Sage et l'aide en ligne

Il est natuellement possible de se documenter sur les fonctions de Sage directement en ligne:

lcalc?

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/lfunctions/lcalc.py

Type: <class ‘sage.lfunctions.lcalc.LCalc’>

Definition: lcalc(args)

Docstring:

Rubinstein’s L-functions Calculator

Type lcalc.[tab] for a list of useful commands that are implemented using the command line interface, but return objects that make sense in Sage. For each command the possible inputs for the L-function are:

" - (default) the Riemann zeta function

'tau' - the L function of the Ramanujan delta function

elliptic curve E - where E is an elliptic curve over \mathbb{Q}; defines L(E,s)

You can also use the complete command-line interface of Rubinstein’s L-functions calculations program via this class. Type lcalc.help() for a list of commands and how to call them.

plot?

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/misc/decorators.py

Type: <type ‘function’>

Definition: plot(funcs, exclude=None, fillalpha=0.5, fillcolor=’automatic’, detect_poles=False, plot_points=200, thickness=1, adaptive_tolerance=0.01, rgbcolor=(0, 0, 1), adaptive_recursion=5, aspect_ratio=’automatic’, alpha=1, legend_label=None, fill=False, *args, **kwds)

Docstring:

Use plot by writing

plot(X, ...)

where X is a Sage object (or list of Sage objects) that either is callable and returns numbers that can be coerced to floats, or has a plot method that returns a GraphicPrimitive object.

There are many other specialized 2D plot commands available in Sage, such as plot_slope_field, as well as various graphics primitives like Arrow; type sage.plot.plot? for a current list.

Type plot.options for a dictionary of the default options for plots. You can change this to change the defaults for all future plots. Use plot.reset() to reset to the default options.

PLOT OPTIONS:

plot_points - (default: 200) the minimal number of plot points.

adaptive_recursion - (default: 5) how many levels of recursion to go before giving up when doing adaptive refinement. Setting this to 0 disables adaptive refinement.

adaptive_tolerance - (default: 0.01) how large a difference should be before the adaptive refinement code considers it significant. See the documentation further below for more information, starting at “the algorithm used to insert”.

xmin - starting x value

xmax - ending x value

color - an RGB tuple (r,g,b) with each of r,g,b between 0 and 1, or a color name as a string (e.g., ‘purple’), or an HTML color such as ‘#aaff0b’.

detect_poles - (Default: False) If set to True poles are detected. If set to “show” vertical asymptotes are drawn.

legend_label - the label for this item in the legend

APPEARANCE OPTIONS:

The following options affect the appearance of the line through the points on the graph of X (these are the same as for the line function):

INPUT:

alpha - How transparent the line is

thickness - How thick the line is

rgbcolor - The color as an RGB tuple

hue - The color given as a hue

Any MATPLOTLIB line option may also be passed in. E.g.,

linestyle - The style of the line, which is one of

"-" (solid) – default

"--" (dashed)

"-." (dash dot)

":" (dotted)

"None" or " " or "" (nothing)

The linestyle can also be prefixed with a drawing style (e.g., "steps--")

"default" (connect the points with straight lines)

"steps" or "steps-pre" (step function; horizontal line is to the left of point)

"steps-mid" (step function; points are in the middle of horizontal lines)

"steps-post" (step function; horizontal line is to the right of point)

marker - The style of the markers, which is one of

"None" or " " or "" (nothing) – default

"," (pixel), "." (point)

"_" (horizontal line), "|" (vertical line)

"o" (circle), "p" (pentagon), "s" (square), "x" (x), "+" (plus), "*" (star)

"D" (diamond), "d" (thin diamond)

"H" (hexagon), "h" (alternative hexagon)

"<" (triangle left), ">" (triangle right), "^" (triangle up), "v" (triangle down)

"1" (tri down), "2" (tri up), "3" (tri left), "4" (tri right)

0 (tick left), 1 (tick right), 2 (tick up), 3 (tick down)

4 (caret left), 5 (caret right), 6 (caret up), 7 (caret down)

" $...$ " (math TeX string)

markersize - the size of the marker in points

markeredgecolor – the color of the marker edge

markerfacecolor – the color of the marker face

markeredgewidth - the size of the marker edge in points

exclude - (Default: None) values which are excluded from the plot range. Either a list of real numbers, or an equation in one variable.

FILLING OPTIONS:

fill - (Default: False) One of:

“axis” or True: Fill the area between the function and the x-axis.

“min”: Fill the area between the function and its minimal value.

“max”: Fill the area between the function and its maximal value.

a number c: Fill the area between the function and the horizontal line y = c.

a function g: Fill the area between the function that is plotted and g.

a dictionary d (only if a list of functions are plotted): The keys of the dictionary should be integers. The value of d[i] specifies the fill options for the i-th function in the list. If d[i] == [j]: Fill the area between the i-th and the j-th function in the list. (But if d[i] == j: Fill the area between the i-th function in the list and the horizontal line y = j.)

fillcolor - (default: ‘automatic’) The color of the fill. Either ‘automatic’ or a color.

fillalpha - (default: 0.5) How transparent the fill is. A number between 0 and 1.

Note that this function does NOT simply sample equally spaced points between xmin and xmax. Instead it computes equally spaced points and add small perturbations to them. This reduces the possibility of, e.g., sampling sin only at multiples of 2\pi, which would yield a very misleading graph.

EXAMPLES: We plot the sin function:
sage: P = plot(sin, (0,10)); print P
Graphics object consisting of 1 graphics primitive
sage: len(P)     # number of graphics primitives
1
sage: len(P[0])  # how many points were computed (random)
225
sage: P          # render
sage: P = plot(sin, (0,10), plot_points=10); print P
Graphics object consisting of 1 graphics primitive
sage: len(P[0])  # random output
32
sage: P          # render
We plot with randomize=False, which makes the initial sample points evenly spaced (hence always the same). Adaptive plotting might insert other points, however, unless adaptive_recursion=0.
sage: p=plot(1, (x,0,3), plot_points=4, randomize=False, adaptive_recursion=0)
sage: list(p[0])
[(0.0, 1.0), (1.0, 1.0), (2.0, 1.0), (3.0, 1.0)]
Some colored functions:
sage: plot(sin, 0, 10, color='purple')
sage: plot(sin, 0, 10, color='#ff00ff')
We plot several functions together by passing a list of functions as input:
sage: plot([sin(n*x) for n in [1..4]], (0, pi))
We can also build a plot step by step from an empty plot:
sage: a = plot([]); a       # passing an empty list returns an empty plot (Graphics() object)
sage: a += plot(x**2); a    # append another plot
sage: a += plot(x**3); a    # append yet another plot
The function \sin(1/x) wiggles wildly near 0. Sage adapts to this and plots extra points near the origin.
sage: plot(sin(1/x), (x, -1, 1))
Via the matplotlib library, Sage makes it easy to tell whether a graph is on both sides of both axes, as the axes only cross if the origin is actually part of the viewing area:
sage: plot(x^3,(x,0,2))  # this one has the origin
sage: plot(x^3,(x,1,2))  # this one does not
Another thing to be aware of with axis labeling is that when the labels have quite different orders of magnitude or are very large, scientific notation (the e notation for powers of ten) is used:
sage: plot(x^2,(x,480,500))  # this one has no scientific notation
sage: plot(x^2,(x,300,500))  # this one has scientific notation on y-axis
You can put a legend with legend_label (the legend is only put once in the case of multiple functions):
sage: plot(exp(x), 0, 2, legend_label='$e^x$')
Sage understands TeX, so these all are slightly different, and you can choose one based on your needs:
sage: plot(sin, legend_label='sin')
sage: plot(sin, legend_label='$sin$')
sage: plot(sin, legend_label='$\sin$')
Note that the independent variable may be omitted if there is no ambiguity:
sage: plot(sin(1/x), (-1, 1))
The algorithm used to insert extra points is actually pretty simple. On the picture drawn by the lines below:
sage: p = plot(x^2, (-0.5, 1.4)) + line([(0,0), (1,1)], color='green')
sage: p += line([(0.5, 0.5), (0.5, 0.5^2)], color='purple')
sage: p += point(((0, 0), (0.5, 0.5), (0.5, 0.5^2), (1, 1)), color='red', pointsize=20)
sage: p += text('A', (-0.05, 0.1), color='red')
sage: p += text('B', (1.01, 1.1), color='red')
sage: p += text('C', (0.48, 0.57), color='red')
sage: p += text('D', (0.53, 0.18), color='red')
sage: p.show(axes=False, xmin=-0.5, xmax=1.4, ymin=0, ymax=2)
You have the function (in blue) and its approximation (in green) passing through the points A and B. The algorithm finds the midpoint C of AB and computes the distance between C and D. If that distance exceeds the adaptive_tolerance threshold (relative to the size of the initial plot subintervals), the point D is added to the curve. If D is added to the curve, then the algorithm is applied recursively to the points A and D, and D and B. It is repeated adaptive_recursion times (5, by default).

The actual sample points are slightly randomized, so the above plots may look slightly different each time you draw them.

We draw the graph of an elliptic curve as the union of graphs of 2 functions.
sage: def h1(x): return abs(sqrt(x^3  - 1))
sage: def h2(x): return -abs(sqrt(x^3  - 1))
sage: P = plot([h1, h2], 1,4)
sage: P          # show the result
We can also directly plot the elliptic curve:
sage: E = EllipticCurve([0,-1])
sage: plot(E, (1, 4), color=hue(0.6))
We can change the line style as well:
sage: plot(sin(x), (x, 0, 10), linestyle='-.')
If we have an empty linestyle and specify a marker, we can see the points that are actually being plotted:
sage: plot(sin(x), (x,0,10), plot_points=20, linestyle='', marker='.')
The marker can be a TeX symbol as well:
sage: plot(sin(x), (x,0,10), plot_points=20, linestyle='', marker=r'$\checkmark$')
Sage currently ignores points that cannot be evaluated
sage: set_verbose(-1)
sage: plot(-x*log(x), (x,0,1))  # this works fine since the failed endpoint is just skipped.
sage: set_verbose(0)
This prints out a warning and plots where it can (we turn off the warning by setting the verbose mode temporarily to -1.)
sage: set_verbose(-1)
sage: plot(x^(1/3), (x,-1,1))
sage: set_verbose(0)
To plot the negative real cube root, use something like the following:
sage: plot(lambda x : RR(x).nth_root(3), (x,-1, 1))
We can detect the poles of a function:
sage: plot(gamma, (-3, 4), detect_poles = True).show(ymin = -5, ymax = 5)
We draw the Gamma-Function with its poles highlighted:
sage: plot(gamma, (-3, 4), detect_poles = 'show').show(ymin = -5, ymax = 5)
The basic options for filling a plot:
sage: p1 = plot(sin(x), -pi, pi, fill = 'axis')
sage: p2 = plot(sin(x), -pi, pi, fill = 'min')
sage: p3 = plot(sin(x), -pi, pi, fill = 'max')
sage: p4 = plot(sin(x), -pi, pi, fill = 0.5)
sage: graphics_array([[p1, p2], [p3, p4]]).show(frame=True, axes=False)

sage: plot([sin(x), cos(2*x)*sin(4*x)], -pi, pi, fill = {0: 1}, fillcolor = 'red', fillalpha = 1)
A example about the growth of prime numbers:
sage: plot(1.13*log(x), 1, 100, fill = lambda x: nth_prime(x)/floor(x), fillcolor = 'red')
Fill the area between a function and its asymptote:
sage: f = (2*x^3+2*x-1)/((x-2)*(x+1))
sage: plot([f, 2*x+2], -7,7, fill = {0: [1]}, fillcolor='#ccc').show(ymin=-20, ymax=20)
Fill the area between a list of functions and the x-axis:
sage: def b(n): return lambda x: bessel_J(n, x)
sage: plot([b(n) for n in [1..5]], 0, 20, fill = 'axis')
Note that to fill between the ith and jth functions, you must use dictionary key-value pairs i:[j]; key-value pairs like i:j will fill between the ith function and the line y=j:
sage: def b(n): return lambda x: bessel_J(n, x) + 0.5*(n-1)
sage: plot([b(c) for c in [1..5]], 0, 40, fill = dict([(i, [i+1]) for i in [0..3]]))
sage: plot([b(c) for c in [1..5]], 0, 40, fill = dict([(i, i+1) for i in [0..3]]))
Extra options will get passed on to show(), as long as they are valid:
sage: plot(sin(x^2), (x, -3, 3), axes_labels=['$x$','$y$']) # These labels will be nicely typeset
sage: plot(sin(x^2), (x, -3, 3), axes_labels=['x','y']) # These will not
sage: plot(sin(x^2), (x, -3, 3), figsize=[8,2])
sage: plot(sin(x^2), (x, -3, 3)).show(figsize=[8,2]) # These are equivalent
This includes options for custom ticks and formatting. See documentation for show() for more details.
sage: plot(sin(pi*x), (x, -8, 8), ticks=[[-7,-3,0,3,7],[-1/2,0,1/2]])
sage: plot(2*x+1,(x,0,5),ticks=[[0,1,e,pi,sqrt(20)],2],tick_formatter="latex")
This is particularly useful when setting custom ticks in multiples of pi.
sage: plot(sin(x),(x,0,2*pi),ticks=pi/3,tick_formatter=pi)
A example with excluded values:
sage: plot(floor(x), (x, 1, 10), exclude = [1..10])
We exclude all points where prime_pi makes a jump:
sage: jumps = [n for n in [1..100] if prime_pi(n) != prime_pi(n-1)]
sage: plot(lambda x: prime_pi(x), (x, 1, 100), exclude = jumps)
Excluded points can also be given by an equation:
sage: g(x) = x^2-2*x-2
sage: plot(1/g(x), (x, -3, 4), exclude = g(x) == 0, ymin = -5, ymax = 5)
exclude and detect_poles can be used together::

sage: f(x) = (floor(x)+0.5) / (1-(x-0.5)^2) sage: plot(f, (x, -3.5, 3.5), detect_poles = ‘show’, exclude = [-3..3], ymin = -5, ymax = 5)

TESTS:

We do not randomize the endpoints:
sage: p = plot(x, (x,-1,1))
sage: p[0].xdata[0] == -1
True
sage: p[0].xdata[-1] == 1
True
We check to make sure that the x/y min/max data get set correctly when there are multiple functions.
sage: d = plot([sin(x), cos(x)], 100, 120).get_minmax_data()
sage: d['xmin']
100.0
sage: d['xmax']
120.0
We check various combinations of tuples and functions, ending with tests that lambda functions work properly with explicit variable declaration, without a tuple.
sage: p = plot(lambda x: x,(x,-1,1))
sage: p = plot(lambda x: x,-1,1)
sage: p = plot(x,x,-1,1)
sage: p = plot(x,-1,1)
sage: p = plot(x^2,x,-1,1)
sage: p = plot(x^2,xmin=-1,xmax=2)
sage: p = plot(lambda x: x,x,-1,1)
sage: p = plot(lambda x: x^2,x,-1,1)
sage: p = plot(lambda x: 1/x,x,-1,1)
sage: f(x) = sin(x+3)-.1*x^3
sage: p = plot(lambda x: f(x),x,-1,1)
We check to handle cases where the function gets evaluated at a point which causes an ‘inf’ or ‘-inf’ result to be produced.
sage: p = plot(1/x, 0, 1)
sage: p = plot(-1/x, 0, 1)
Bad options now give better errors:
sage: P = plot(sin(1/x), (x,-1,3), foo=10)
Traceback (most recent call last):
...
RuntimeError: Error in line(): option 'foo' not valid.
sage: P = plot(x, (x,1,1)) # trac ticket #11753
Traceback (most recent call last):
...
ValueError: plot start point and end point must be different
We test that we can plot f(x)=x (see Trac 10246):
sage: f(x)=x; f
x |--> x
sage: plot(f,(x,-1,1))