def intersect(subspaces):
if len(subspaces) == 0:
raise ValueError("There must be subspaces to intersect.")
#If parent dimension is different it should be treated
vars = [var('x' + str(i)) for i in range(matrices[0].ncols() - 1)]
eqs = []
for s in subspaces:
eqs.append(equations(s, vars))
return solve(eqs, vars)
def solve_one_var(self, eqs, var):
"""
Solve the equations with respect to the given variable
Returns a list of numerical values.
"""
liste = solve(eqs, var, explicit_solutions=True)
a = []
for soluce in liste:
a.append(numerical_approx(soluce.rhs()))
return a
def solve_ext(list_of_conditions,*args,**kwds):
# takes multiple conditions and gives them to solve.
# Each condition is an input for solve, ususally a conjunct of
# equations. The list of conditions is interpreted as a disjunct of conditions.
solutions = []
for cond in list_of_conditions:
sol=S.solve(cond,*args,**kwds)
# if sol != []:
# print "cons:", cond
# print "sol :", sol
solutions.extend(sol)
return solutions
def fit_inside(self, xmin, xmax, ymin, ymax):
"""
return the largest segment that fits into the given bounds
"""
if self.is_horizontal:
k = self.I.y
return Segment(Point(xmin, k), Point(xmax, k))
if self.is_vertical:
k = self.I.x
return Segment(Point(x, ymin), Point(x, ymax))
x = var("x")
f = self.phyFunction()
x1 = solve([f(x) == ymax], x)[0].rhs()
x2 = solve([f(x) == ymin], x)[0].rhs()
x1 = QQ(x1)
x2 = QQ(x2)
X = [xmin, x1, x2, xmax]
X.sort()
A = Point(X[1], f(X[1]))
B = Point(X[2], f(X[2]))
return Segment(Point(X[1], f(X[1])), Point(X[2], f(X[2])))
def substitute(e1, e2):
"""
Examples:
sage: var('x t q b y a')
(x, t, q, b, y, a)
sage: IeqDeduce.substitute(t-2*b>=0,x==q*y+t)
[-q*y - 2*b + x >= 0]
sage: IeqDeduce.substitute(t-2*b>=0,b-y*a==0)
[-2*a*y + t >= 0]
sage: IeqDeduce.substitute(t-2*b>=0,b-4==0)
[t - 8 >= 0]
sage: IeqDeduce.substitute(t-2*b>=0,b+4==0)
[t + 8 >= 0]
sage: IeqDeduce.substitute(t-2*b>=0,b^2+4==0)
[t + 4*I >= 0, t - 4*I >= 0]
sage: IeqDeduce.substitute(t-2*b>=0,b^2-4==0)
[t + 4 >= 0, t - 4 >= 0]
#todo: cannot do when e2 is not equation
sage: IeqDeduce.substitute(2*b>=0,b>=5)
dig_polynomials:Warn:substitution fails on b >= 5
2*b >= 0
sage: IeqDeduce.substitute(2*b==0,b>=5)
dig_polynomials:Warn:substitution fails on b >= 5
2*b == 0
"""
e1_vs = get_vars(e1)
e2_vs = get_vars(e2)
rs = [
solve(e2, e2_v, solution_dict=True) for e2_v in e2_vs
if e2_v in e1_vs
]
rs = flatten(rs)
try:
rs = [e1.subs(rs_) for rs_ in rs]
return rs
except Exception:
logger.warn('substitution fails on {}'.format(e2))
return e1
def solve_more_vars(self, eqs, *vars):
"""
Solve the equations with respect to the given variables
Returns a list like [ [1,2],[3,4] ] if the solutions are (1,2) and (3,4)
"""
liste = solve(eqs, vars, explicit_solutions=True)
a = []
for soluce in liste:
sol = []
for variable in soluce:
sol.append(numerical_approx(variable.rhs()))
a.append(sol)
return a
def substitute(e1,e2):
"""
Examples:
sage: var('x t q b y a')
(x, t, q, b, y, a)
sage: IeqDeduce.substitute(t-2*b>=0,x==q*y+t)
[-q*y - 2*b + x >= 0]
sage: IeqDeduce.substitute(t-2*b>=0,b-y*a==0)
[-2*a*y + t >= 0]
sage: IeqDeduce.substitute(t-2*b>=0,b-4==0)
[t - 8 >= 0]
sage: IeqDeduce.substitute(t-2*b>=0,b+4==0)
[t + 8 >= 0]
sage: IeqDeduce.substitute(t-2*b>=0,b^2+4==0)
[t + 4*I >= 0, t - 4*I >= 0]
sage: IeqDeduce.substitute(t-2*b>=0,b^2-4==0)
[t + 4 >= 0, t - 4 >= 0]
#todo: cannot do when e2 is not equation
sage: IeqDeduce.substitute(2*b>=0,b>=5)
dig_polynomials:Warn:substitution fails on b >= 5
2*b >= 0
sage: IeqDeduce.substitute(2*b==0,b>=5)
dig_polynomials:Warn:substitution fails on b >= 5
2*b == 0
"""
e1_vs = get_vars(e1)
e2_vs = get_vars(e2)
rs = [solve(e2,e2_v,solution_dict=True)
for e2_v in e2_vs if e2_v in e1_vs]
rs = flatten(rs)
try:
rs = [e1.subs(rs_) for rs_ in rs]
return rs
except Exception:
logger.warn('substitution fails on {}'.format(e2))
return e1
# Solve for the eigenvalues and then manually
# construct the right eigenvectors of A.
eigenvalues = A.eigenvalues()
v1 = var('v1')
v2 = var('v2')
evector_solutions = []
for eval in eigenvalues:
characteristic_product = Matrix([v1, v2
]) * (A - eval * Matrix([[1, 0], [0, 1]]))
evector_solutions.append(
solve([characteristic_product[0, 0], characteristic_product[0, 1]],
[v1, v2],
solution_dict=True))
# We are interested in the eigenvector corresponding to
# eigenvalue 1.
idx = eigenvalues.index(1)
soln = evector_solutions[idx]
assert len(soln) == 1 # sometimes more here?
vec = Matrix([v1, v2])
vec = vec.subs(soln[0])
# There is a free variable in the solution, name begins with 'r'.
free = [v for v in vec[0, 0].variables() if str(v)[0] == 'r']
assert len(free) == 1
free_var = var(free[0])
def solve_single(list_of_conditions,vars,**kwds):
sol = []
for v in vars:
sol += S.solve(list_of_conditions,vars,**kwds)
return sol
def Intersection(f,g,a=None,b=None,numerical=False):
##
# When f and g are objects with an attribute equation, return the list of points of intersections.
#
# - The list of point is sorted by order of `x` coordinates.
# - Return only real solutions.
#
# Only numerical approximations are returned as there are some errors
# otherwise. As an example the following #solving return points that
# are not even near from the circle \f$ x^2+y^2=9 \f$ :
# ```
# solve( [ -1/3*sqrt(3)*y + 1/3*sqrt(3)*(-0.19245008972987399*sqrt(3) - 3) + x == 0,x^2 + y^2 - 9 == 0 ],[x,y] )
# ```
#
# ## Examples
#
# ```
# sage: from phystricks import *
# sage: fun=phyFunction(x**2-5*x+6)
# sage: droite=phyFunction(2)
# sage: pts = Intersection(fun,droite)
# sage: for P in pts:print P
# <Point(1,2)>
# <Point(4,2)>
#```
#
#```
# sage: f=phyFunction(sin(x))
# sage: g=phyFunction(cos(x))
# sage: pts=Intersection(f,g,-2*pi,2*pi,numerical=True)
# sage: for P in pts:print P
# <Point(-5.497787143782138,0.707106781186548)>
# <Point(-2.3561944901923466,-0.707106781186546)>
# <Point(0.7853981633974484,0.707106781186548)>
# <Point(3.926990816987241,-0.707106781186547)>
#```
#
# If 'numerical' is True, it search for the intersection points of the functions 'f' and 'g' (it only work with functions). In this case an interval is required.
from AffineVectorGraph import AffineVectorGraph
if isinstance(f,AffineVectorGraph):
f=f.segment
if isinstance(g,AffineVectorGraph):
g=g.segment
if numerical and "sage" in dir(f) :
import SmallComputations
k=f-g
xx=SmallComputations.find_roots_recursive(k.sage,a,b)
pts=[ Point(x,f(x)) for x in xx ]
return pts
x,y=var('x,y')
pts=[]
if numerical :
soluce=solve([f.equation(numerical=True),g.equation(numerical=True)],[x,y])
else :
soluce=solve([f.equation(),g.equation()],[x,y])
for s in soluce:
a=s[0].rhs()
b=s[1].rhs()
z=a**2+b**2
ok1,a=test_imaginary_part(a)
ok2,b=test_imaginary_part(b)
if ok1 and ok2 :
pts.append(Point(a,b))
return pts
def points(self):
vars = [var('x' + str(i)) for i in range(self.dim)]
x = vector(vars)
return solve(x * self.matrix * x == 0, vars)
def Intersection(f, g, a=None, b=None, numerical=False):
"""
Return the list of intersection point between `f` and `g`.
When f and g are objects with an attribute equation, return
the list of points of intersections.
- The list of point is sorted by order of `x` coordinates.
- Return only real solutions.
Only numerical approximations are returned as there are some errors
otherwise. As an example the following #solving return points that
are not even near from the circle \f$ x^2+y^2=9 \f$ :
``` #pylint:disable=line-too-long
solve( [ -1/3*sqrt(3)*y + 1/3*sqrt(3)*(-0.19245008972987399*sqrt(3) - 3) + x == 0,x^2 + y^2 - 9 == 0 ],[x,y] )
```
Examples
```
sage: from yanntricks import *
sage: fun=phyFunction(x**2-5*x+6)
sage: droite=phyFunction(2)
sage: pts = Intersection(fun,droite)
sage: for P in pts:print P
<Point(1,2)>
<Point(4,2)>
```
```
sage: f=phyFunction(sin(x))
sage: g=phyFunction(cos(x))
sage: pts=Intersection(f,g,-2*pi,2*pi,numerical=True)
sage: for P in pts:print P
<Point(-5.497787143782138,0.707106781186548)>
<Point(-2.3561944901923466,-0.707106781186546)>
<Point(0.7853981633974484,0.707106781186548)>
<Point(3.926990816987241,-0.707106781186547)>
```
If 'numerical' is True, it searchs for the intersection points
of the functions 'f' and 'g' (it only work with functions).
In this case an interval is required.
"""
from yanntricks.src.affine_vector import AffineVector
from yanntricks.src.SmallComputations import find_roots_recursive
from yanntricks.src.point import Point
if isinstance(f, AffineVector):
f = f.segment
if isinstance(g, AffineVector):
g = g.segment
if numerical and "sage" in dir(f):
k = f - g
xx = find_roots_recursive(k.sage, a, b)
pts = [Point(x, f(x)) for x in xx]
return pts
x, y = var('x,y')
pts = []
if numerical:
soluce = solve(
[f.equation(numerical=True),
g.equation(numerical=True)], [x, y])
else:
soluce = solve([f.equation(), g.equation()], [x, y])
for s in soluce:
a = s[0].rhs()
b = s[1].rhs()
ok1, a = test_imaginary_part(a)
ok2, b = test_imaginary_part(b)
if ok1 and ok2:
pts.append(Point(a, b))
return pts
