Inleiding (Numerieke) Analyse in Sage -- Sage

var("x, y") art = exp(1/(1+x^2))/(1+y^2)*(1/x^2 + 3/y^3)+x art_plot = implicit_plot(art == 4, (x, -4, 4), (y, -4, 4)) art_plot.show() art

x + (1/x^2 + 3/y^3)*e^(1/(x^2 + 1))/(y^2 + 1)

x + (1/x^2 + 3/y^3)*e^(1/(x^2 + 1))/(y^2 + 1)

nice_plot = art_plot for y0 in [-.8,-1..-3]: art_point = (find_root(art(y=y0)-4, -2, -.1), y0) print art_point nice_plot += point(art_point, color="red") nice_plot

(-0.33788948043637007, -0.800000000000000)
(-0.3870029905047995, -1.00000000000000)
(-0.39943523857520247, -1.20000000000000)
(-0.3906906092695037, -1.40000000000000)
(-0.3726358466847202, -1.60000000000000)
(-0.3515548349816437, -1.80000000000000)
(-0.33034158206351877, -2.00000000000000)
(-0.3102133576683931, -2.20000000000000)
(-0.29160905682373867, -2.40000000000000)
(-0.2746166380255696, -2.60000000000000)
(-0.2591720909304286, -2.80000000000000)
(-0.24515183627871762, -3.00000000000000)

(-0.33788948043637007, -0.800000000000000)
(-0.3870029905047995, -1.00000000000000)
(-0.39943523857520247, -1.20000000000000)
(-0.3906906092695037, -1.40000000000000)
(-0.3726358466847202, -1.60000000000000)
(-0.3515548349816437, -1.80000000000000)
(-0.33034158206351877, -2.00000000000000)
(-0.3102133576683931, -2.20000000000000)
(-0.29160905682373867, -2.40000000000000)
(-0.2746166380255696, -2.60000000000000)
(-0.2591720909304286, -2.80000000000000)
(-0.24515183627871762, -3.00000000000000)

var("x") def f1(x): return sin(x)^2 f2 = lambda x: sin(x)^2 f3(x) = sin(x)^2 # of f3 = sin(x)^2 f4 = function("f4")(x) f1 ; f2 ; f3 ; f4

$\newcommand{\Bold}[1]{\mathbf{#1}}\verb|<function|\phantom{\verb!x!}\verb|f1|\phantom{\verb!x!}\verb|at|\phantom{\verb!x!}\verb|0x7f1affdf3cf8>|$

$\newcommand{\Bold}[1]{\mathbf{#1}}\verb|<function|\phantom{\verb!x!}\verb|<lambda>|\phantom{\verb!x!}\verb|at|\phantom{\verb!x!}\verb|0x7f1affdf3c08>|$

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \sin\left(x\right)^{2}$

$\newcommand{\Bold}[1]{\mathbf{#1}}f_{4}\left(x\right)$

x0 = 3.0 # in RR, RealField(2048), RDF f1(x0) ; f2(x0) ; f3(x=x0) ; f4(x=x0)

$\newcommand{\Bold}[1]{\mathbf{#1}}0.0199148566748170$

$\newcommand{\Bold}[1]{\mathbf{#1}}f_{4}\left(3.00000000000000\right)$

parent(f1) ; parent(f2) ; parent(f3) ; parent(f4)

$\newcommand{\Bold}[1]{\mathbf{#1}}\verb|<type|\phantom{\verb!x!}\verb|'function'>|$

$\newcommand{\Bold}[1]{\mathbf{#1}}\text{SR}$

diff(f3^2, x); diff(f4^2, x)

$\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 4 \, \cos\left(x\right) \sin\left(x\right)^{3}$

$\newcommand{\Bold}[1]{\mathbf{#1}}2 \, f_{4}\left(x\right) \frac{\partial}{\partial x}f_{4}\left(x\right)$

var("y") diff(f4, y)

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

dddf3 = diff(e^f3(x=x*sin(y)), x, y, x) dddf3; dddf3.simplify_full()

$\newcommand{\Bold}[1]{\mathbf{#1}}8 \, x \cos\left(x \sin\left(y\right)\right)^{3} \cos\left(y\right) e^{\left(\sin\left(x \sin\left(y\right)\right)^{2}\right)} \sin\left(x \sin\left(y\right)\right)^{3} \sin\left(y\right)^{2} + 12 \, x \cos\left(x \sin\left(y\right)\right)^{3} \cos\left(y\right) e^{\left(\sin\left(x \sin\left(y\right)\right)^{2}\right)} \sin\left(x \sin\left(y\right)\right) \sin\left(y\right)^{2} - 12 \, x \cos\left(x \sin\left(y\right)\right) \cos\left(y\right) e^{\left(\sin\left(x \sin\left(y\right)\right)^{2}\right)} \sin\left(x \sin\left(y\right)\right)^{3} \sin\left(y\right)^{2} + 8 \, \cos\left(x \sin\left(y\right)\right)^{2} \cos\left(y\right) e^{\left(\sin\left(x \sin\left(y\right)\right)^{2}\right)} \sin\left(x \sin\left(y\right)\right)^{2} \sin\left(y\right) - 8 \, x \cos\left(x \sin\left(y\right)\right) \cos\left(y\right) e^{\left(\sin\left(x \sin\left(y\right)\right)^{2}\right)} \sin\left(x \sin\left(y\right)\right) \sin\left(y\right)^{2} + 4 \, \cos\left(x \sin\left(y\right)\right)^{2} \cos\left(y\right) e^{\left(\sin\left(x \sin\left(y\right)\right)^{2}\right)} \sin\left(y\right) - 4 \, \cos\left(y\right) e^{\left(\sin\left(x \sin\left(y\right)\right)^{2}\right)} \sin\left(x \sin\left(y\right)\right)^{2} \sin\left(y\right)$

$\newcommand{\Bold}[1]{\mathbf{#1}}-4 \, {\left(2 \, \cos\left(x \sin\left(y\right)\right)^{4} \cos\left(y\right) e \sin\left(y\right) - 4 \, \cos\left(x \sin\left(y\right)\right)^{2} \cos\left(y\right) e \sin\left(y\right) + \cos\left(y\right) e \sin\left(y\right) + {\left(2 \, x \cos\left(x \sin\left(y\right)\right)^{5} \cos\left(y\right) e \sin\left(y\right)^{2} - 8 \, x \cos\left(x \sin\left(y\right)\right)^{3} \cos\left(y\right) e \sin\left(y\right)^{2} + 5 \, x \cos\left(x \sin\left(y\right)\right) \cos\left(y\right) e \sin\left(y\right)^{2}\right)} \sin\left(x \sin\left(y\right)\right)\right)} e^{\left(-\cos\left(x \sin\left(y\right)\right)^{2}\right)}$

dddf4 = diff(e^f4(x=x*sin(y)), x, y, x) dddf4; dddf4.simplify_full()

$\newcommand{\Bold}[1]{\mathbf{#1}}x \cos\left(y\right) e^{\left(f_{4}\left(x \sin\left(y\right)\right)\right)} \sin\left(y\right)^{2} \mathrm{D}_{0}\left(f_{4}\right)\left(x \sin\left(y\right)\right)^{3} + 3 \, x \cos\left(y\right) e^{\left(f_{4}\left(x \sin\left(y\right)\right)\right)} \sin\left(y\right)^{2} \mathrm{D}_{0}\left(f_{4}\right)\left(x \sin\left(y\right)\right) \mathrm{D}_{0, 0}\left(f_{4}\right)\left(x \sin\left(y\right)\right) + x \cos\left(y\right) e^{\left(f_{4}\left(x \sin\left(y\right)\right)\right)} \sin\left(y\right)^{2} \mathrm{D}_{0, 0, 0}\left(f_{4}\right)\left(x \sin\left(y\right)\right) + 2 \, \cos\left(y\right) e^{\left(f_{4}\left(x \sin\left(y\right)\right)\right)} \sin\left(y\right) \mathrm{D}_{0}\left(f_{4}\right)\left(x \sin\left(y\right)\right)^{2} + 2 \, \cos\left(y\right) e^{\left(f_{4}\left(x \sin\left(y\right)\right)\right)} \sin\left(y\right) \mathrm{D}_{0, 0}\left(f_{4}\right)\left(x \sin\left(y\right)\right)$

$\newcommand{\Bold}[1]{\mathbf{#1}}{\left({\left(\mathrm{D}_{0}\left(f_{4}\right)\left(x \sin\left(y\right)\right)^{3} + 3 \, \mathrm{D}_{0}\left(f_{4}\right)\left(x \sin\left(y\right)\right) \mathrm{D}_{0, 0}\left(f_{4}\right)\left(x \sin\left(y\right)\right) + \mathrm{D}_{0, 0, 0}\left(f_{4}\right)\left(x \sin\left(y\right)\right)\right)} x \cos\left(y\right) \sin\left(y\right)^{2} + 2 \, {\left(\mathrm{D}_{0}\left(f_{4}\right)\left(x \sin\left(y\right)\right)^{2} + \mathrm{D}_{0, 0}\left(f_{4}\right)\left(x \sin\left(y\right)\right)\right)} \cos\left(y\right) \sin\left(y\right)\right)} e^{\left(f_{4}\left(x \sin\left(y\right)\right)\right)}$

f = sin(x)^2*e^x plot(f, (x, -3,3))

F_e = integrate(f, x) # exact plot(F_e - F_e(x=0), (x, -3, 3)).show() F_e

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{10} \, \cos\left(2 \, x\right) e^{x} - \frac{1}{5} \, e^{x} \sin\left(2 \, x\right) + \frac{1}{2} \, e^{x}$

integral_numerical(lambda x: f(x=x), 0, 3) # numeriek (antwoord en fout)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(8.83665694828, 9.8106600047 \times 10^{-14}\right)$

def F_n(x0): return integral_numerical(lambda x: f(x=x), 0, x0)[0] plot(F_n, (-3,3))

g = e^cos(x^2) g

$\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(\cos\left(x^{2}\right)\right)}$

integrate(g, x, 0, 1)

$\newcommand{\Bold}[1]{\mathbf{#1}}\int_{0}^{1} e^{\left(\cos\left(x^{2}\right)\right)}\,{d x}$

numerical_integral(lambda x: g(x=x), 0, 1)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(2.4890884753, 2.76344333561 \times 10^{-14}\right)$

var("x, y") f = ( 5000-(x*x+y*y+x*y)/200+25*(x+y)/2 )*exp(-(x*x+y*y)/1000000+3*(x+y)/2000-7/10) # PE 262 f

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{200} \, {\left(x^{2} + x y + y^{2} - 2500 \, x - 2500 \, y - 1000000\right)} e^{\left(-\frac{1}{1000000} \, x^{2} - \frac{1}{1000000} \, y^{2} + \frac{3}{2000} \, x + \frac{3}{2000} \, y - \frac{7}{10}\right)}$

plot3d(f, (x, 0, 1600), (y, 0, 1600))

x0 = 0 plot(lambda y: f(x=x0, y=y), (0, 1600))

dfdy = diff(f(x=x0), y) for sol in solve(dfdy==0, y): # exact pretty_print(sol.rhs()) pretty_print(sol.rhs().simplify_full()) pretty_print(sol.rhs().simplify_full().n()) print "\n\n"

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{2} \, {\left(\frac{15625000}{9} i \, \sqrt{185777} \sqrt{3} + \frac{9156250000}{27}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{5468750 i \, \sqrt{3} - 5468750}{9 \, {\left(\frac{15625000}{9} i \, \sqrt{185777} \sqrt{3} + \frac{9156250000}{27}\right)}^{\frac{1}{3}}} + \frac{3250}{3}$

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{250}{9} \, \sqrt{3} {\left(5 \, \sqrt{7} \sqrt{3} \cos\left(\frac{1}{3} \, \arctan\left(\frac{3}{586} \, \sqrt{185777} \sqrt{3}\right)\right) - 15 \, \sqrt{7} \sin\left(\frac{1}{3} \, \arctan\left(\frac{3}{586} \, \sqrt{185777} \sqrt{3}\right)\right) - 13 \, \sqrt{3}\right)}$

$\newcommand{\Bold}[1]{\mathbf{#1}}895.483409898745$

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{2} \, {\left(\frac{15625000}{9} i \, \sqrt{185777} \sqrt{3} + \frac{9156250000}{27}\right)}^{\frac{1}{3}} {\left(-i \, \sqrt{3} + 1\right)} + \frac{-5468750 i \, \sqrt{3} - 5468750}{9 \, {\left(\frac{15625000}{9} i \, \sqrt{185777} \sqrt{3} + \frac{9156250000}{27}\right)}^{\frac{1}{3}}} + \frac{3250}{3}$

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{250}{9} \, \sqrt{3} {\left(5 \, \sqrt{7} \sqrt{3} \cos\left(\frac{1}{3} \, \arctan\left(\frac{3}{586} \, \sqrt{185777} \sqrt{3}\right)\right) + 15 \, \sqrt{7} \sin\left(\frac{1}{3} \, \arctan\left(\frac{3}{586} \, \sqrt{185777} \sqrt{3}\right)\right) - 13 \, \sqrt{3}\right)}$

$\newcommand{\Bold}[1]{\mathbf{#1}}-725.205275699822$

$\newcommand{\Bold}[1]{\mathbf{#1}}{\left(\frac{15625000}{9} i \, \sqrt{185777} \sqrt{3} + \frac{9156250000}{27}\right)}^{\frac{1}{3}} + \frac{10937500}{9 \, {\left(\frac{15625000}{9} i \, \sqrt{185777} \sqrt{3} + \frac{9156250000}{27}\right)}^{\frac{1}{3}}} + \frac{3250}{3}$

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2500}{3} \, \sqrt{7} \cos\left(\frac{1}{3} \, \arctan\left(\frac{3}{586} \, \sqrt{185777} \sqrt{3}\right)\right) + \frac{3250}{3}$

$\newcommand{\Bold}[1]{\mathbf{#1}}3079.72186580108$

find_root(dfdy, 0, 1600) # numeriek

$\newcommand{\Bold}[1]{\mathbf{#1}}895.483409899$

find_local_maximum(lambda y: f(x=x0, y=y).n(), 0, 1600) # numeriek

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(10396.4621932841, 895.48340921\right)$

for sol in solve([diff(f, x)==0, diff(f, y)==0], x, y, solution_dict=True): print "f(%.3f, %.3f) = %.3f" % (sol[x], sol[y], f(x=sol[x], y=sol[y])) # exact

f(2766.593, -1016.593) = -2.891
f(-1016.593, 2766.593) = -2.891
f(-417.059, -417.059) = -806.398
f(777.329, 777.329) = 23474.132
f(2056.397, 2056.397) = -353.683

f(2766.593, -1016.593) = -2.891
f(-1016.593, 2766.593) = -2.891
f(-417.059, -417.059) = -806.398
f(777.329, 777.329) = 23474.132
f(2056.397, 2056.397) = -353.683

minimize(lambda xy: -f(x=xy[0], y=xy[1]), (800,800)) # numeriek

Optimization terminated successfully.
         Current function value: -23474.131917
         Iterations: 47
         Function evaluations: 89

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(777.328847716,\,777.328812135\right)$

Optimization terminated successfully.
         Current function value: -23474.131917
         Iterations: 47
         Function evaluations: 89

minimize(lambda xy: (diff(f, x)^2 + diff(f, y)^2)(x=xy[0], y=xy[1]), (800, 800)) # numeriek

Optimization terminated successfully.
         Current function value: 0.000000
         Iterations: 46
         Function evaluations: 89

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(777.32886444,\,777.328847823\right)$

Optimization terminated successfully.
         Current function value: 0.000000
         Iterations: 46
         Function evaluations: 89

g = function("g")(x) my_ode = x^2 * diff(g, x, x) - x * diff(g, x) == 3*g my_ode

$\newcommand{\Bold}[1]{\mathbf{#1}}x^{2} \frac{\partial^{2}}{(\partial x)^{2}}g\left(x\right) - x \frac{\partial}{\partial x}g\left(x\right) = 3 \, g\left(x\right)$

g_e = desolve(my_ode, g) # exact g_e; plot(g_e, (x, 1, 5))

$\newcommand{\Bold}[1]{\mathbf{#1}}K_{2} x^{3} - \frac{K_{1}}{4 \, x}$

Traceback (click to the left of this block for traceback)
...
ValueError: Variable '_K2' not found

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_15.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("Z19lID0gZGVzb2x2ZShteV9vZGUsIGcpICMgZXhhY3QKZ19lOyBwbG90KGdfZSwgKHgsIDEsIDUpKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/var/sage/tmpXwql3w/___code___.py", line 4, in <module>
    exec compile(u'g_e; plot(g_e, (x, _sage_const_1 , _sage_const_5 ))
  File "", line 1, in <module>
    
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/misc/decorators.py", line 554, in wrapper
    return func(*args, **options)
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/plot/plot.py", line 1921, in plot
    G = funcs.plot(*args, **original_opts)
  File "sage/symbolic/expression.pyx", line 11335, in sage.symbolic.expression.Expression.plot (/opt/sage/sage-7.4/src/build/cythonized/sage/symbolic/expression.cpp:59135)
  File "sage/symbolic/expression.pyx", line 11381, in sage.symbolic.expression.Expression._plot_fast_callable (/opt/sage/sage-7.4/src/build/cythonized/sage/symbolic/expression.cpp:59564)
  File "sage/ext/fast_callable.pyx", line 456, in sage.ext.fast_callable.fast_callable (/opt/sage/sage-7.4/src/build/cythonized/sage/ext/fast_callable.c:4242)
  File "sage/symbolic/expression.pyx", line 11218, in sage.symbolic.expression.Expression._fast_callable_ (/opt/sage/sage-7.4/src/build/cythonized/sage/symbolic/expression.cpp:58356)
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1580, in fast_callable
    return FastCallableConverter(ex, etb)()
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 219, in __call__
    return self.arithmetic(ex, operator)
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1508, in arithmetic
    return reduce(lambda x,y: self.etb.call(operator, x,y), operands)
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1508, in <lambda>
    return reduce(lambda x,y: self.etb.call(operator, x,y), operands)
  File "sage/ext/fast_callable.pyx", line 734, in sage.ext.fast_callable.ExpressionTreeBuilder.call (/opt/sage/sage-7.4/src/build/cythonized/sage/ext/fast_callable.c:6591)
  File "sage/ext/fast_callable.pyx", line 609, in sage.ext.fast_callable.ExpressionTreeBuilder.__call__ (/opt/sage/sage-7.4/src/build/cythonized/sage/ext/fast_callable.c:5685)
  File "sage/symbolic/expression.pyx", line 11218, in sage.symbolic.expression.Expression._fast_callable_ (/opt/sage/sage-7.4/src/build/cythonized/sage/symbolic/expression.cpp:58356)
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1580, in fast_callable
    return FastCallableConverter(ex, etb)()
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 218, in __call__
    return self.arithmetic(div, div.operator())
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1508, in arithmetic
    return reduce(lambda x,y: self.etb.call(operator, x,y), operands)
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1508, in <lambda>
    return reduce(lambda x,y: self.etb.call(operator, x,y), operands)
  File "sage/ext/fast_callable.pyx", line 734, in sage.ext.fast_callable.ExpressionTreeBuilder.call (/opt/sage/sage-7.4/src/build/cythonized/sage/ext/fast_callable.c:6591)
  File "sage/ext/fast_callable.pyx", line 609, in sage.ext.fast_callable.ExpressionTreeBuilder.__call__ (/opt/sage/sage-7.4/src/build/cythonized/sage/ext/fast_callable.c:5685)
  File "sage/symbolic/expression.pyx", line 11218, in sage.symbolic.expression.Expression._fast_callable_ (/opt/sage/sage-7.4/src/build/cythonized/sage/symbolic/expression.cpp:58356)
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1580, in fast_callable
    return FastCallableConverter(ex, etb)()
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 213, in __call__
    return self.symbol(ex)
  File "/opt/sage/sage-7.4/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1529, in symbol
    return self.etb.var(SR(ex))
  File "sage/ext/fast_callable.pyx", line 681, in sage.ext.fast_callable.ExpressionTreeBuilder.var (/opt/sage/sage-7.4/src/build/cythonized/sage/ext/fast_callable.c:6190)
ValueError: Variable '_K2' not found

solver = ode_solver() solver.function = lambda x,y: [y[1], (3*y[0]+x*y[1])/x^2] solver.y_0 = [1, -38/39] solver.ode_solve(t_span=[1,5], num_points=300) solver.plot_solution(interpolate=True)

solver = ode_solver() solver.function = lambda t,y: [y[1], -sin(y[0])] solver.y_0 = [pi-0.001, 0] solver.ode_solve(t_span=[0,100], num_points=300) solver.plot_solution(interpolate=True)

Inleiding (Numerieke) Analyse in Sage

2036 days ago by Toon.Baeyens

Inleiding (Numerieke) Analyse in Sage

Symbolisch rekenen

Functies

Integreren

Extrema's en nulpunten

1 dimensie

Meer dimensies

Gewone differentiaal vergelijkingen (ODE)