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hw7.py
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hw7.py
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# version code 1049
# Please fill out this stencil and submit using the provided submission script.
from orthogonalization import orthogonalize
import orthonormalization
from mat import Mat
from vec import Vec
from vecutil import list2vec
from matutil import listlist2mat, mat2rowdict
## Problem 1
def basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- a list of linearly independent Vecs with equal span to vlist
'''
return [v for v in orthogonalize(vlist) if v*v > 1e-20]
## Problem 2
def subset_basis(vlist):
'''
Input:
- vlist: a list of Vecs
Output:
- linearly independent subset of vlist with the same span as vlist
'''
return [vlist[i] for i,v in enumerate(orthogonalize(vlist)) if v*v > 1e-20]
## Problem 3
# Hint: orthogonal matrix's inverse equals its transpose.
# Q.transpose()*x = b
def orthogonal_vec2rep(Q, b):
'''
Input:
- Q: an orthogonal Mat
- b: Vec whose domain equals the column-label set of Q.
Output:
- The coordinate representation of b in terms of the rows of Q.
Example:
>>> Q = Mat(({0, 1}, {0, 1}), {(0, 1): 0, (1, 0): 0, (0, 0): 2, (1, 1): 2})
>>> b = Vec({0, 1},{0: 4, 1: 2})
>>> orthogonal_vec2rep(Q, b) == Vec({0, 1},{0: 8, 1: 4})
True
'''
return Q*b
## Problem 4
def orthogonal_change_of_basis(A, B, a):
'''
Input:
- A: an orthogonal Mat (inversed)
- B: an orthogonal Mat whose column labels are the row labels of A
- a: the coordinate representation in terms of rows of A of some vector v
Output:
- the Vec b such that b is the coordinate representation of v in terms of columns of B
Example:
>>> A = Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): 0, (1, 2): 0, (0, 0): 1, (2, 0): 0, (1, 0): 0, (2, 2): 1, (0, 2): 0, (2, 1): 0, (1, 1): 1})
>>> B = Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): 0, (1, 2): 0, (0, 0): 2, (2, 0): 0, (1, 0): 0, (2, 2): 2, (0, 2): 0, (2, 1): 0, (1, 1): 2})
>>> a = Vec({0, 1, 2},{0: 4, 1: 1, 2: 3})
>>> orthogonal_change_of_basis(A, B, a) == Vec({0, 1, 2},{0: 8, 1: 2, 2: 6})
True
'''
return a*A*B
## Problem 5
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
'''
return b-W.transpose()*W*b
## Problem 6
# Write your solution for this problem in orthonormalization.py.
## Problem 7
# Write your solution for this problem in orthonormalization.py.
## Problem 8
# Please give each solution as a Vec
least_squares_A1 = listlist2mat([[8, 1], [6, 2], [0, 6]])
least_squares_Q1 = listlist2mat([[.8,-0.099],[.6, 0.132],[0,0.986]])
least_squares_R1 = listlist2mat([[10,2],[0,6.08]])
least_squares_b1 = list2vec([10, 8, 6])
x_hat_1 = Vec({0, 1},{0: 1.0832432432432433, 1: 0.9837837837837838})
least_squares_A2 = listlist2mat([[3, 1], [4, 1], [5, 1]])
least_squares_Q2 = listlist2mat([[.424, .808],[.566, .115],[.707, -.577]])
least_squares_R2 = listlist2mat([[7.07, 1.7],[0,.346]])
least_squares_b2 = list2vec([10,13,15])
x_hat_2 = Vec({0, 1},{0: 2.499999999999997, 1: 2.666666666666679})
## Problem 9
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
'''
from triangular import triangular_solve
from QR import factor
from dictutil import dict2list
from vecutil import vec2list
Q,R = factor(A)
b = Q.transpose()*b
rowdict = mat2rowdict(R)
rowlist = dict2list(rowdict, list(rowdict.keys()))
x = triangular_solve(rowlist, list(rowlist[0].D), vec2list(b))
return x