def 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 factor(A):
""" Implementation of the Gram-Schmidt procedure.
"""
col_labels = sorted(A.D[1], key=hash)
Acols = dict2list(matrix2coldict(A), col_labels)
Qlist, Rlist = aug_orthonormalize(Acols)
# Now make Matrices
Q = coldict2mat(Qlist)
R = coldict2mat(list2dict(Rlist, col_labels))
return Q, R
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
'''
Qlist,Rlist=orthonormalization.aug_orthonormalize([v for v in mat2coldict(A).values()])
Q = coldict2mat(Qlist)
R = coldict2mat(Rlist)
return solve(R, transpose(Q)*b)
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
'''
Qlist, Rlist = orthonormalization.aug_orthonormalize(
[v for v in mat2coldict(A).values()])
Q = coldict2mat(Qlist)
R = coldict2mat(Rlist)
return solve(R, transpose(Q) * b)
from hw7 import QR_solve
from mat import coldict2mat
from mat import Mat
from orthonormalization import aug_orthonormalize
from QR import factor
from vec import Vec
from vecutil import list2vec
print('Augmented Orthonormalize')
L = [list2vec(v) for v in [[4,3,1,2],[8,9,-5,-5],[10,1,-1,5]]]
print(coldict2mat(L))
Qlist, Rlist = aug_orthonormalize(L)
print(coldict2mat(Qlist))
print(coldict2mat(Rlist))
print((coldict2mat(Qlist)*coldict2mat(Rlist)))
print('QR Solve')
A=Mat(({'a','b','c'},{'A','B'}), {('a','A'):-1, ('a','B'):2, ('b','A'):5, ('b','B'):3,('c','A'):1, ('c','B'):-2})
print(A)
Q, R = factor(A)
print(Q)
print(R)
b = Vec({'a','b','c'}, {'a':1,'b':-1})
x = QR_solve(A,b)
print(x)
residual = A.transpose()*(b-A*x)
print(residual)