way? from itertools import permutations from numpy import * def matchpeople(q): # number of people (jobs) n = len(q) jobs = arange(n) bestassignment = [] bestqualitysofar = 0 for assignment in permutations(jobs): totalquality = 0 for j in range(n): k = assignment[j] totalquality += q[j][k] if totalquality > bestqualitysofar: bestqualitysofar = totalquality bestassignment = assignment return bestassignment

18 8.5 Element-by-Element operations >>> from numpy import * >>> a = array([1,2,3,4,5]) >>> mysqrt(a) Traceback (most recent call last): blah blah blah ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all() >>> mysqrtv = vectorize(mysqrt) >>> mysqrtv(a) array([ 1., , , 2., ])

19 8.6 Outputs of unknown size Size of a function's output isn't always the same as (or directly related to) its input Example: translate ASCII text to Morse code (use 1 for dot, 2 for dash,.etc.)

20 8.6 Outputs of unknown size Solution: start with [] and use append How would we write chartomorse()?

21 8.7 Loop Termination Sometimes we should break out of a loop before it's done: Can you think of slightly shorter version?

22 8.8 Conditional Looping with while Sometimes we don't even know how many times to loop Classic example: Compute the prime factors of a number n

Fall 2017 Discussion 7: October 25, 2017 Solutions. 1 Introduction. 2 Primitives

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