Recall from last time that in Python:
v = [1, 2, 3]
w = v
w[0] = 10
v
[10, 2, 3] [10, 2, 3] |
Show examples in book of Matlab and Mathematica (probably) copy-on-write. So watch out! Code that looks identical in Sage and Matlab can do very different things.
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Control Flow
- if -- choose what code to evaluate
- while -- evaluate code as long as expression evaluates to True
- for -- iterate over something
if
a = 2; b = 3
if a > b:
c = 5
print (1)
elif a == b:
c = 10
print (2)
else:
c = 20
print (3)
print "-----"
3 ----- 3 ----- |
c
20 20 |
Discussion: Indentation instead of brackets (etc.).
Dangerous code in C or Java:
if (a > b)
c = 5;
x = 10;
The above misleading code can't be written in Python.
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while
i = 5
while i > 0:
print(i)
i = i - 1
5 4 3 2 1 5 4 3 2 1 |
break (from a while loop)
i = 25
while i > 0:
print(i)
i = i - 1
if i == 20:
break
25 24 23 22 21 25 24 23 22 21 |
Python does not have a "do ... while <expr>" construction.
for
Iterate over something.
for i in [1, 2, 7, 15]:
print i, i*i
1 1 2 4 7 49 15 225 1 1 2 4 7 49 15 225 |
You can also use break in for loops (just like with while loops):
for i in [1, 2, 7, 15]:
print i, i*i
if i == 7: break
1 1 2 4 7 49 1 1 2 4 7 49 |
Examples of things to iterate over:
range(10)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] |
range(5, 20)
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] |
range(5, 21, 2)
[5, 7, 9, 11, 13, 15, 17, 19] [5, 7, 9, 11, 13, 15, 17, 19] |
[1..10]
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10] [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] |
[5,7, .., 21]
[5, 7, 9, 11, 13, 15, 17, 19, 21] [5, 7, 9, 11, 13, 15, 17, 19, 21] |
(1,3,..,9)
<generator object ellipsis_iter at 0x5766050> <generator object ellipsis_iter at 0x5766050> |
for i in (1,3,..,21):
print i,
1 3 5 7 9 11 13 15 17 19 21 1 3 5 7 9 11 13 15 17 19 21 |
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List Comprehensions
A way to create lists that looks like a for loop.
[i*i for i in [1,2,7,15]]
[1, 4, 49, 225] [1, 4, 49, 225] |
v = [1/3, 2, 3, 'abc', 'xyz']
w = [type(z) for z in v]
w
[<type 'sage.rings.rational.Rational'>, <type 'sage.rings.integer.Integer'>, <type 'sage.rings.integer.Integer'>, <type 'str'>, <type 'str'>] [<type 'sage.rings.rational.Rational'>, <type 'sage.rings.integer.Integer'>, <type 'sage.rings.integer.Integer'>, <type 'str'>, <type 'str'>] |
v = [n for n in [1..250] if is_prime(n^2+1)]
v
[1, 2, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40, 54, 56, 66, 74, 84, 90, 94, 110, 116, 120, 124, 126, 130, 134, 146, 150, 156, 160, 170, 176, 180, 184, 204, 206, 210, 224, 230, 236, 240, 250] [1, 2, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40, 54, 56, 66, 74, 84, 90, 94, 110, 116, 120, 124, 126, 130, 134, 146, 150, 156, 160, 170, 176, 180, 184, 204, 206, 210, 224, 230, 236, 240, 250] |
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Functions
Use def
def f(a, b, c=10):
print "a=", a, "b=", b, "c=",c
return a+b+c
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f(1,2,5)
a= 1 b= 2 c= 5 8 a= 1 b= 2 c= 5 8 |
f(1,2)
a= 1 b= 2 c= 10 13 a= 1 b= 2 c= 10 13 |
f(a=1, c=3, b=2)
a= 1 b= 2 c= 3 6 a= 1 b= 2 c= 3 6 |
Variable Scope
c = 1; d = 1
def bar(a, b):
global d
c = a; d = b
print c, d
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bar(5,10)
5 10 5 10 |
c,d
(1, 10) (1, 10) |
Python "functions" are not functions in the sense of mathematics. They depend on more than their input, and can have side effects.
def f(x):
import time
if int(time.time()) % 2 == 0:
return x^2
else:
return x^3
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f(2)
4 4 |
f(2)
4 4 |
f(2)
8 8 |
plot(f, figsize=[8,2])
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plot(f, figsize=[8,2])
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Arguments of functions are just new references to an object (just like assignments):
def f(v):
print "v -->", id(v)
v[0] = 10
print v
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v = [1,2,7]
print id(v)
f(v)
90500488 v --> 90500488 [10, 2, 7] 90500488 v --> 90500488 [10, 2, 7] |
v
[10, 2, 7] [10, 2, 7] |
v = [1,2,7]
print id(v)
f(copy(v))
108833536 v --> 90694144 [10, 2, 7] 108833536 v --> 90694144 [10, 2, 7] |
Standard mistake that you may never make: a default list as input
def f(a, L=[]):
print "L -->", id(L)
L.append(a)
print L
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f(1)
L --> 106352584 [1] L --> 106352584 [1] |
f(2)
L --> 106352584 [1, 2] L --> 106352584 [1, 2] |
f(3)
L --> 106352584 [1, 2, 3] L --> 106352584 [1, 2, 3] |
Recursion
It works, but by default it is limited.
def fac(n): # assume input >= 1
return n if n==1 else n*fac(n-1)
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fac(5)
120 120 |
fac(1100)
WARNING: Output truncated! full_output.txt Traceback (click to the left of this block for traceback) ... RuntimeError: maximum recursion depth exceeded in cmp WARNING: Output truncated! full_output.txt Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_101.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("ZmFjKDExMDAp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module> File "/tmp/tmpanid85/___code___.py", line 3, in <module> exec compile(u'fac(_sage_const_1100 ) File "", line 1, in <module> File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac ... return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) File "/tmp/tmp8es5Q5/___code___.py", line 4, in fac return n if n==_sage_const_1 else n*fac(n-_sage_const_1 ) RuntimeError: maximum recursion depth exceeded in cmp |
import sys
sys.getrecursionlimit()
1000 1000 |
sys.setrecursionlimit(1000000)
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fac(1100)
534370848809263770342421558229505611830801307688668954088274588180366180\ 507230142593359743515667883274154675144889870336145824766456944642617389\ 104733808858036231887137145451981246007462271428980768726544820445627938\ 521384696779273305727495769481668074212168774982084587690054053663052207\ 974765307461854174761407828535465327192407514146840446037688836558176964\ 615337214194379121252115480979065230041337053263382381903878504838477803\ 724588229236052052802737146250787678631254934255034752507195831922837186\ 030239875807777800355948130376137959023725202218865036777462816124054696\ 067849623033545571878068013262711078162177228752949470718358225524298555\ 062659304726235529400439683750995457229825772063311815720595326782817708\ 920420936288168056236465773117812572473654112151753182299528872746983872\ 340629193606959156392223691790438526372745935906675613488166812498098047\ 697386214801382532597457501623884675983655617265690103925964794592333024\ 323536810983017987978944342677864199822342261966341544427566312125083062\ 324656190016824389531140071376929493673248560712179251586301228003899683\ 281401139943312168491213607465195878608574071289179201422918723640246484\ 582489779977603613604718849834315506364003926574273569421105427234464838\ 229595036372921748183592701478838442454491279935703996801454263268591217\ 411293039868708405683574547193689070618929850174592801197517541507435597\ 289982955567323602505100388178442747542851341998969076383964477727378271\ 071547961192019215362150550579304549119593219720589371606769113577207430\ 589945046720705494848261132473585053881617998656177083673499543703808532\ 192432883186291945808989805781924389370357686568396818659379312091885082\ 282670831506561018936080655136970279976803222392921356379936817819124988\ 714512823673236915860472014331087898518943799844045922160923039163416282\ 747378951005309376545654483238563097799645410120160626688816981494456970\ 335134673327992503771447719262504033078194051236272349251363303333031943\ 725505672666478934559485065930513420665737267456485791075859859536589310\ 456140360381555965691258129225797502605868499261246665965245515699452002\ 476606332869836526626211811389591512469040731053588029798552054188230707\ 892213626619862738697277632967887268727487919402518596797342957419685013\ 036891174178271574199813848317234052091159529637811519976390901131118896\ 757105267934486097267301995911743575358800329297051491721051468122364760\ 559105886494816369278793601900859787526572270771248827682569134013497568\ 169550577062064755085844944156336082567893030480431935935274066793521165\ 881294443099150565491295279276352340940609556759543421730501252476349532\ 078080000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000000000000000 53437084880926377034242155822950561183080130768866895408827458818036618050723014259335974351566788327415467514488987033614582476645694464261738910473380885803623188713714545198124600746227142898076872654482044562793852138469677927330572749576948166807421216877498208458769005405366305220797476530746185417476140782853546532719240751414684044603768883655817696461533721419437912125211548097906523004133705326338238190387850483847780372458822923605205280273714625078767863125493425503475250719583192283718603023987580777780035594813037613795902372520221886503677746281612405469606784962303354557187806801326271107816217722875294947071835822552429855506265930472623552940043968375099545722982577206331181572059532678281770892042093628816805623646577311781257247365411215175318229952887274698387234062919360695915639222369179043852637274593590667561348816681249809804769738621480138253259745750162388467598365561726569010392596479459233302432353681098301798797894434267786419982234226196634154442756631212508306232465619001682438953114007137692949367324856071217925158630122800389968328140113994331216849121360746519587860857407128917920142291872364024648458248977997760361360471884983431550636400392657427356942110542723446483822959503637292174818359270147883844245449127993570399680145426326859121741129303986870840568357454719368907061892985017459280119751754150743559728998295556732360250510038817844274754285134199896907638396447772737827107154796119201921536215055057930454911959321972058937160676911357720743058994504672070549484826113247358505388161799865617708367349954370380853219243288318629194580898980578192438937035768656839681865937931209188508228267083150656101893608065513697027997680322239292135637993681781912498871451282367323691586047201433108789851894379984404592216092303916341628274737895100530937654565448323856309779964541012016062668881698149445697033513467332799250377144771926250403307819405123627234925136330333303194372550567266647893455948506593051342066573726745648579107585985953658931045614036038155596569125812922579750260586849926124666596524551569945200247660633286983652662621181138959151246904073105358802979855205418823070789221362661986273869727763296788726872748791940251859679734295741968501303689117417827157419981384831723405209115952963781151997639090113111889675710526793448609726730199591174357535880032929705149172105146812236476055910588649481636927879360190085978752657227077124882768256913401349756816955057706206475508584494415633608256789303048043193593527406679352116588129444309915056549129527927635234094060955675954342173050125247634953207808000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 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By the way, Sage's builtin factorial function (which is part of the MPIR library) is pretty damn fast (far faster to compute the answer than convert it to decimal):
time n = factorial(10^6)
Time: CPU 0.94 s, Wall: 0.94 s Time: CPU 0.94 s, Wall: 0.94 s |
time len(str(n))
5565709 Time: CPU 5.50 s, Wall: 5.47 s 5565709 Time: CPU 5.50 s, Wall: 5.47 s |
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Symbolic Functions
These are used for Calculus. They are also called "functions" but have little to do with Python functions.
f(x) = sin(x)*log(x)
f
x |--> log(x)*sin(x) x |--> log(x)*sin(x) |
type(f)
<type 'sage.symbolic.expression.Expression'> <type 'sage.symbolic.expression.Expression'> |
f(pi/2)
log(1/2*pi) log(1/2*pi) |
f.integrate(x)
x |--> -log(x)*cos(x) + 1/2*Ei(-I*x) + 1/2*Ei(I*x) x |--> -log(x)*cos(x) + 1/2*Ei(-I*x) + 1/2*Ei(I*x) |
f.plot((x,0,10), figsize=[8,2])
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Classes (your own data types)
class MyRational:
def __init__(self, n, d):
self._n = Integer(n); self._d = Integer(d)
def __repr__(self):
return '%s/%s'%(self._n, self._d)
def __add__(self, right):
return MyRational(self._n*right._d + self._d*right._n, self._d*right._d)
def __mul__(self, right):
return MyRational(self._n*right._n, self._d*right._d)
def reduced_form(self):
"""Return the reduced form of this rational number."""
a = self._n / self._d
return MyRational(a.numerator(), a.denominator())
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a = MyRational(2,6); b = MyRational(2, 3)
print a, b
2/6 2/3 2/6 2/3 |
a.reduced_form()
1/3 1/3 |
c = a + b; c
18/18 18/18 |
c.reduced_form()
1/1 1/1 |
How do you think we should get the following to work?
a - b
Traceback (click to the left of this block for traceback) ... TypeError: unsupported operand type(s) for -: 'instance' and 'instance' Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_52.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("YSAtIGI="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/private/var/folders/4z/h5m3q9b94zs641jlbxj79sv00000gn/T/tmp5CsvSO/___code___.py", line 2, in <module>
exec compile(u'a - b
File "", line 1, in <module>
TypeError: unsupported operand type(s) for -: 'instance' and 'instance'
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Next: Extensive discussion of Python data types such as lists, tuples, strings, etc.
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