test

33 days 전, ABC 작성

plot(sin(x)) 
       

avuh
plot? 
       

File: /usr/local/sage-5.12/local/lib/python2.7/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”.

  • base - (default: 10) the base of the logarithm if a logarithmic scale is set. This must be greater than 1. The base can be also given as a list or tuple (basex, basey). basex sets the base of the logarithm along the horizontal axis and basey sets the base along the vertical axis.

  • scale – (default: "linear") string. The scale of the axes. Possible values are "linear", "loglog", "semilogx", "semilogy".

    The scale can be also be given as single argument that is a list or tuple (scale, base) or (scale, basex, basey).

    The "loglog" scale sets both the horizontal and vertical axes to logarithmic scale. The "semilogx" scale sets the horizontal axis to logarithmic scale. The "semilogy" scale sets the vertical axis to logarithmic scale. The "linear" scale is the default value when Graphics is initialized.

  • xmin - starting x value

  • xmax - ending x value

  • ymin - starting y value in the rendered figure

  • ymax - ending y value in the rendered figure

  • 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_color - the color of the text for this item in the legend

  • legend_label - the label for this item in the legend

Note

  • If the scale is "linear", then irrespective of what base is set to, it will default to 10 and will remain unused.
  • If you want to limit the plot along the horizontal axis in the final rendered figure, then pass the xmin and xmax keywords to the show() method. To limit the plot along the vertical axis, ymin and ymax keywords can be provided to either this plot command or to the show command.
  • For the other keyword options that the plot function can take, refer to the method show().

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 - (default: “-”) The style of the line, which is one of
    • "-" or "solid"
    • "--" or "dashed"
    • "-." or "dash dot"
    • ":" or "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='$\sin$')

It is possible to use a different color for the text of each label:

sage: p1 = plot(sin, legend_label='sin', legend_color='red')
sage: p2 = plot(cos, legend_label='cos', legend_color='green')
sage: p1 + p2

Note that the independent variable may be omitted if there is no ambiguity:

sage: plot(sin(1/x), (-1, 1))

Plotting in logarithmic scale is possible for 2D plots. There are two different syntaxes supported:

sage: plot(exp, (1, 10), scale='semilogy') # log axis on vertical

sage: plot_semilogy(exp, (1, 10)) # same thing

sage: plot_loglog(exp, (1, 10))   # both axes are log

sage: plot(exp, (1, 10), scale='loglog', base=2) # base of log is 2

We can also change the scale of the axes in the graphics just before displaying:

sage: G = plot(exp, 1, 10)
sage: G.show(scale=('semilogy', 2))

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))

Another way to avoid getting complex numbers for negative input is to calculate for the positive and negate the answer:

sage: plot(sign(x)*abs(x)^(1/3),-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), title='Plot of $\sin(x^2)$', axes_labels=['$x$','$y$']) # These labels will be nicely typeset
sage: plot(sin(x^2), (x, -3, 3), title='Plot of sin(x^2)', 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)

You can even have custom tick labels along with custom positioning.

sage: plot(x**2, (x,0,3), ticks=[[1,2.5],[0.5,1,2]], tick_formatter=[["$x_1$","$x_2$"],["$y_1$","$y_2$","$y_3$"]])

You can force Type 1 fonts in your figures by providing the relevant option as shown below. This also requires that LaTeX, dvipng and Ghostscript be installed:

sage: plot(x, typeset='type1') # optional - latex

A example with excluded values:

sage: plot(floor(x), (x, 1, 10), exclude = [1..10])

We exclude all points where PrimePi 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 (click to the left of this block for traceback)
...

File: /usr/local/sage-5.12/local/lib/python2.7/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”.

  • base - (default: 10) the base of the logarithm if a logarithmic scale is set. This must be greater than 1. The base can be also given as a list or tuple (basex, basey). basex sets the base of the logarithm along the horizontal axis and basey sets the base along the vertical axis.

  • scale – (default: "linear") string. The scale of the axes. Possible values are "linear", "loglog", "semilogx", "semilogy".

    The scale can be also be given as single argument that is a list or tuple (scale, base) or (scale, basex, basey).

    The "loglog" scale sets both the horizontal and vertical axes to logarithmic scale. The "semilogx" scale sets the horizontal axis to logarithmic scale. The "semilogy" scale sets the vertical axis to logarithmic scale. The "linear" scale is the default value when Graphics is initialized.

  • xmin - starting x value

  • xmax - ending x value

  • ymin - starting y value in the rendered figure

  • ymax - ending y value in the rendered figure

  • 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_color - the color of the text for this item in the legend

  • legend_label - the label for this item in the legend

Note

  • If the scale is "linear", then irrespective of what base is set to, it will default to 10 and will remain unused.
  • If you want to limit the plot along the horizontal axis in the final rendered figure, then pass the xmin and xmax keywords to the show() method. To limit the plot along the vertical axis, ymin and ymax keywords can be provided to either this plot command or to the show command.
  • For the other keyword options that the plot function can take, refer to the method show().

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 - (default: “-”) The style of the line, which is one of
    • "-" or "solid"
    • "--" or "dashed"
    • "-." or "dash dot"
    • ":" or "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='$\sin$')

It is possible to use a different color for the text of each label:

sage: p1 = plot(sin, legend_label='sin', legend_color='red')
sage: p2 = plot(cos, legend_label='cos', legend_color='green')
sage: p1 + p2

Note that the independent variable may be omitted if there is no ambiguity:

sage: plot(sin(1/x), (-1, 1))

Plotting in logarithmic scale is possible for 2D plots. There are two different syntaxes supported:

sage: plot(exp, (1, 10), scale='semilogy') # log axis on vertical

sage: plot_semilogy(exp, (1, 10)) # same thing

sage: plot_loglog(exp, (1, 10))   # both axes are log

sage: plot(exp, (1, 10), scale='loglog', base=2) # base of log is 2

We can also change the scale of the axes in the graphics just before displaying:

sage: G = plot(exp, 1, 10)
sage: G.show(scale=('semilogy', 2))

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))

Another way to avoid getting complex numbers for negative input is to calculate for the positive and negate the answer:

sage: plot(sign(x)*abs(x)^(1/3),-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), title='Plot of $\sin(x^2)$', axes_labels=['$x$','$y$']) # These labels will be nicely typeset
sage: plot(sin(x^2), (x, -3, 3), title='Plot of sin(x^2)', 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)

You can even have custom tick labels along with custom positioning.

sage: plot(x**2, (x,0,3), ticks=[[1,2.5],[0.5,1,2]], tick_formatter=[["$x_1$","$x_2$"],["$y_1$","$y_2$","$y_3$"]])

You can force Type 1 fonts in your figures by providing the relevant option as shown below. This also requires that LaTeX, dvipng and Ghostscript be installed:

sage: plot(x, typeset='type1') # optional - latex

A example with excluded values:

sage: plot(floor(x), (x, 1, 10), exclude = [1..10])

We exclude all points where PrimePi 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 ticket #10246):

sage: f(x)=x; f
x |--> x
sage: plot(f,(x,-1,1))