josef teichmann :: personal homepage at ETH Zurich

Lecture Analysis 1/2 electrical engineering

We mainly use Sage for numerical implementations, code can be found here. You can either install Sage locally on your computer, for instructions look at the installation guide, or use sage online here by creating a working account. To run the code, just copy and paste it into the first cell of your newly created worksheet and evaluate it. Sage is based on the general purpose scripting language Python and thus inherits its syntax, so be aware to reproduce the line identations in the code correctly after pasting it into the notebook cell:

Interactive worksheet for numerical, analytical and MC-integration of a 1d real-valued function

We provide a worksheet file on basic methods for integration, differentiation, plotting and handling functions in SAGE (analysis2_100308.sws).

We provide a worksheet file on numerical, analytical and MC integration (mci_100226.sws), which can be uploaded and unpacked via the sage notebook. Thanks to Georg Grafendorfer who did this implementation.

We provide a worksheet file on gradient flows(gradient-flows.sws), which demonstrates how a local search for extrema via gradient flows works.

Simulation of 2d-ODEs by explicit Euler scheme and plotting of the first coordinate as a function of time

x,y,t = var('x,y,t')
from sage.ext.fast_eval import fast_float
@interact
def _(f = input_box(default=y), g=input_box(default=-x-y+sin(t)),
      xmin=input_box(default=0), xmax=input_box(default=10),
      ymin=input_box(default=-10), ymax=input_box(default=10),
      start_x=input_box(default=0.5), start_y=input_box(default=-0.5),
      step_size=(0.01,(0.001, 0.2)), steps=(600,(0, 1400)) ):
    ff = fast_float(f, 'x', 'y', 't')
    gg = fast_float(g, 'x', 'y', 't')
    steps = int(steps)

points = [ (0, start_x) ]
    yy = start_y
    for i in range(steps):
        tt, xx = points[-1]
        try:
            points.append( (tt + step_size, xx +  step_size * ff(xx,yy,tt)) )
            yy = yy + step_size * gg(xx,yy,tt)
        except (ValueError, ArithmeticError, TypeError):
            break

solution = line(points)

result = solution
    html(r"$\displaystyle\frac{dx}{dt} = %s$  $ \displaystyle\frac{dy}{dt} = %s$" % (latex(f),latex(g)))
    result.show(xmin=xmin,xmax=xmax,ymin=ymin,ymax=ymax)

Simulation of 2d-ODEs by explicit Euler scheme and plotting of the integral curve together with the vector field

x,y = var('x,y')
from sage.ext.fast_eval import fast_float
@interact
def _(f = input_box(default=1), g=input_box(default=y),
      xmin=input_box(default=-1), xmax=input_box(default=1),
      ymin=input_box(default=-1), ymax=input_box(default=1),
      start_x=input_box(default=0.5), start_y=input_box(default=0.5),
      step_size=(0.01,(0.001, 0.2)), steps=(600,(0, 1400)) ):
    ff = fast_float(f, 'x', 'y')
    gg = fast_float(g, 'x', 'y')
    steps = int(steps)

points = [ (start_x, start_y) ]
    for i in range(steps):
        xx, yy = points[-1]
        try:
            points.append( (xx +  step_size * ff(xx,yy), yy + step_size * gg(xx,yy)) )
        except (ValueError, ArithmeticError, TypeError):
            break

starting_point = point(points[0], pointsize=50)
    solution = line(points)
    vector_field = plot_vector_field( (f,g), (x,xmin,xmax), (y,ymin,ymax) )

result = vector_field + starting_point + solution

html(r"$\displaystyle\frac{dx}{dt} = %s$  $ \displaystyle\frac{dy}{dt} = %s$" % (latex(f),latex(g)))
    result.show(xmin=xmin,xmax=xmax,ymin=ymin,ymax=ymax)