16.11: Chapter 12
- Page ID
- 117507
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)12.1 Recursion basics
12.2 Simple math recursion
n
depends on the summation to n - 1
.
1
.
rec_fact()
uses a base case of n == 0
, which works because, just like 1!
, 0! = 1
. However, the base case of n == 0
returns n
, which at this stage is 0
, and therefore zeroes out the rest of the computation.
n == 0
. 0! = 1
.
n == 1
is not used in the function. When n
is 0
, the overall multiplication becomes 0
. The recursion begins returning once the n < 0
base case is reached, initially returning -1
. But the previous recursive call would return 0
, thereby zeroing out all computations and leaving an overall result of 0
.
12.3 Recursion with strings and lists
"madam"
is a palindrome, so the function will recognize the word as a palindrome.
len(word) == 0
, which is resolved with the condition len(word) <= 1
.
strip()
function removes a given letter from the beginning and end of the string, including repeated letters. When strip('m')
is used with "madamm"
, the resultant string is "ada"
.
list_num
has an extra 15, and other_list
has an extra 22.
12.4 More math recursion
fib(9)
= fib(8)
+ fib(7)
. fib(9)
= fib(8) + fib(7) = 21 + 13 = 34
.
5 * 3 = 15
and 5 * 7 = 35
, so 5
is a the greatest common divisor of 15
and 35
.
gcd(24, 30)
results in 30 - 24 = 6
, so the next call is gcd(24, 6)
. Thereafter, other calls are gcd(18, 6)
, gcd(12, 6)
, and gcd (6,6)
for a total of four recursive calls after the initial function call.
gcd(13, 23)
results in 23 - 13 = 10
, so the next call is gcd(13, 10)
. Thereafter, other calls are gcd(10, 3)
, gcd(7, 3)
, gcd (4,3)
, gcd(3,2)
, gcd(2,1)
, and gcd(1,1)
for a total of eight calls.
12.5 Using recursion to solve problems
mid = (4 + 7) // 2 = 11 // 2 = 5
. The operator // is used for floor division.
mid
being 3
, next mid
becomes 5
, and finally, mid
becomes 4
. The element found at index 4 is equal to the key 16
.
mid
being (0 + 13) // 2 = 6
. The next call has mid (7 + 13) // 2 = 10
. The next call has mid (11 + 13) // 2 = 12
. At this point, 86 < 99
, so the final call is to mid 11
, after which there is nowhere else to go.