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- """
- Illustrate forward, backward and centered finite differences
- in four figures.
- """
- from pysketcher import *
- #test_test()
- xaxis = 2
- drawing_tool.set_coordinate_system(0, 7, 1, 6, axis=False)
- f = SketchyFunc1('$u(t)$')
- x = 3 # center point where we want the derivative
- xb = 2 # x point used for backward difference
- xf = 4 # x point used for forward difference
- p = (x, f(x)) # center point
- pf = (xf, f(xf)) # forward point
- pb = (xb, f(xb)) # backward point
- r = 0.1 # radius of circles placed at key points
- c = Circle(p, r).set_linecolor('blue')
- cf = Circle(pf, r).set_linecolor('red')
- cb = Circle(pb, r).set_linecolor('green')
- # Points in the mesh
- p0 = point(x, xaxis) # center point
- pf0 = point(xf, xaxis) # forward point
- pb0 = point(xb, xaxis) # backward point
- tick = 0.05
- # 1D mesh with three points
- mesh = Composition({
- 'tnm1': Text('$t_{n-1}$', pb0 - point(0, 0.3)),
- 'tn': Text('$t_{n}$', p0 - point(0, 0.3)),
- 'tnp1': Text('$t_{n+1}$', pf0 - point(0, 0.3)),
- 'axis': Composition({
- 'hline': Line(pf0-point(3,0), pb0+point(3,0)).\
- set_linecolor('black').set_linewidth(1),
- 'tick_m1': Line(pf0+point(0,tick), pf0-point(0,tick)).\
- set_linecolor('black').set_linewidth(1),
- 'tick_n': Line(p0+point(0,tick), p0-point(0,tick)).\
- set_linecolor('black').set_linewidth(1),
- 'tick_p1': Line(pb0+point(0,tick), pb0-point(0,tick)).\
- set_linecolor('black').set_linewidth(1)}),
- })
- # Vertical dotted lines at each mesh point
- vlinec = Line(p, p0).set_linestyle('dotted').\
- set_linecolor('blue').set_linewidth(1)
- vlinef = Line(pf, pf0).set_linestyle('dotted').\
- set_linecolor('red').set_linewidth(1)
- vlineb = Line(pb, pb0).set_linestyle('dotted').\
- set_linecolor('green').set_linewidth(1)
- # Compose vertical lines for each type of difference
- forward_lines = Composition({'center': vlinec, 'right': vlinef})
- backward_lines = Composition({'center': vlinec, 'left': vlineb})
- centered_lines = Composition({'left': vlineb, 'right': vlinef})
- centered_lines2 = Composition({'left': vlineb, 'right': vlinef,
- 'center': vlinec})
- # Tangents illustrating the derivative
- domain = [1, 5]
- domain2 = [2, 5]
- forward_tangent = Line(p, pf).new_interval(x=domain2).\
- set_linestyle('dashed').set_linecolor('red')
- backward_tangent = Line(pb, p).new_interval(x=domain).\
- set_linestyle('dashed').set_linecolor('green')
- centered_tangent = Line(pb, pf).new_interval(x=domain).\
- set_linestyle('dashed').set_linecolor('blue')
- h = 1E-3 # h in finite difference approx used to compute the exact tangent
- exact_tangent = Line((x+h, f(x+h)), (x-h, f(x-h))).\
- new_interval(x=domain).\
- set_linestyle('dotted').set_linecolor('black')
- forward = Composition(
- dict(tangent=forward_tangent,
- point1=c, point2=cf, coor=forward_lines,
- name=Text('forward',
- forward_tangent.geometric_features()['end'] + \
- point(0.1,0), alignment='left')))
- backward = Composition(
- dict(tangent=backward_tangent,
- point1=c, point2=cb, coor=backward_lines,
- name=Text('backward',
- backward_tangent.geometric_features()['end'] + \
- point(0.1,0), alignment='left')))
- centered = Composition(
- dict(tangent=centered_tangent,
- point1=cb, point2=cf, point=c, coor=centered_lines2,
- name=Text('centered',
- centered_tangent.geometric_features()['end'] + \
- point(0.1,0), alignment='left')))
- exact = Composition(dict(graph=f, tangent=exact_tangent))
- forward = Composition(dict(difference=forward, exact=exact)).\
- set_name('forward')
- backward = Composition(dict(difference=backward, exact=exact)).\
- set_name('backward')
- centered = Composition(dict(difference=centered, exact=exact)).\
- set_name('centered')
- all = Composition(
- dict(exact=exact, forward=forward, backward=backward,
- centered=centered)).set_name('all')
- for fig in forward, backward, centered, all:
- drawing_tool.erase()
- fig.draw()
- mesh.draw()
- drawing_tool.display()
- drawing_tool.savefig('fd_'+fig.get_name())
- raw_input()
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