SAGE tip: Fourier Series Approximation with Interactive mode

After my last post, about Fouries Series, I decided to give a try to the interactive mode of SAGE.

What did I do?

First I check out the examples worked at the Wiki page. In the section of graphics I got inspired from the post Interactive 2D Plotting by Timothy Clemans and Newton’s Method by William Stein post in the Calculus section. So… there it is!

The code is too long to be posted here… but here it goes anyway

1	var('x,n,i')

1 2	def error_msg(msg): .. print '<p style="font-family: Arial, sans-serif; color: #000;"><span     style="color: red; font-weight: bold;">Error</span>: %s</p>' % msg

1

...

1 2 3 4 5

def a(f, n, init, final): .. x = f.variables()[0] .. L = (final - init)/2 .. coeff = integral(f(x)*cos(n*pi*x/L), (x,init,final))/L .. return coeff

…

def b(f, n, init, final):
.. x = f.variables()[0]
.. L = (final – init)/2
.. coeff = integral(f(x)*sin(n*pi*x/L), (x,init,final))/L
.. return coeff

…

def FS(f, n, init, final):
.. x = f.variables()[0]
.. L = (final – init)/2
.. return a(f, 0, init, final)/2 + sum(a(f, i, init, final)*cos(i*x*pi/L)+b(f, i, init, final)*sin(i*x*pi/L), i, 1, n)

…

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@interact def interactive_2d_plotter(clr=Color('green'), expression=input_box('x^2', 'Expression', str), x_range=range_slider(-6,6,1,(-4,4), label='X Range'), order=slider([0..100],None,None,3), square=checkbox(True, 'Square'), axes=checkbox(False, 'Show Axes')): .. if expression: ..   try: ..      expression = SR(expression) # turn string into a Sage expression ..   except TypeError: ..      print error_msg('This is not an expression.') ..      return ..   try: ..      xmin, xmax = x_range ..      n = order ..      if square or not axes: ..         print "var('%s')nplot(%s).show(%s%s%s)" % (expression.variables()[0], repr(expression), 'aspect_ratio=1' if square else '', ', ' if square and not axes else '', 'axes=False' if not axes else '') ..         if square: ..            P = plot(FS(expression, n, xmin, xmax), xmin, xmax, color=clr) + plot(expression, xmin, xmax) ..            P.show(aspect_ratio=1, axes=axes) ..         else: ..            P = plot(FS(expression, n, xmin, xmax), xmin, xmax, color=clr) + plot(expression, xmin, xmax) ..            P.show(axes=axes) ..      else: ..         print "var('%s')nplot(%s)" % (expression.variables()[0], repr(expression)) ..         P = plot(FS(expression, n, xmin, xmax), xmin, xmax, color=clr) + plot(expression, xmin, xmax) ..         P.show(axes=axes) ..   except ValueError: ..      print error_msg('This expression has more than one variable.') ..      return ..   except TypeError: ..      print error_msg("This expression contains an unknown function.") ..      return

because it’s almost in-understandable

Some pictures!!!

The code is set to give the Fourier approximation of the with aspect_ratio = (1,1).

Initial Plot. Approx. of x^2

One can set a non one-to-one aspect_ratio, by un-checking the box,

Initial Plot with-no-square

And we can view the frame as well by checking the other box,

Initial plot no-square and frame

The first box allows us to change the colour, by using a palette,

Initial plot with color-palette

for example, set the colour to red,

Initial plot in red

We can also change the function, to an exponential, ,

Fourier approximation of the exponential

and change the domain of the function,

Changing the interval

or the order of the approximation,

Exponential changing the order of the approximation

I published my sage file on my page (at the end of the page in the attached files)… for the sake of clarity (and completeness).

Enjoy,

Dox

SAGE tip: Graphs of Elementary Numerical Integrals

Looking on the notebook worksheets publish on sagenb I found a very interesting application of the interactive mode, Interactive.

Since one can not look the result of the published notebooks, I use its code to playing around a little bit, so here are the plots.

The default function to be integrated is , the interval of integration is set , the number of partitions of the interval is 4, and the method is the mid-point,

We can see the visual integration, the analitic result, the numerical one, and the error. Nice, Isn’t it?

Let’s change the number of partitions, say to 10,

Much better! But now, let’s try the other methods… left-point,

The right-point,

The upper-point,

The lower-point,

The trapezoid method,

which is one of the best.

But finally the Simpson method,

Impressive!!!

So, I think it’s a good idea to give a try to the interactive mode in SAGE, and if you go thru the code you’ll find out it’s not as difficult as it seems.

Enjoy!

Dox