Type some Sage code below and press Compute (Test examples). Choose a different language from the dropdown menu (say Macaulay2), write some code in the editor, and click the "Compute" button to run it (reload browser if you do not see the editor).
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Computing with SageMath Directly from the Forum
- Eli
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Try to integrate 1/(1+x^4) with Sage or Maxima, i.e., integrate(1/(1 + x^4), x);
The answer should be
\begin{equation}
\begin{split}
& \int\frac{1}{1 + x^4}{\rm d}x \\
& = 1/4\sqrt{2}\times{\rm tan}^{-1}\left(1/2\sqrt{2}\times(2x + \sqrt{2})\right) + 1/4\sqrt{2}\times {\rm tan}^{-1}\left(1/2\times \sqrt{2}\times(2x - \sqrt{2})\right)\\
& + 1/8\sqrt{2}\times{\rm log}(x^2 + \sqrt{2}\times x + 1) - 1/8\sqrt{2}\times{\rm log}(x^2 - \sqrt{2}\times x + 1)
\end{split}
\end{equation}
Try this in Macaulay2:
The answer should be
\begin{equation}
\begin{split}
& \int\frac{1}{1 + x^4}{\rm d}x \\
& = 1/4\sqrt{2}\times{\rm tan}^{-1}\left(1/2\sqrt{2}\times(2x + \sqrt{2})\right) + 1/4\sqrt{2}\times {\rm tan}^{-1}\left(1/2\times \sqrt{2}\times(2x - \sqrt{2})\right)\\
& + 1/8\sqrt{2}\times{\rm log}(x^2 + \sqrt{2}\times x + 1) - 1/8\sqrt{2}\times{\rm log}(x^2 - \sqrt{2}\times x + 1)
\end{split}
\end{equation}
Try this in Macaulay2:
- R = QQ[x,y,z]
- curve = ideal( x^4-y^5, x^3-y^7 )
- gb curve
- dim curve
- degree curve
- curve1 = saturate(curve,ideal(x))
TSSFL -- A Creative Journey Towards Infinite Possibilities!
- Eli
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Test these solutions in Sage (see reference manual here):
The output is:
Solve system of ODEs:
- x,y,z=var('x,y,z')
- # Next we define the parameters
- sigma=10
- rho=28
- beta=8/3
- # The Lorenz equations
- lorenz=[sigma*(y-x),x*(rho-z)-y,x*y-beta*z]
- # Time and initial conditions
- times=srange(0,50.05,0.005)
- ics=[0,1,1]
- sol=desolve_odeint(lorenz,ics,times,[x,y,z],rtol=1e-13,atol=1e-14)
- vis=line(zip(sol[:,0],sol[:,1]))
- vis.show()
The output is:
- #from sage.calculus.desolvers import desolve_odeint -- optional
- x,y=var('x,y')
- f=[x*(1-y),-y*(1-x)]
- sol=desolve_odeint(f,[0.5,2],srange(0,10,0.1),[x,y])
- vis=line(zip(sol[:,0],sol[:,1]))
- print(sol)
- vis.show()
- #y = function('y')(x)
- #eq = diff(y,x) - (x - 2)/(y^2 + 1)*sin(x - 2) - sin(y) == 0
- #desolve_laplace(eq,y,ics=[0,3])
- y1,y2,y3=var('y1,y2,y3')
- f1=77.27*(y2+y1*(1-8.375*1e-6*y1-y2))
- f2=1/77.27*(y3-(1+y1)*y2)
- f3=0.16*(y1-y3)
- f=[f1,f2,f3]
- ci=[0.2,0.4,0.7]
- t=srange(0,10,0.01)
- v=[y1,y2,y3]
- sol=desolve_odeint(f,ci,t,v,rtol=1e-3,atol=1e-4,h0=0.1,hmax=1,hmin=1e-4,mxstep=1000,mxords=17)
- print(sol)
Solve system of ODEs:
- t = var('t')
- x = function('x')(t)
- y = function('y')(t)
- de1 = diff(x,t) + y - 1 == 0
- de2 = diff(y,t) - x + 1 == 0
- sol = desolve_system([de1, de2], [x,y])
- #Add ics if you want
- #sol = desolve_system([de1, de2], [x,y],ics=[0,0.5,1],ivar=t)
- print(sol)
- from sage.calculus.desolvers import desolve_system_rk4
- x,y,t=var('x y t')
- P=desolve_system_rk4([x*(1-y),-y*(1-x)],[x,y],ics=[0,0.5,2],ivar=t,end_points=20)
- Q=[ [i,j] for i,j,k in P]
- LP=list_plot(Q)
- Q=[ [j,k] for i,j,k in P]
- LP=list_plot(Q)
- plot(LP)
- var('t,x,y,X,Y')
- f(t,x,y,X,Y)=[X, Y, -x/(x^2+y^2)^(3/2), -y/(x^2+y^2)^(3/2)]
- ics = [0.8, 0, 0, 1.22474487139159]
- t = 100*pi
- sol = desolve_mintides(f, ics, 0, t, t, 1e-12, 1e-12) # optional -tides
- sol # optional -tides # abs tol 1e-5
- Attachments
-
- Lorentz.png
- (99.22 KiB) Not downloaded yet
- Lorentz.png
- (99.22 KiB) Not downloaded yet
0
TSSFL -- A Creative Journey Towards Infinite Possibilities!
- Eli
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Plotting in Sage (see more from here):
See the attached outputs.
Code: Select all
f(x) = (x-3)*(x-5)*(x-7)+40
P = line([(2,0),(2,f(2))], color='black')
P += line([(8,0),(8,f(8))], color='black')
P += polygon([(2,0),(2,f(2))] + [(x, f(x)) for x in [2,2.1,..,8]] + [(8,0),(2,0)], rgbcolor=(0.8,0.8,0.8),aspect_ratio='automatic')
P += text("$\\int_{a}^b f(x) dx$", (5, 20), fontsize=16, color='black')
P += plot(f, (1, 8.5), thickness=3)
P # show the result
- var("x y")
- G = Graphics()
- counter = 0
- for col in colors.keys(): # Long time
- G += implicit_plot(x^2 + y^2 == 1 + counter*.1, (x,-4,4),(y,-4,4), color=col)
- counter += 1
- G
- var("x y")
- contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), fill=False, cmap='hsv', labels=True)
- var("x y")
- parametric_plot([cos(x) + 2 * cos(x/4), sin(x) - 2 * sin(x/4)], (x,0, 8*pi), fill=True)
- def b(n): return lambda x: bessel_J(n, x)
- plot([b(n) for n in [1..5]], 0, 20, fill='axis')
See the attached outputs.
- Attachments
-
- Contour.png (359.73 KiB) Viewed 44533 times
- Contour.png (359.73 KiB) Viewed 44533 times
-
- Contour2.png (38.63 KiB) Viewed 44533 times
- Contour2.png (38.63 KiB) Viewed 44533 times
-
- Parametric_plot.png (26.95 KiB) Viewed 44530 times
- Parametric_plot.png (26.95 KiB) Viewed 44530 times
-
- Bessel.png
- (84.12 KiB) Not downloaded yet
- Bessel.png
- (84.12 KiB) Not downloaded yet
0
TSSFL -- A Creative Journey Towards Infinite Possibilities!
- Eli
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Try to solve
\begin{equation}
\frac{{\rm d}y}{{\rm d}x} = \frac{x - 2}{y^2 + 1}\times {\rm sin}(x - 2) - {\rm sin}(y).
\end{equation}
\begin{equation}
\frac{{\rm d}y}{{\rm d}x} = \frac{x - 2}{y^2 + 1}\times {\rm sin}(x - 2) - {\rm sin}(y).
\end{equation}
0
TSSFL -- A Creative Journey Towards Infinite Possibilities!
- Eli
- Senior Expert Member
- Reactions: 183
- Posts: 5387
- Joined: 9 years ago
- Location: Tanzania
- Has thanked: 75 times
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- Contact:
We can label and decorate our plots:
- x,y,z=var('x,y,z')
- # Next we define the parameters
- sigma=10
- rho=28
- beta=8/3
- # The Lorenz equations
- lorenz=[sigma*(y-x),x*(rho-z)-y,x*y-beta*z]
- # Time and initial conditions
- times=srange(0,50.05,0.005)
- ics=[0,1,1]
- sol=desolve_odeint(lorenz,ics,times,[x,y,z],rtol=1e-13,atol=1e-14)
- vis=line(zip(sol[:,0],sol[:,1]), color='purple')
- vis.axes_labels(['X axis', 'Y axis'])
- vis.legend(False)
- vis.show(title='Lorenz Equations', frame=True, legend_loc="lower right")
- vis.show()
- Attachments
-
- Lorenz_systems.png
- (120.12 KiB) Not downloaded yet
- Lorenz_systems.png
- (120.12 KiB) Not downloaded yet
-
- Lorenz_system2.png
- (124.85 KiB) Not downloaded yet
- Lorenz_system2.png
- (124.85 KiB) Not downloaded yet
0
TSSFL -- A Creative Journey Towards Infinite Possibilities!
- Eli
- Senior Expert Member
- Reactions: 183
- Posts: 5387
- Joined: 9 years ago
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- Has thanked: 75 times
- Been thanked: 88 times
- Contact:
Try the following Algebra problems in Sage:
Example 1: Transpose of a Matrix
Example 2: LU Factorization of a Matrix
Example 3: Modified Gram-Schmidt Algorithm -- QR decomposition
See more examples here.
Example 1: Transpose of a Matrix
- A = matrix(QQ, 5, range(25))
- A.T
Example 2: LU Factorization of a Matrix
- A = matrix(QQ, [[ 2, 1, 1],
- [ 4,-6, 0],
- [-2, 7, 2]])
- (P,L,U) = A.LU()
- #table([[A, '=', P, '$\\cdot$', L, '$\\cdot$', U]])
- print(P, L, U, sep='\n')
- A = matrix(QQ, [[ 2, 1, 1],
- [ 4,-6, 0],
- [-2, 7, 2]])
- (P,L,U) = A.LU(pivot='nonzero')
- #table([[A, '=', P, '$\\cdot$', L, '$\\cdot$', U]])
- print(P, L, U, sep='\n')
- A = matrix(QQ, [[ 1, -4, 1, 0, -2, 1, 3, 3, 2],
- [-1, 4, 0, -4, 0, -4, 5, -7, -7],
- [ 0, 0, 1, -4, -1, -3, 6, -5, -6],
- [-2, 8, -1, -4, 2, -4, 1, -8, -7],
- [ 1, -4, 2, -4, -3, 2, 5, 6, 4]])
- P, L, U = A.LU()
- U
- A.rref()
- A.pivots()
- D = matrix(QQ, [[ 1, 0, 2, 0, -2, -1],
- [ 3, -2, 3, -1, 0, 6],
- [-4, 2, -3, 1, -1, -8],
- [-2, 2, -3, 2, 1, 0],
- [ 0, -1, -1, 0, 2, 5],
- [-1, 2, -4, -1, 5, -3]])
- P, L, U = D.LU(pivot='nonzero')
- P
- L
- U
- D == L*U
Example 3: Modified Gram-Schmidt Algorithm -- QR decomposition
- A = matrix(QQbar, [[-2, 0, -4, -1, -1],
- [-2, 1, -6, -3, -1],
- [1, 1, 7, 4, 5],
- [3, 0, 8, 3, 3],
- [-1, 1, -6, -6, 5]])
- Q, R = A.QR()
- Q
- R
- Q.conjugate_transpose()*Q
- Q*R == A
See more examples here.
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TSSFL -- A Creative Journey Towards Infinite Possibilities!
- Eli
- Senior Expert Member
- Reactions: 183
- Posts: 5387
- Joined: 9 years ago
- Location: Tanzania
- Has thanked: 75 times
- Been thanked: 88 times
- Contact:
Knots Using Sage:
- B = BraidGroup(4)
- K = Knot(B([1,1,1,2,-1,2,-3,2,-3]))
- plot(K)
- K.alexander_polynomial()
- K.jones_polynomial()
- K.determinant()
- K.signature()
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TSSFL -- A Creative Journey Towards Infinite Possibilities!
- Eli
- Senior Expert Member
- Reactions: 183
- Posts: 5387
- Joined: 9 years ago
- Location: Tanzania
- Has thanked: 75 times
- Been thanked: 88 times
- Contact:
Interactive Fourier Series with Sage (see source):
- ## Interactive Fourier Series, via: http://www.walkingrandomly.com/?p=1879
- def ftermSquare(n):
- return(1/n*sin(n*x*pi/3))
- def ftermSawtooth(n):
- return(1/n*sin(n*x*pi/3))
- def ftermParabola(n):
- return((-1)^n/n^2 * cos(n*x))
- def fseriesSquare(n):
- return(4/pi*sum(ftermSquare(i) for i in range (1,2*n,2)))
- def fseriesSawtooth(n):
- return(1/2-1/pi*sum(ftermSawtooth(i) for i in range (1,n)))
- def fseriesParabola(n):
- return(pi^2/3 + 4*sum(ftermParabola(i) for i in range(1,n)))
- @interact
- def plotFourier(n=slider(1, 30,1,3,'Number of terms')
- ,Function=['Square Wave','Saw Tooth','Periodic Parabola']):
- if Function=='Saw Tooth':
- show(plot(fseriesSawtooth(n),x,-6,6,figsize=(7,3)))
- if Function=='Square Wave':
- show(plot(fseriesSquare(n),x,-6,6,figsize=(7,3)))
- if Function=='Periodic Parabola':
- show(plot(fseriesParabola(n),x,-6,6,figsize=(7,3)))
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