How to draw graph of Gauss function?

问题

Gauss function has an infinite number of jump discontinuities at x = 1/n, for positive integers.

I want to draw diagram of Gauss function.

Using Maxima cas I can draw it with simple command :

  f(x):= 1/x - floor(1/x);  plot2d(f(x),[x,0,1]);

but the result is not good ( near x=0 it should be like here)

Also Maxima claims:

plot2d: expression evaluates to non-numeric value somewhere in plotting range.

I can define picewise function ( jump discontinuities at x = 1/n, for positive integers )

so I tried :

define( g(x), for i:2 thru 20 step 1 do if (x=i) then x else (1/x) - floor(1/x));

but it don't works.

I can also use chebyshew polynomials to aproximate function ( like in : A Graduate Introduction to Numerical Methods From the Viewpoint of Backward Error Analysis by Corless, Robert, Fillion, Nicolas)

How to do it properly ?

回答1:

For plot2d you can set the adapt_depth and nticks parameters. The default values are 5 and 29, respectively. set_plot_option() (i.e. with no argument) returns the current list of option values. If you increase adapt_depth and/or nticks, then plot2d will use more points for plotting. Perhaps that makes the figure look good enough.

Another way is to use the draw2d function (in the draw package) and explicitly tell it to plot each segment. We know that there are discontinuities at 1/k, for k = 1, 2, 3, .... We have to decide how many segments to plot. Let's say 20.

(%i6) load (draw) $
(%i7) f(x):= 1/x - floor(1/x) $
(%i8) makelist (explicit (f, x, 1/(k + 1), 1/k), k, 1, 20);
(%o8) [explicit(f,x,1/2,1),explicit(f,x,1/3,1/2),
       explicit(f,x,1/4,1/3),explicit(f,x,1/5,1/4),
       explicit(f,x,1/6,1/5),explicit(f,x,1/7,1/6),
       explicit(f,x,1/8,1/7),explicit(f,x,1/9,1/8),
       explicit(f,x,1/10,1/9),explicit(f,x,1/11,1/10),
       explicit(f,x,1/12,1/11),explicit(f,x,1/13,1/12),
       explicit(f,x,1/14,1/13),explicit(f,x,1/15,1/14),
       explicit(f,x,1/16,1/15),explicit(f,x,1/17,1/16),
       explicit(f,x,1/18,1/17),explicit(f,x,1/19,1/18),
       explicit(f,x,1/20,1/19),explicit(f,x,1/21,1/20)]
(%i9) apply (draw2d, %);