Python: function plot

plot(*args, **kwds): 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. 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 '--' (dashed), '-.' (dash dot), '-' (solid), 'steps', ':' (dotted) - ``marker`` - "'0' (tickleft), '1' (tickright), '2' (tickup), '3' (tickdown), '' (nothing), ' ' (nothing), '+' (plus), ',' (pixel), '.' (point), '1' (tri_down), '3' (tri_left), '2' (tri_up), '4' (tri_right), '<' (triangle_left), '>' (triangle_right), 'None' (nothing), 'D' (diamond), 'H' (hexagon2), '_' (hline), '\^' (triangle_up), 'd' (thin_diamond), 'h' (hexagon1), 'o' (circle), 'p' (pentagon), 's' (square), 'v' (triangle_down), 'x' (x), '|' (vline)" - ``markersize`` - the size of the marker in points - ``markeredgecolor`` - the markerfacecolor can be any color arg - ``markeredgewidth`` - the size of the marker edge in points FILLING OPTIONS: INPUT: - ``fill`` - (Default: None) 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 dictonary 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, rgbcolor='#ff00ff') sage: plot(sin, 0, 10, rgbcolor='purple') 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)) 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)) 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)], rgbcolor='green') sage: p += line([(0.5, 0.5), (0.5, 0.5^2)], rgbcolor='purple') sage: p += point(((0, 0), (0.5, 0.5), (0.5, 0.5^2), (1, 1)), rgbcolor='red', pointsize=20) sage: p += text('A', (-0.05, 0.1), rgbcolor='red') sage: p += text('B', (1.01, 1.1), rgbcolor='red') sage: p += text('C', (0.48, 0.57), rgbcolor='red') sage: p += text('D', (0.53, 0.18), rgbcolor='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), rgbcolor=hue(0.6)) We can change the line style to one of ``'--'`` (two hyphens, yielding dashed), ``'-.'`` (dash dot), ``'-'`` (solid), ``'steps'``, ``':'`` (dotted):: sage: plot(sin(x), 0, 10, linestyle='-.') 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]])) 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.