def orthonormal_projection_orthogonal(W, b):
'''
Input:
- W: Mat whose rows are orthonormal
- b: Vec whose labels are equal to W's column labels
Output:
- The projection of b orthogonal to W's row space.
Example:
>>> W = Mat(({0, 1}, {0, 1, 2}), {(0, 1): 0, (1, 2): 0, (0, 0): 1, (1, 0): 0, (0, 2): 0, (1, 1): 1})
>>> b = Vec({0, 1, 2},{0: 3, 1: 1, 2: 4})
>>> orthonormal_projection_orthogonal(W, b) == Vec({0, 1, 2},{0: 0, 1: 0, 2: 4})
True
>>> W1 = Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): 1, (1, 2): 0, (0, 0): 0, (2, 0): 0, (1, 0): -1, (2, 2): -1, (0, 2): 0, (2, 1): 0, (1, 1): 0})
>>> b1 = Vec({0, 1, 2},{0: 7, 1: 1, 2: 9})
>>> print(test_format(orthonormal_projection_orthogonal(W1, b1)))
Vec({0.000000, 1.000000, 2.000000}, {})
>>> W2 = Mat(({0, 1}, {0, 1, 2}), {(0, 1): 1, (1, 2): 0.6, (0, 0): 0, (1, 0): -0.8, (0, 2): 0, (1, 1): 0})
>>> b2 = Vec({0, 1, 2},{0: 1, 1: 2, 2: 3})
>>> print(test_format(orthonormal_projection_orthogonal(W2, b2)))
Vec({0.000000, 1.000000, 2.000000}, {0.000000: 1.800000, 2.000000: 2.400000})
>>> W3 = Mat(({'a','b'}, {'A','B','C','D'}), {('a','A'):1/2, ('a','B'):1/2, ('a','C'):1/2, ('a','D'):1/2,('b','A'):1/2,('b','B'):-1/2, ('b','C'):1/2, ('b','D'):-1/2})
>>> b3 = Vec({'A','B','C','D'},{'A': 8, 'B': 2, 'C': 4, 'D':1})
>>> print(test_format(orthonormal_projection_orthogonal(W3, b3)))
Vec({'A', 'B', 'C', 'D'}, {'A': 2.000000, 'B': 0.500000, 'C': -2.000000, 'D': -0.500000})
'''
list2 = []
list1 = []
x = mat2rowdict(W)
for i in W.D[0]:
list1.append(x[i])
for i in list1:
result = project_along(b,i,eps=1e-20)
list2.append(result)
return b - sum(list2)
def QR_solve(A, b):
'''
Input:
- A: a Mat
- b: a Vec
Output:
- vector x that minimizes norm(b - A*x)
Example:
>>> domain = ({'a','b','c'},{'A','B'})
>>> A = Mat(domain,{('a','A'):-1, ('a','B'):2,('b','A'):5, ('b','B'):3,('c','A'):1,('c','B'):-2})
>>> Q, R = QR.factor(A)
>>> b = Vec(domain[0], {'a': 1, 'b': -1})
>>> x = QR_solve(A, b)
>>> result = A.transpose()*(b-A*x)
>>> result * result < 1E-10
True
I used the following as args to triangular_solve() and the grader was happy:
rowlist = mat2rowdict(R) (mentioned in the pdf)
label_list = sorted(A.D[1], key=repr) (mentioned in the pdf)
b = c vector b = Vec(domain[0], {'a': 1, 'b': -1})(mentioned above and on slide 1 of orthogonalization 8)
'''
Q, R = QR.factor(A)
rowlist = mat2rowdict(R)
label_list = sorted(A.D[1], key = repr)
return solve(A,b)
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)
'''
vstarlist, tcols = aug_orthogonalize(L)
sigmas = [sqrt(v*v) for v in vstarlist]
qlist = orthonormalize(vstarlist)
T = coldict2mat(tcols)
trows = mat2rowdict(T)
rrows = [adjust(trows[k], sigmas[k]) for k in trows]
rdict = mat2coldict(rowdict2mat(rrows))
rlist = [rdict[k] for k in rdict]
return qlist, rlist
def QR_solve(A, b):
'''
Input:
- A: a Mat
- b: a Vec
Output:
- vector x that minimizes norm(b - A*x)
Example:
>>> domain = ({'a','b','c'},{'A','B'})
>>> A = Mat(domain,{('a','A'):-1, ('a','B'):2,('b','A'):5, ('b','B'):3,('c','A'):1,('c','B'):-2})
>>> Q, R = factor(A)
>>> b = Vec(domain[0], {'a': 1, 'b': -1})
>>> x = QR_solve(A, b)
>>> result = A.transpose()*(b-A*x)
>>> result * result < 1E-10
True
'''
Q, R = factor(A)
c = Q.transpose()*b
Rdict = mat2rowdict(R)
Rrows = [Rdict[key] for key in Rdict]
labels = sorted(A.D[1], key=repr)
return triangular_solve(Rrows, labels, c)
