Sage: Unifying Mathematical Software

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The Sage Notebook

Runs in a web browser: connects to the Sage server.
You can easily edit and markup text in between cells for documentation, explanation, exposition, ....
Includes jsmath to render basic $\TeX$ code. For example, $$\sqrt{{\rm Var}(X)} \qquad {\textstyle x_{\rm max} - x_{\rm max}} \qquad {\rm LL}(k) \Rightarrow {\rm LR}(k)$$ $$\int_0^\infty e^{3x}sin(x) dx$$

{{{id=23| R. = QQ[] p = ((y^2 - x^3)^2 - 4*x^5*y - x^7) * (y^2 - x^3) * (y^3 - x^2) p /// }}} {{{id=11| show(p) /// }}}

The Notebook interfaces with your software

Here is an example of using GAP from the notebook.

{{{id=3| %gap a := AlternatingGroup(8) /// }}} {{{id=9| %gap Size(a) /// }}}

GAP commands are also available as gap.<gapcommand> which returns a Python object.

{{{id=12| S8 = gap.Group('(1,2)','(1,2,3,4,5,6,7,8)') S8 /// }}} {{{id=13| type(S8) /// }}} {{{id=14| S8.Size() /// }}} {{{id=16| G = gap.SylowSubgroup(S8, 2); G /// }}} {{{id=15| G.IsNormal(S8) /// }}} {{{id=17| /// }}}

Sage Includes Extensive and Beautiful Documentation

3600 pages of documentation: html docs
Context sensitive help: introspection and tab completion

{{{id=20| p = Partition([4,3,1,1,1]) /// }}}

Type p. and then hit TAB.

{{{id=21| p. /// }}} {{{id=28| print p.ferrers_diagram() /// }}}

To view the documentation and examples for the core method, type p.core? and hit TAB.

{{{id=27| p.core? /// }}}

To view the source code for the core method, type p.core?? and hit TAB.

{{{id=30| p.core?? /// }}}

Yoda!

50,000 triangles!

{{{id=34| from scipy import io x = io.loadmat(DATA + 'yodapose.mat') from sage.plot.plot3d.index_face_set import IndexFaceSet V = x['V']; F3 = x['F3']-1; F4 = x['F4']-1 Y = (IndexFaceSet(F3, V, color = Color('#00aa00')) + IndexFaceSet(F4, V, color = Color('#00aa00'))) Y = Y.rotateX(-1) Y.show(aspect_ratio = [1,1,1], frame = False, figsize = 4) /// }}} {{{id=33| /// }}} {{{id=32| /// }}}

@interact with your mathematics!

{{{id=40| def plottaylor(order=(1..15)): var('x') f = sin(x) * e^(-x) g = f.taylor(x, 0, order) F = plot(f,-1, 5, thickness=2) G = plot(g,-1, 5, color='green', thickness=2) show(F+G, ymin = -.5, ymax = 1) /// }}} {{{id=39| plottaylor(1) /// }}} {{{id=38| @interact def plottaylor(order=(1..15)): var('x') f = sin(x) * e^(-x) g = f.taylor(x, 0, order) html('$f(x)\;=\;%s$'%latex(f)) html('$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(0,latex(g),order+1)) F = plot(f,-1, 5, thickness=2) G = plot(g,-1, 5, color='green', thickness=2) show(F+G, ymin = -.5, ymax = 1) /// }}}

@interact with your mathematics!

{{{id=37| %hide import urllib class Day: def __init__(self, date, open, high, low, close, volume): self.date = date self.open=float(open); self.high=float(high); self.low=float(low); self.close=float(close) self.volume=int(volume) def __repr__(self): return '%10s %4.2f %4.2f %4.2f %4.2f %10d'%(self.date, self.open, self.high, self.low, self.close, self.volume) class Stock: def __init__(self, symbol): self.symbol = symbol.upper() def __repr__(self): return "%s (%s)"%(self.symbol, self.yahoo()['price']) def yahoo(self): url = 'http://finance.yahoo.com/d/quotes.csv?s=%s&f=%s' % (self.symbol, 'l1c1va2xj1b4j4dyekjm3m4rr5p5p6s7') values = urllib.urlopen(url).read().strip().strip('"').split(',') data = {} data['price'] = values[0] data['change'] = values[1] data['volume'] = values[2] data['avg_daily_volume'] = values[3] data['stock_exchange'] = values[4] data['market_cap'] = values[5] data['book_value'] = values[6] data['ebitda'] = values[7] data['dividend_per_share'] = values[8] data['dividend_yield'] = values[9] data['earnings_per_share'] = values[10] data['52_week_high'] = values[11] data['52_week_low'] = values[12] data['50day_moving_avg'] = values[13] data['200day_moving_avg'] = values[14] data['price_earnings_ratio'] = values[15] data['price_earnings_growth_ratio'] = values[16] data['price_sales_ratio'] = values[17] data['price_book_ratio'] = values[18] data['short_ratio'] = values[19] return data def historical(self): try: return self.__historical except AttributeError: pass symbol = self.symbol def get_data(exchange): name = get_remote_file('http://finance.google.com/finance/historical?q=%s:%s&output=csv'%(exchange, symbol.upper()), verbose=False) return open(name).read() R = get_data('NASDAQ') if "Bad Request" in R: R = get_data("NYSE") R = R.splitlines() headings = R[0].split(',') self.__historical = [] try: for x in reversed(R[1:]): date, opn, high, low, close, volume = x.split(',') self.__historical.append(Day(date, opn,high,low,close,volume)) except ValueError: pass self.__historical = Sequence(self.__historical,cr=True,universe=lambda x:x) return self.__historical def plot_average(self, spline_samples=10): d = self.historical() if len(d) == 0: return text('no historical data at Google Finance about %s'%self.symbol, (0,3)) avg = list(enumerate([(z.high+z.low)/2 for z in d])) P = line(avg) + points(avg, rgbcolor='black', pointsize=4) + \ text(self.symbol, (len(d)*1.05, d[-1].low), horizontal_alignment='right', rgbcolor='black') if spline_samples > 0: k = 250//spline_samples spl = spline([avg[i*k] for i in range(len(d)//k)] + [avg[-1]]) P += plot(spl, (0,len(d)+30), color=(0.7,0.7,0.7)) P.xmax(260) return P def plot_diff(self): d = self.historical() if len(d) == 0: return text('no historical data at Google Finance about %s'%self.symbol, (0,3)) diff = [] for i in range(1, len(d)): z1 = d[i]; z0 = d[i-1] diff.append((i, (z1.high+z1.low)/2 - (z0.high + z0.low)/2)) P = line(diff,thickness=0.5) + points(diff, rgbcolor='black', pointsize=4) + \ text(self.symbol, (len(d)*1.05, 0), horizontal_alignment='right', rgbcolor='black') P.xmax(260) return P symbols = ['bsc', 'vmw', 'sbux', 'aapl', 'amzn', 'goog', 'wfmi', 'msft', 'yhoo', 'ebay', 'java', 'rht', ]; symbols.sort() stocks = dict([(s,Stock(s)) for s in symbols]) @interact def data(symbol = symbols, other_symbol='', spline_samples=(8,[0..15])): if other_symbol != '': symbol = other_symbol S = Stock(symbol) html('

%s

'%S) S.plot_average(spline_samples).save('avg.png', figsize=[10,2]) S.plot_diff().save('diff.png', figsize=[10,2]) Y = S.yahoo() k = Y.keys(); k.sort() html('Price during last 52 weeks:
Grey line is a spline through %s points (do not take seriously!):

'%spline_samples) html('Difference from previous day:

') html('' + '\n'.join(''%(k[i], Y[k[i]]) for i in range(len(k))) + '

') /// }}} {{{id=36| /// }}}

Cython is a Python-to-C compiler and a way to very efficiently use C/C++ constructions and libraries.
Over a third of Sage core library written in Cython.

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A Cython Example

Pure Python: sum the first $N$ positive integers: $\sum_{k=1}^N k$

{{{id=62| def sum_first_N_python(N): s = int(0) for k in range(1,N+1): s += k return s /// }}} {{{id=59| time sum_first_N_python(10^7) /// }}}

TWO CHANGES: declare $k$ to be a C int; declare $s$ to be a C long long.

{{{id=43| %cython def sum_first_N_cython(N): cdef long long s = 0 cdef int k for k in range(1,N+1): s += k return s /// }}} {{{id=73| time sum_first_N_cython(10^7) /// }}} {{{id=42| /// }}}

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Community

Asking Questions

Definitely do ask a lot of questions!
sage-devel: 900 members, 1000 messages per month
sage-support: 1220 members, 800 messages per month
sage-combinat-devel: 77 members, 100 messages per month
#sage-devel on freenode: 30 members, medium activity, usually at least two ‘core’ developers around
real people: this might be a good moment for the Sage developers in the room to stand up and introduce themselves.

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Netiquette

Many Sage developers are volunteers and have day jobs or other obligations.
Usually, questions are answered quickly if someone knows the answer straight away.
If your question goes unanswered, it is probably just because people didn’t get around to it yet. Don’t hesitate to ask again!
Keep it friendly though, we take pride in the fact that our mailing lists hardly see any insults and flame wars.

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