222 - Topic 11 - Lagrange Multipliers

%auto var('x,y') # To start things off, enter the function f that you want to use. global func @interact def _(func = [1,2,3,4,5,6]): global f if func==1: f = x^2-y^2 elif func==2: f = x^2/4+10*x+3*y^2/20+12*y elif func==3: f = x^3*y^2 elif func==3: f = x^2-y^2+x*y elif func==4: f = y^3-3*y*x^2-3*y^2-3*x^2+1 elif func==5: f = x+y # This one is interesting with constraint 5 since constraint no continuous # This one also works with the more complicated constraint 6. elif func==6: f = 3*x+2*y+4 #func=8 #f = (x+y)/(x^2+y^2+1) #func=9 #f = x^2+3*x*y+y^2 #func=10 #f = 8*x^2+9*y^2 show(f) 

f 

x^3*y^2

x^3*y^2

%auto # Next, we need a constraint equation of the form g(x,y)= 0. (Note, "C" is inside g.) # Also, the constraint curve needs to be entered as a vector valued function r(t) = <u(t),v(t)> var('t') func=3 if func==1: cons=1 g(x,y) = x^2+y^2-1 u(t)=cos(t) v(t)=sin(t) tstart=0 tend=2*pi elif func==2: cons=2 g(x,y)=x^2/9+y^2/4-900 u(t) = 90*cos(t) v(t) = 60*sin(t) tstart=0 tend=2*pi elif func==3: cons=3 g(x,y)=x+y-100 u(t)=t v(t)=100-t tstart=120 tend=180 elif func==4: cons=4 g(x,y)=x^2+y^2-2 u(t) = sqrt(2)*cos(t) v(t) = sqrt(2)*sin(t) tstart = 0 tend = 2*pi elif func==5: cons=5 g(x,y)=x*y-185 u(t) = t v(t) = 185/t tstart=3 tend=40 elif func==6: cons=6 # A figure 8 g(x,y)=4*x^2-4*x^4-y^2 u(t) = cos(t) v(t) = sin(2*t) tstart=0 tend=2*pi else: print "Need to evaluate function cell first to pick a function to use." print "Using constraint",cons print "Using g = ",g print "Using u = ",u print "Using v = ",v print "Using interval ",tstart,"->",tend 

Using constraint 3
Using g =  (x, y) |--> x + y - 100
Using u =  t |--> t
Using v =  t |--> -t + 100
Using interval  120 -> 180

Using constraint 3
Using g =  (x, y) |--> x + y - 100
Using u =  t |--> t
Using v =  t |--> -t + 100
Using interval  120 -> 180

%auto fx = diff(f,x) fy = diff(f,y) gradf = [fx,fy] gx = diff(g,x) gy = diff(g,y) gradg = [gx,gy] var('lamb') solns_boundary = solve([gx==lamb*fx, gy==lamb*fy, g==0],(x,y,lamb),solution_dict=True) print 'Extrema on the boundary may occur at the lagrange points' show((sol[x],sol[y]) for sol in solns_boundary) for sol in solns_boundary: print "(",sol[x],",",sol[y],")" 

Extrema on the boundary may occur at the lagrange points

( 60 , 40 )

Extrema on the boundary may occur at the lagrange points

( 60 , 40 )

lagrange=[] xs=[] ys=[] for soln in solns_boundary: # prune off all complex roots if soln[x] not in RR: print 'Not using complex critical value at (',soln[x],',',soln[y],')' else: lagrange.append((soln[x],soln[y])) xs.append(soln[x]) ys.append(soln[y]) print print "Using" print "x = ",xs print "y = ",ys print "Lag = ",lagrange for sol in lagrange: print sol[0],sol[1],f(x=sol[0],y=sol[1]) 

Using
x =  [60]
y =  [40]
Lag =  [(60, 40)]
60 40 345600000

Using
x =  [60]
y =  [40]
Lag =  [(60, 40)]
60 40 345600000

# Need to make these dependent upon the ranges of soln_boundary delx = (max(xs)-min(xs))/2 dely = (max(ys)-min(ys))/2 print delx print dely if delx==0: delx = abs(min(xs)/2) if dely==0: dely = abs(min(ys)/2) xmin = min(xs)-delx xmax = max(xs)+delx ymin = min(ys)-dely ymax = max(ys)+dely pretty_print(html('Function = %s'%str(func)+' and Constraint = %s'%str(cons))) F = plot3d(f,(x,xmin,xmax),(y,ymin,ymax),color='yellow') Curve = parametric_plot3d( (u(t),v(t),f(x=u(t),y=v(t))),(t,tstart,tend),color='red') Points = sum(point3d((sol[0],sol[1],f(x=sol[0],y=sol[1])),size=20) for sol in lagrange) # Points += point3d((-20,-40,f(x=-20,y=-40)),size=50) @interact def _(In3D = checkbox(default=True),Spin=checkbox(default=True)): show(F+Curve+Points,spin=Spin) if In3D: print 'You will want your 3D glasses...' show(F+Curve+Points,stereo='redcyan',spin=Spin,title='A plot') 

 

H = contour_plot(f,(x,xmin,xmax),(y,ymin,ymax),contours=50,fill = false) H += parametric_plot( (u(t),v(t)),(t,tstart,tend),color='red') H.show() print 'Numerical Approximations to the exact Lagrange solutions:' for k in range(len(solns_boundary)): x0 = N(solns_boundary[k][x]) y0 = N(solns_boundary[k][y]) show((x0,y0,f(x=x0,y=y0))) 

Numerical Approximations to the exact Lagrange solutions:

Numerical Approximations to the exact Lagrange solutions: