def direct_sum_decompose(U_basis, V_basis, w):
'''
input: A list of Vecs, U_basis, containing a basis for a vector space, U.
A list of Vecs, V_basis, containing a basis for a vector space, V.
A Vec, w, that belongs to the direct sum of these spaces.
output: A pair, (u, v), such that u+v=w and u is an element of U and
v is an element of V.
>>> U_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> direct_sum_decompose(U_basis, V_basis, w) == (Vec({0, 1, 2, 3, 4, 5},{0: 2.0, 1: 4.999999999999972, 2: 0.0, 3: 0.0, 4: 1.0, 5: 0.0}), Vec({0, 1, 2, 3, 4, 5},{0: 0.0, 1: 0.0, 2: 0.0, 3: 0.0, 4: 0.0, 5: 0.0}))
True
'''
rep = vec2rep(w, U_basis + V_basis)
rep_u = Vec(set(range(len(U_basis))),
{k: v
for k, v in rep.f.items() if k < len(U_basis)})
rep_v = Vec(set(range(len(V_basis))),
{(k - len(U_basis)): v
for k, v in rep.f.items() if k - len(U_basis) >= 0})
u = coldict2mat(U_basis) * rep_u
v = coldict2mat(V_basis) * rep_v
return (u, v)
def direct_sum_decompose(U_basis, V_basis, w):
'''
input: A list of Vecs, U_basis, containing a basis for a vector space, U.
A list of Vecs, V_basis, containing a basis for a vector space, V.
A Vec, w, that belongs to the direct sum of these spaces.
output: A pair, (u, v), such that u+v=w and u is an element of U and
v is an element of V.
>>> U_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> direct_sum_decompose(U_basis, V_basis, w) == (Vec({0, 1, 2, 3, 4, 5},{0: 2.0, 1: 4.999999999999972, 2: 0.0, 3: 0.0, 4: 1.0, 5: 0.0}), Vec({0, 1, 2, 3, 4, 5},{0: 0.0, 1: 0.0, 2: 0.0, 3: 0.0, 4: 0.0, 5: 0.0}))
True
'''
from hw4 import vec2rep
U = coldict2mat(U_basis)
V = coldict2mat(V_basis)
sum = U_basis + V_basis
sol_w = vec2rep(sum,w)
lenU = len(U_basis)
wu = list2vec ([ v for i, v in sol_w.f.items () if i < lenU ])
wv = list2vec ([ v for i, v in sol_w.f.items () if i >= lenU ])
u = U*wu
v = V*wv
return (u,v)
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
if len(L) == 1:
return False
tList = copy.deepcopy(L)
v_i = tList.pop(i)
x = solve(coldict2mat(tList),v_i)
res = v_i - coldict2mat(tList) * x
if res * res < pow(10,-14) :
return True
return False
pass
def aug_orthonormalize(L):
'''
Input:
- L: a list of Vecs
Output:
- A pair Qlist, Rlist such that:
* coldict2mat(L) == coldict2mat(Qlist) * coldict2mat(Rlist)
* Qlist = orthonormalize(L)
>>> from vec import Vec
>>> D={'a','b','c','d'}
>>> L = [Vec(D, {'a':4,'b':3,'c':1,'d':2}), Vec(D, {'a':8,'b':9,'c':-5,'d':-5}), Vec(D, {'a':10,'b':1,'c':-1,'d':5})]
>>> Qlist, Rlist = aug_orthonormalize(L)
'''
Ql, Rl = aug_orthogonalize(L)
l=len(Ql)
Qlist=[]
Am=[]
for v in Ql:
A=sqrt(sum([v[i]**2 for i in range(len(v.D))]))
Am.append(A)
Qlist.append(1/A*v)
D=set(range(l))
AmV=[Vec(D,{i:Am[i]}) for i in range(l)]
AmM=coldict2mat(AmV)
Rlist=AmM*coldict2mat(Rl)
return Qlist, Rlist
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> D={'a','b','c','d'}
>>> L = [Vec(D, {'a':1,'b':-1}), Vec(D, {'c':-1,'b':1}), Vec(D, {'c':1,'d':-1}), Vec(D, {'a':-1,'d':1}), Vec(D, {'b':1, 'c':1, 'd':-1})]
>>> is_superfluous(L,4)
False
>>> is_superfluous(L,3)
True
>>> is_superfluous(L,2)
True
>>> L == [Vec(D,{'a':1,'b':-1}),Vec(D,{'c':-1,'b':1}),Vec(D,{'c':1,'d':-1}),Vec(D, {'a':-1,'d':1}),Vec(D,{'b':1, 'c':1, 'd':-1})]
True
>>> is_superfluous([Vec({0,1}, {})], 0)
True
>>> is_superfluous([Vec({0,1}, {0:1})], 0)
False
'''
if len(L) == 1:
return L[0] == -L[0]
L = L.copy()
b = L.pop(i)
r = b - coldict2mat(L) * solve(coldict2mat(L), b)
return r * r < 10e-14
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
b = L[i]
if i == 0:
A = coldict2mat(L[i+1:])
elif i == len(L)-1:
A = coldict2mat(L[0:i])
else:
A = coldict2mat(L[0:i-1]+L[i+1:])
u = solve(A, b)
r = b - A * u
return True if r*r < 1.0e-14 else False
def direct_sum_decompose(U_basis, V_basis, w):
'''
input: A list of Vecs, U_basis, containing a basis for a vector space, U.
A list of Vecs, V_basis, containing a basis for a vector space, V.
A Vec, w, that belongs to the direct sum of these spaces.
output: A pair, (u, v), such that u+v=w and u is an element of U and
v is an element of V.
>>> U_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> direct_sum_decompose(U_basis, V_basis, w) == (Vec({0, 1, 2, 3, 4, 5},{0: 2.0, 1: 4.999999999999972, 2: 0.0, 3: 0.0, 4: 1.0, 5: 0.0}), Vec({0, 1, 2, 3, 4, 5},{0: 0.0, 1: 0.0, 2: 0.0, 3: 0.0, 4: 0.0, 5: 0.0}))
True
'''
UV = coldict2mat(U_basis + V_basis)
W = solve(UV, w)
U = coldict2mat(U_basis)
V = coldict2mat(V_basis)
Wu = Vec(set(range(len(U_basis))), {x: W[x] for x in range(len(U_basis))})
Wv = Vec(set(range(len(V_basis))),
{x: W[len(U_basis) + x]
for x in range(len(V_basis))})
u = U * Wu
v = V * Wv
return (u, v)
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
L_aux = list(L)
vector = L_aux.pop(i)
sol = solve(coldict2mat(L_aux), vector)
residual = vector - coldict2mat(L_aux) * sol
if residual * residual < 10**-14: return True
else: return False
def direct_sum_decompose(U_basis, V_basis, w):
'''
input: A list of Vecs, U_basis, containing a basis for a vector space, U.
A list of Vecs, V_basis, containing a basis for a vector space, V.
A Vec, w, that belongs to the direct sum of these spaces.
output: A pair, (u, v), such that u+v=w and u is an element of U and
v is an element of V.
>>> U_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> direct_sum_decompose(U_basis, V_basis, w) == (Vec({0, 1, 2, 3, 4, 5},{0: 2.0, 1: 4.999999999999972, 2: 0.0, 3: 0.0, 4: 1.0, 5: 0.0}), Vec({0, 1, 2, 3, 4, 5},{0: 0.0, 1: 0.0, 2: 0.0, 3: 0.0, 4: 0.0, 5: 0.0}))
True
'''
uv_basis = U_basis + V_basis #basis of space U+V
x = vec2rep(uv_basis,w) #x is the coordination representation in space U+V : uv_basis*x = w
#then split the coordinations : the first len(U_basis) are coordinates in U, the rest are coordinates in V
# coordinates in U
uc = list2vec([x[i] for i in range(len(U_basis))])
# coordinates in V
uv = list2vec([x[i] for i in range(len(U_basis),len(uv_basis))])
u = coldict2mat(U_basis)*uc
v = coldict2mat(V_basis)*uv
return (u,v)
def direct_sum_decompose(U_basis, V_basis, w):
'''
input: A list of Vecs, U_basis, containing a basis for a vector space, U.
A list of Vecs, V_basis, containing a basis for a vector space, V.
A Vec, w, that belongs to the direct sum of these spaces.
output: A pair, (u, v), such that u+v=w and u is an element of U and
v is an element of V.
>>> U_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> direct_sum_decompose(U_basis, V_basis, w) == (Vec({0, 1, 2, 3, 4, 5},{0: 2.0, 1: 4.999999999999972, 2: 0.0, 3: 0.0, 4: 1.0, 5: 0.0}), Vec({0, 1, 2, 3, 4, 5},{0: 0.0, 1: 0.0, 2: 0.0, 3: 0.0, 4: 0.0, 5: 0.0}))
True
'''
from hw4 import vec2rep
U = coldict2mat(U_basis)
V = coldict2mat(V_basis)
sum = U_basis + V_basis
sol_w = vec2rep(sum, w)
lenU = len(U_basis)
wu = list2vec([v for i, v in sol_w.f.items() if i < lenU])
wv = list2vec([v for i, v in sol_w.f.items() if i >= lenU])
u = U * wu
v = V * wv
return (u, v)
def QR_solve(A):
col_labels = sorted(A.D[1], key=repr)
Acolms = dict2list(mat2coldict(A), col_labels)
(Qlist, Rlist) = aug_orthogonalize(Acolms)
Q = coldict2mat(list2dict(Qlist, col_labels))
R = coldict2mat(list2dict(Rlist, col_labels))
return Q, R
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([ a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
b = L.pop(i)
u = solve(coldict2mat(L),b)
residual = b - coldict2mat(L)*u
if residual * residual < 10e-14:
return True
else:
return False
def direct_sum_decompose(U_basis, V_basis, w):
'''
input: A list of Vecs, U_basis, containing a basis for a vector space, U.
A list of Vecs, V_basis, containing a basis for a vector space, V.
A Vec, w, that belongs to the direct sum of these spaces.
output: A pair, (u, v), such that u+v=w and u is an element of U and
v is an element of V.
>>> U_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> direct_sum_decompose(U_basis, V_basis, w) == (Vec({0, 1, 2, 3, 4, 5},{0: 2.0, 1: 4.999999999999972, 2: 0.0, 3: 0.0, 4: 1.0, 5: 0.0}), Vec({0, 1, 2, 3, 4, 5},{0: 0.0, 1: 0.0, 2: 0.0, 3: 0.0, 4: 0.0, 5: 0.0}))
True
'''
UV = coldict2mat(U_basis + V_basis)
W = solve(UV, w)
U = coldict2mat(U_basis)
V = coldict2mat(V_basis)
Wu = Vec(set(range(len(U_basis))),{x: W[x] for x in range(len(U_basis))})
Wv = Vec(set(range(len(V_basis))),{x: W[len(U_basis) + x] for x in range(len(V_basis))})
u = U * Wu
v = V * Wv
return (u,v)
def QR_factor(A):
col_labels = sorted(A.D[1], key=repr)
Acols = dict2list(mat2coldict(A),col_labels)
Qlist, Rlist = aug_orthonormalize(Acols)
#Now make Mats
Q = coldict2mat(Qlist)
R = coldict2mat(list2dict(Rlist, col_labels))
return Q,R
def QR_factor(A):
col_labels = sorted(A.D[1], key=repr)
Acols = dict2list(mat2coldict(A), col_labels)
Qlist, Rlist = aug_orthonormalize(Acols)
#Now make Mats
Q = coldict2mat(Qlist)
R = coldict2mat(list2dict(Rlist, col_labels))
return Q, R
def direct_sum_decompose(U, V, w):
U_copy = U.copy()
U_copy.extend(V)
linear_comb = solve(coldict2mat(U_copy), w)
u_linear_comb = list2vec([linear_comb[i] for i in range(len(U))])
v_linear_comb = list2vec(
[linear_comb[i] for i in range(len(U),
len(V) + len(U))])
return (coldict2mat(U) * u_linear_comb, coldict2mat(V) * v_linear_comb)
def direct_sum_decompose(U_basis, V_basis, w):
UV = coldict2mat(U_basis+V_basis)
U = coldict2mat(U_basis)
V = coldict2mat(V_basis)
W = solve(UV,w)
Wu = list2vec([v for i, v in W.f.items() if i < len(U_basis)])
Wv = list2vec([v for i, v in W.f.items() if i >= len(U_basis)])
u = U * Wu
v = V * Wv
return (u,v)
def direct_sum_decompose(U_basis, V_basis, w):
'''
Input:
- U_basis: a list of Vecs forming a basis for a vector space U
- V_basis: a list of Vecs forming a basis for a vector space V
- w: a Vec in the direct sum of U and V
Output:
- a pair (u, v) such that u + v = w, u is in U, v is in V
Example:
>>> D = {0,1,2,3,4,5}
>>> U_basis = [Vec(D,{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec(D,{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec(D,{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec(D,{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec(D,{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec(D,{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, w)
>>> (u + v - w).is_almost_zero()
True
>>> U_matrix = coldict2mat(U_basis)
>>> V_matrix = coldict2mat(V_basis)
>>> (u - U_matrix*solve(U_matrix, u)).is_almost_zero()
True
>>> (v - V_matrix*solve(V_matrix, v)).is_almost_zero()
True
>>> ww = Vec(D,{0: 2, 1: 5, 2: 51, 4: 1, 5: 7})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, ww)
>>> (u + v - ww).is_almost_zero()
True
>>> (u - U_matrix*solve(U_matrix, u)).is_almost_zero()
True
>>> (v - V_matrix*solve(V_matrix, v)).is_almost_zero()
True
>>> U_basis == [Vec(D,{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec(D,{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec(D,{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
True
>>> V_basis == [Vec(D,{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec(D,{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
True
>>> w == Vec(D,{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
True
'''
union_basis = U_basis.copy()
union_basis.extend(V_basis)
union_matrix = coldict2mat(union_basis)
U_matrix = coldict2mat(U_basis)
V_matrix = coldict2mat(V_basis)
# calculate the w representation in u and v terms
w_rep = solve(union_matrix, w)
# extracting the u an v representation on U_basis and V_basis respectively
u_rep = Vec(U_matrix.D[1], {idx:w_rep[idx] for idx in range(len(U_basis))})
v_rep = Vec(V_matrix.D[1], {idx:w_rep[idx + len(U_basis)] for idx in range(len(V_basis))})
# calculate u and v vectors
u = U_matrix * u_rep
v = V_matrix * v_rep
return (u, v)
def direct_sum_decompose(U_basis, V_basis, w):
'''
Input:
- U_basis: a list of Vecs forming a basis for a vector space U
- V_basis: a list of Vecs forming a basis for a vector space V
- w: a Vec in the direct sum of U and V
Output:
- a pair (u, v) such that u + v = w, u is in U, v is in V
Example:
>>> D = {0,1,2,3,4,5}
>>> U_basis = [Vec(D,{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec(D,{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec(D,{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec(D,{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec(D,{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec(D,{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, w)
>>> (u + v - w).is_almost_zero()
True
>>> U_matrix = coldict2mat(U_basis)
>>> V_matrix = coldict2mat(V_basis)
>>> (u - U_matrix*solve(U_matrix, u)).is_almost_zero()
True
>>> (v - V_matrix*solve(V_matrix, v)).is_almost_zero()
True
>>> ww = Vec(D,{0: 2, 1: 5, 2: 51, 4: 1, 5: 7})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, ww)
>>> (u + v - ww).is_almost_zero()
True
>>> (u - U_matrix*solve(U_matrix, u)).is_almost_zero()
True
>>> (v - V_matrix*solve(V_matrix, v)).is_almost_zero()
True
>>> U_basis == [Vec(D,{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec(D,{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec(D,{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
True
>>> V_basis == [Vec(D,{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec(D,{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
True
>>> w == Vec(D,{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
True
>>> ww = Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 5, 2: 51, 4: 1, 5: 7})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, ww)
>>> (u + v - ww).is_almost_zero()
True
'''
uvmat = coldict2mat(U_basis+V_basis)
uv = solve(uvmat,w)
U_matrix = coldict2mat(U_basis)
V_matrix = coldict2mat(V_basis)
u = U_matrix*Vec(U_matrix.D[1], {i:uv[i] for i in range(len(U_matrix.D[1]))})
v = V_matrix*Vec(V_matrix.D[1], {i:uv[i+len(U_matrix.D[1])] for i in range(len(V_matrix.D[1]))})
return (u,v)
def direct_sum_decompose(U,V,w): basis_vectors_W = get_basis(U) UV = coldict2mat(basis_vectors_W) W = solve(UV, w) print(W) basis_vectors_W = get_basis(V) UV = coldict2mat(basis_vectors_W) W = solve(UV, w) print(W) basis_vectors_W = get_basis(U+V) UV = coldict2mat(basis_vectors_W) W = solve(UV, w) print(W)
def aug_orthonormalize(L):
'''
Input:
- L: a list of Vecs
Output:
- A pair Qlist, Rlist such that:
* coldict2mat(L) == coldict2mat(Qlist) * coldict2mat(Rlist)
* Qlist = orthonormalize(L)
'''
Qlist = orthonormalize(L)
Rlist = mat2coldict(transpose(coldict2mat(Qlist)) * coldict2mat(L))
newRlist = [Rlist[k] for k in Rlist]
return Qlist,newRlist
def aug_orthonormalize(L):
'''
Input:
- L: a list of Vecs
Output:
- A pair Qlist, Rlist such that:
* coldict2mat(L) == coldict2mat(Qlist) * coldict2mat(Rlist)
* Qlist = orthonormalize(L)
'''
Qlist = orthonormalize(L)
Rlist = mat2coldict(transpose(coldict2mat(Qlist)) * coldict2mat(L))
newRlist = [Rlist[k] for k in Rlist]
return Qlist, newRlist
def aug_orthonormalize(L):
'''
Input:
- L: a list of Vecs
Output:
- A pair Qlist, Rlist such that:
* coldict2mat(L) == coldict2mat(Qlist) * coldict2mat(Rlist)
* Qlist = orthonormalize(L)
'''
V, S = aug_orthogonalize(L)
Q = orthonormalize(L)
R = mat2coldict(transpose(coldict2mat(V)) * coldict2mat(Q) * coldict2mat(S))
return (Q,[x for x in R.values()])
def aug_orthonormalize(L):
'''
Input:
- L: a list of Vecs
Output:
- A pair Qlist, Rlist such that:
* coldict2mat(L) == coldict2mat(Qlist) * coldict2mat(Rlist)
* Qlist = orthonormalize(L)
'''
V, S = aug_orthogonalize(L)
Q = orthonormalize(L)
R = mat2coldict(
transpose(coldict2mat(V)) * coldict2mat(Q) * coldict2mat(S))
return (Q, [x for x in R.values()])
def aug_orthonormalize(L):
"""
Input:
- L: a list of Vecs
Output:
- A pair Qlist, Rlist such that:
* coldict2mat(L) == coldict2mat(Qlist) * coldict2mat(Rlist)
* Qlist = orthonormalize(L)
"""
from orthogonalization import aug_orthogonalize
from matutil import mat2coldict, coldict2mat
Qlist = orthonormalize(L)
Rlist = mat2coldict(coldict2mat(Qlist).transpose() * coldict2mat(L))
return Qlist, list(Rlist.values())
def direct_sum_decompose(U_basis, V_basis, w):
'''
Input:
- U_basis: a list of Vecs forming a basis for a vector space U
- V_basis: a list of Vecs forming a basis for a vector space V
- w: a Vec in the direct sum of U and V
Output:
- a pair (u, v) such that u + v = w, u is in U, v is in V
Example:
>>> D = {0,1,2,3,4,5}
>>> U_basis = [Vec(D,{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec(D,{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec(D,{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec(D,{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec(D,{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec(D,{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, w)
>>> (u + v - w).is_almost_zero()
True
>>> U_matrix = coldict2mat(U_basis)
>>> V_matrix = coldict2mat(V_basis)
>>> (u - U_matrix*solve(U_matrix, u)).is_almost_zero()
True
>>> (v - V_matrix*solve(V_matrix, v)).is_almost_zero()
True
>>> ww = Vec(D,{0: 2, 1: 5, 2: 51, 4: 1, 5: 7})
>>> (u, v) = direct_sum_decompose(U_basis, V_basis, ww)
>>> (u + v - ww).is_almost_zero()
True
>>> (u - U_matrix*solve(U_matrix, u)).is_almost_zero()
True
>>> (v - V_matrix*solve(V_matrix, v)).is_almost_zero()
True
>>> U_basis == [Vec(D,{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec(D,{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec(D,{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
True
>>> V_basis == [Vec(D,{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec(D,{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
True
>>> w == Vec(D,{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
True
'''
M = coldict2mat(U_basis+V_basis)
solution = solve(M, w)
u, v = [], []
U_matrix, V_matrix = coldict2mat(U_basis), coldict2mat(V_basis)
for i in range(len(U_basis) + len(V_basis)):
if i < len(U_basis):
u.append(solution.f[i])
else:
v.append(solution.f[i])
return U_matrix*list2vec(u), V_matrix*list2vec(v)
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
coldict_i = {k: L[k] for k in range(len(L)) if k != i}
mat_i = coldict2mat(coldict_i)
v = solve(mat_i, L[i])
res = mat_i * v - L[i]
res_2 = res * res
##print(res_2)
return res_2 < 1e-14
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
coldict_i = {k:L[k] for k in range(len(L)) if k!=i }
mat_i = coldict2mat(coldict_i)
v = solve(mat_i, L[i])
res = mat_i * v - L[i]
res_2 = res*res
##print(res_2)
return res_2 < 1e-14
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
if len(L) == 1:
return False
copyL = L[:]
b = copyL[i]
del copyL[i]
matrix = coldict2mat(copyL)
u = solve(matrix, b)
residual = b - matrix * u
return (residual * residual) < 10e-14
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
assert i in range(len(L))
A = coldict2mat({c:L[c] for c in range(len(L)) if c!=i})
residual = L[i] - A*solve(A, L[i])
if residual * residual < 1e-14:
return True
return False
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
assert(len(L) > 0)
assert(i < len(L))
if len(L) == 1:
return True
mtx = coldict2mat([ L[j] for j in range(len(L)) if j != i ])
u = solve(mtx,L[i])
residual = mtx * u - L[i]
return residual * residual < 10e-14
def direct_sum_decompose(U_basis, V_basis, w):
'''
input: A list of Vecs, U_basis, containing a basis for a vector space, U.
A list of Vecs, V_basis, containing a basis for a vector space, V.
A Vec, w, that belongs to the direct sum of these spaces.
output: A pair, (u, v), such that u+v=w and u is an element of U and
v is an element of V.
>>> U_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> direct_sum_decompose(U_basis, V_basis, w) == (Vec({0, 1, 2, 3, 4, 5},{0: 2.0, 1: 4.999999999999972, 2: 0.0, 3: 0.0, 4: 1.0, 5: 0.0}), Vec({0, 1, 2, 3, 4, 5},{0: 0.0, 1: 0.0, 2: 0.0, 3: 0.0, 4: 0.0, 5: 0.0}))
True
'''
combined = U_basis + V_basis
separation_boundary = len(U_basis)
soln = solve(coldict2mat(combined), w)
u = Vec(w.D, {})
v = Vec(w.D, {})
for key, val in soln.f.items():
if key < separation_boundary:
u += val * U_basis[key]
else:
v += val * combined[key]
return (u, v)
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> D={'a','b','c','d'}
>>> L = [Vec(D, {'a':1,'b':-1}), Vec(D, {'c':-1,'b':1}), Vec(D, {'c':1,'d':-1}), Vec(D, {'a':-1,'d':1}), Vec(D, {'b':1, 'c':1, 'd':-1})]
>>> is_superfluous(L,4)
False
>>> is_superfluous(L,3)
True
>>> is_superfluous(L,2)
True
>>> L == [Vec(D,{'a':1,'b':-1}),Vec(D,{'c':-1,'b':1}),Vec(D,{'c':1,'d':-1}),Vec(D, {'a':-1,'d':1}),Vec(D,{'b':1, 'c':1, 'd':-1})]
True
>>> is_superfluous([Vec({0,1}, {})], 0)
True
>>> is_superfluous([Vec({0,1}, {0:1})], 0)
False
'''
assert i <= len(L)
if len(L) == 1: return False or len(L[i].f) == 0
A = coldict2mat([L[j] for j in range(len(L)) if j != i])
u = solve(A, L[i])
res = L[i] - A * u
return res.is_almost_zero()
def direct_sum_decompose(U_basis, V_basis, w):
'''
input: A list of Vecs, U_basis, containing a basis for a vector space, U.
A list of Vecs, V_basis, containing a basis for a vector space, V.
A Vec, w, that belongs to the direct sum of these spaces.
output: A pair, (u, v), such that u+v=w and u is an element of U and
v is an element of V.
>>> U_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> direct_sum_decompose(U_basis, V_basis, w) == (Vec({0, 1, 2, 3, 4, 5},{0: 2.0, 1: 4.999999999999972, 2: 0.0, 3: 0.0, 4: 1.0, 5: 0.0}), Vec({0, 1, 2, 3, 4, 5},{0: 0.0, 1: 0.0, 2: 0.0, 3: 0.0, 4: 0.0, 5: 0.0}))
True
'''
combined = U_basis + V_basis
separation_boundary = len(U_basis)
soln = solve(coldict2mat(combined), w)
u = Vec(w.D, {})
v = Vec(w.D, {})
for key, val in soln.f.items():
if key < separation_boundary:
u += val * U_basis[key]
else:
v += val * combined[key]
return (u, v)
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
if len(L) > 1:
A = coldict2mat(L[:i] + L[i + 1:])
v = solve(A, L[i])
r = L[i] - A * v
if r * r < 10 ** -14:
return True
return False
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
if len(L)<=1:
return False
A = coldict2mat([L[index] for index in range(len(L)) if index != i])
b = solve(A,L[i])
residue = L[i] - A*b
return residue*residue < 1e-14
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
if len(L) == 1:
return False
Li = L.pop(i)
mL = coldict2mat(L)
sol = solve(mL,Li)
res = Li - mL * sol
if abs(res*res) < 10**-14:
return True
else:
return False
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
if len(L) <= 1:
return False
# remove L[i] without changing the list (i.e. w/o using L.pop())
b = L[i] # RHS of A*u = b
A = coldict2mat(L[:i] + L[i+1:]) # coefficient matrix L w/o L[i]
u = solve(A, b)
e = b - A*u
return e*e < 1e-14
def Mv_mat_mat_mult(A, B):
assert A.D[1] == B.D[0]
# print (str(A.D) + str(A.f))
# print (str(B.D) + str(B.f))
# print('\n')
colsB = utils.mat2coldict(B)
return utils.coldict2mat({k:A*colsB[k] for k in colsB})
def direct_sum_decompose(U_basis, V_basis, w):
'''
input: A list of Vecs, U_basis, containing a basis for a vector space, U.
A list of Vecs, V_basis, containing a basis for a vector space, V.
A Vec, w, that belongs to the direct sum of these spaces.
output: A pair, (u, v), such that u+v=w and u is an element of U and
v is an element of V.
>>> U_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> direct_sum_decompose(U_basis, V_basis, w) == (Vec({0, 1, 2, 3, 4, 5},{0: 2.0, 1: 4.999999999999972, 2: 0.0, 3: 0.0, 4: 1.0, 5: 0.0}), Vec({0, 1, 2, 3, 4, 5},{0: 0.0, 1: 0.0, 2: 0.0, 3: 0.0, 4: 0.0, 5: 0.0}))
True
'''
testMat = coldict2mat(U_basis + V_basis)
coef = solve(testMat, w)
Nu = len(U_basis)
Nv = len(V_basis)
retU = Vec(U_basis[0].D, {})
retV = Vec(V_basis[0].D, {})
for i in range(Nu):
retU += coef[i] * U_basis[i]
for i in range(Nv):
retV += coef[i + Nu] * V_basis[i]
return (retU, retV)
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
if len(L) == 1:
return False
elif i >= len(L):
return False
else:
u = L[i]
z = Vec(L[0].D, {})
A = coldict2mat(L[:i] + [z] + L[i + 1:])
b = solve(A, u)
r = u - A * b
return r * r < 1e-14
def Mv_mat_mat_mult(A, B):
assert A.D[1] == B.D[0]
col_dict = mat2coldict(B)
res = dict()
for c in col_dict.keys():
res[c] = A * col_dict[c]
return coldict2mat(res)
def find_triangular_matrix_inverse(A):
'''
Supporting GF2 is not required.
Input:
- A: an upper triangular Mat with nonzero diagonal elements
Output:
- Mat that is the inverse of A
Example:
>>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
>>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
True
'''
I = {i: Vec(A.D[0], {i: 1}) for i in A.D[1]}
rl = list()
ll = list()
cd = dict()
rd = mat2rowdict(A)
for k, i in rd.items():
ll.append(k)
rl.append(rd[k])
for k in ll:
cd[k] = triangular_solve(rl, ll, I[k])
return coldict2mat(cd)
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
if len(L) == 1:
return False
copyL = L[:]
b = copyL[i]
del copyL[i]
matrix = coldict2mat(copyL)
u = solve(matrix, b)
residual = b - matrix*u
return (residual * residual) < 10e-14
def is_superfluous(L, i):
"""
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
"""
if len(L) > 1:
l = L.copy()
b = l.pop(i)
M = coldict2mat(l)
solution = solve(M, b)
residual = M * solution - b
if residual * residual < 10e-14:
return True
return False
def mat_move2board(Y):
'''
Input:
- Y: a Mat each column of which is a {'y1', 'y2', 'y3'}-Vec
giving the whiteboard coordinates of a point q.
Output:
- a Mat each column of which is the corresponding point in the
whiteboard plane (the point of intersection with the whiteboard plane
of the line through the origin and q).
Example:
>>> Y_in = Mat(({'y1', 'y2', 'y3'}, {0,1,2,3}),
{('y1',0):2, ('y2',0):4, ('y3',0):8,
('y1',1):10, ('y2',1):5, ('y3',1):5,
('y1',2):4, ('y2',2):25, ('y3',2):2,
('y1',3):5, ('y2',3):10, ('y3',3):4})
>>> print(Y_in)
0 1 2 3
------------
y1 | 2 10 4 5
y2 | 4 5 25 10
y3 | 8 5 2 4
>>> print(mat_move2board(Y_in))
0 1 2 3
------------------
y1 | 0.25 2 2 1.25
y2 | 0.5 1 12.5 2.5
y3 | 1 1 1 1
'''
y_in = mat2coldict(Y)
y_out = {x:move2board(y_in[x]) for x in y_in.keys()}
return coldict2mat(y_out)
def exchange(S, A, z):
'''
Input:
- S: a list of vectors, as instances of your Vec class
- A: a list of vectors, each of which are in S, with len(A) < len(S)
- z: an instance of Vec such that A+[z] is linearly independent
Output: a vector w in S but not in A such that Span S = Span ({z} U S - {w})
Example:
>>> S = [list2vec(v) for v in [[0,0,5,3],[2,0,1,3],[0,0,1,0],[1,2,3,4]]]
>>> A = [list2vec(v) for v in [[0,0,5,3],[2,0,1,3]]]
>>> z = list2vec([0,2,1,1])
>>> exchange(S, A, z) == Vec({0, 1, 2, 3},{0: 0, 1: 0, 2: 1, 3: 0})
True
'''
for i in range(len(S)):
w = S[i]
if w not in A:
tmpL = list(S)
tmpV = tmpL.pop(i)
tmpL.insert(i,z)
tmpMat = coldict2mat(tmpL)
sol = solve(tmpMat,tmpV)
residual = tmpV - tmpMat * sol
if residual * residual < 1.0e-14:
return w
return None
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
if len(L) > 1:
l = L.copy()
b = l.pop(i)
M = coldict2mat(l)
solution = solve(M, b)
residual = M * solution - b
if residual * residual < 10e-14:
return True
return False
def direct_sum_decompose(U_basis, V_basis, w):
'''
input: A list of Vecs, U_basis, containing a basis for a vector space, U.
A list of Vecs, V_basis, containing a basis for a vector space, V.
A Vec, w, that belongs to the direct sum of these spaces.
output: A pair, (u, v), such that u+v=w and u is an element of U and
v is an element of V.
>>> U_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> direct_sum_decompose(U_basis, V_basis, w) == (Vec({0, 1, 2, 3, 4, 5},{0: 2.0, 1: 4.999999999999972, 2: 0.0, 3: 0.0, 4: 1.0, 5: 0.0}), Vec({0, 1, 2, 3, 4, 5},{0: 0.0, 1: 0.0, 2: 0.0, 3: 0.0, 4: 0.0, 5: 0.0}))
True
'''
sum_base = U_basis + V_basis
# find coefficients
composed_result = solve(coldict2mat(sum_base),w)
# calculate u
u = Vec(U_basis[0].D,{})
for i in range(len(U_basis)):
u += composed_result[i] * U_basis[i]
# calculate v
v = Vec(V_basis[0].D,{})
for i in range(len(U_basis), len(sum_base)):
v += composed_result[i] * V_basis[i-len(U_basis)]
return (u,v)
def is_superfluous(L, i):
"""
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
"""
if len(L) > 1:
b = L.pop(i)
A = coldict2mat(L)
u = solve(A, b)
residual = b - (A * u)
condition = u != Vec(A.D[1], {}) and residual * residual < 10 ** -14
L.insert(i, b)
else:
return False
return condition
def exchange(S, A, z):
'''
Input:
- S: a list of vectors, as instances of your Vec class
- A: a list of vectors, each of which are in S, with len(A) < len(S)
- z: an instance of Vec such that A+[z] is linearly independent
Output: a vector w in S but not in A such that Span S = Span ({z} U S - {w})
Example:
>>> S = [list2vec(v) for v in [[0,0,5,3],[2,0,1,3],[0,0,1,0],[1,2,3,4]]]
>>> A = [list2vec(v) for v in [[0,0,5,3],[2,0,1,3]]]
>>> z = list2vec([0,2,1,1])
>>> exchange(S, A, z) == Vec({0, 1, 2, 3},{0: 0, 1: 0, 2: 1, 3: 0})
True
'''
for i in range(len(S)):
w = S[i]
if w not in A:
tmpL = list(S)
tmpV = tmpL.pop(i)
tmpL.insert(i, z)
tmpMat = coldict2mat(tmpL)
sol = solve(tmpMat, tmpV)
residual = tmpV - tmpMat * sol
if residual * residual < 1.0e-14:
return w
return None
def mat_move2board(Y):
'''
Input:
- Y: a Mat each column of which is a {'y1', 'y2', 'y3'}-Vec
giving the whiteboard coordinates of a point q.
Output:
- a Mat each column of which is the corresponding point in the
whiteboard plane (the point of intersection with the whiteboard plane
of the line through the origin and q).
Example:
>>> Y_in = Mat(({'y1', 'y2', 'y3'}, {0,1,2,3}),
{('y1',0):2, ('y2',0):4, ('y3',0):8,
('y1',1):10, ('y2',1):5, ('y3',1):5,
('y1',2):4, ('y2',2):25, ('y3',2):2,
('y1',3):5, ('y2',3):10, ('y3',3):4})
>>> print(Y_in)
0 1 2 3
------------
y1 | 2 10 4 5
y2 | 4 5 25 10
y3 | 8 5 2 4
>>> print(mat_move2board(Y_in))
0 1 2 3
------------------
y1 | 0.25 2 2 1.25
y2 | 0.5 1 12.5 2.5
y3 | 1 1 1 1
'''
y_in = mat2coldict(Y)
y_out = {x: move2board(y_in[x]) for x in y_in.keys()}
return coldict2mat(y_out)
def mat_move2board(Y):
'''
Input:
- Y: a Mat each column of which is a {'y1', 'y2', 'y3'}-Vec
giving the whiteboard coordinates of a point q.
Output:
- a Mat each column of which is the corresponding point in the
whiteboard plane (the point of intersection with the whiteboard plane
of the line through the origin and q).
Example:
>>> Y_in = Mat(({'y1', 'y2', 'y3'}, {0,1,2,3}),
... {('y1',0):2, ('y2',0):4, ('y3',0):8,
... ('y1',1):10, ('y2',1):5, ('y3',1):5,
... ('y1',2):4, ('y2',2):25, ('y3',2):2,
... ('y1',3):5, ('y2',3):10, ('y3',3):4})
>>> print(Y_in)
<BLANKLINE>
0 1 2 3
------------
y1 | 2 10 4 5
y2 | 4 5 25 10
y3 | 8 5 2 4
<BLANKLINE>
>>> print(mat_move2board(Y_in))
<BLANKLINE>
0 1 2 3
------------------
y1 | 0.25 2 2 1.25
y2 | 0.5 1 12.5 2.5
y3 | 1 1 1 1
<BLANKLINE>
'''
coldict = mat2coldict(Y)
return coldict2mat({key:move2board(val) for key, val in coldict.items()})
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
L = L[:]
v = L.pop(i)
M = coldict2mat(L)
u = solve(M, v)
r = v - M * u
return r * r < 10e-14
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
dist= len(L)
if dist == 1: return False
A= coldict2mat([L[j] for j in range(dist) if i != j])
b= L[i]
u= solve(A,b)
residual= b - A*u
return (residual*residual) < 1E-14
def direct_sum_decompose(U_basis, V_basis, w):
"""
input: A list of Vecs, U_basis, containing a basis for a vector space, U.
A list of Vecs, V_basis, containing a basis for a vector space, V.
A Vec, w, that belongs to the direct sum of these spaces.
output: A pair, (u, v), such that u+v=w and u is an element of U and
v is an element of V.
>>> U_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
>>> V_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
>>> w = Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
>>> direct_sum_decompose(U_basis, V_basis, w) == (Vec({0, 1, 2, 3, 4, 5},{0: 2.0, 1: 4.999999999999972, 2: 0.0, 3: 0.0, 4: 1.0, 5: 0.0}), Vec({0, 1, 2, 3, 4, 5},{0: 0.0, 1: 0.0, 2: 0.0, 3: 0.0, 4: 0.0, 5: 0.0}))
True
"""
testMat = coldict2mat(U_basis + V_basis)
coef = solve(testMat, w)
Nu = len(U_basis)
Nv = len(V_basis)
retU = Vec(U_basis[0].D, {})
retV = Vec(V_basis[0].D, {})
for i in range(Nu):
retU += coef[i] * U_basis[i]
for i in range(Nv):
retV += coef[i + Nu] * V_basis[i]
return (retU, retV)
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
assert i in range(len(L))
if len(L) > 1:
colVecs = coldict2mat({ x : L[x] for x in range(len(L)) if x != i })
u = solve(colVecs, L[i])
residual = L[i] - colVecs * u
else:
residual = 1
return residual * residual < 10e-14
def mat_move2board(Y):
'''
Input:
- Y: a Mat each column of which is a {'y1', 'y2', 'y3'}-Vec
giving the whiteboard coordinates of a point q.
Output:
- a Mat each column of which is the corresponding point in the
whiteboard plane (the point of intersection with the whiteboard plane
of the line through the origin and q).
Example:
>>> Y_in = Mat(({'y1', 'y2', 'y3'}, {0,1,2,3}),
... {('y1',0):2, ('y2',0):4, ('y3',0):8,
... ('y1',1):10, ('y2',1):5, ('y3',1):5,
... ('y1',2):4, ('y2',2):25, ('y3',2):2,
... ('y1',3):5, ('y2',3):10, ('y3',3):4})
>>> print(Y_in)
<BLANKLINE>
0 1 2 3
------------
y1 | 2 10 4 5
y2 | 4 5 25 10
y3 | 8 5 2 4
<BLANKLINE>
>>> print(mat_move2board(Y_in))
<BLANKLINE>
0 1 2 3
------------------
y1 | 0.25 2 2 1.25
y2 | 0.5 1 12.5 2.5
y3 | 1 1 1 1
<BLANKLINE>
'''
return coldict2mat({c: move2board(r) for c, r in mat2coldict(Y).items()})
def is_superfluous(L, i):
'''
Input:
- L: list of vectors as instances of Vec class
- i: integer in range(len(L))
Output:
True if the span of the vectors of L is the same
as the span of the vectors of L, excluding L[i].
False otherwise.
Examples:
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> is_superfluous(L, 3)
True
>>> is_superfluous([a0,a1,a2,a3], 3)
True
>>> is_superfluous([a0,a1,a2,a3], 0)
True
>>> is_superfluous([a0,a1,a2,a3], 1)
False
'''
if len(L) == 1:
return False
subL = [ L[x] for x in range(len(L)) if x != i]
u = vec2rep(subL, L[i])
residual = L[i] - coldict2mat({i:subL[i] for i in range(len(subL)) })*u
error = residual*residual
return True if error < 10e-14 else False
