def generate_random_matrix(rows, columns, rank):
# TODO: change 3911 to q
m = randMatrix(rows, columns, min=0, max=Params.FINITE_FIELD)
# print("m: {}".format(m))
while m.rank() != rank:
m = randMatrix(rows, columns, min=0, max=Params.FINITE_FIELD)
return m
def _compare_tensorflow_matrix(variables, expr, use_float=False):
f = lambdify(variables, expr, 'tensorflow')
if not use_float:
random_matrices = [randMatrix(v.rows, v.cols) for v in variables]
else:
random_matrices = [
randMatrix(v.rows, v.cols) / 100. for v in variables
]
graph = tf.Graph()
r = None
with graph.as_default():
random_variables = [eval(tensorflow_code(i)) for i in random_matrices]
session = tf.compat.v1.Session(graph=graph)
r = session.run(f(*random_variables))
e = expr.subs({k: v for k, v in zip(variables, random_matrices)})
e = e.doit()
if e.is_Matrix:
if not isinstance(e, MatrixBase):
e = e.as_explicit()
e = e.tolist()
if not use_float:
assert (r == e).all()
else:
r = [i for row in r for i in row]
e = [i for row in e for i in row]
assert all(
abs(a - b) < 10**-(4 - int(log(abs(a), 10))) for a, b in zip(r, e))
def Det4x4(i):
A = randMatrix(4, 4, -4, 4, prng = prng)
text = fr'''
Calcule o determinante da matriz \[ {latex(A)} .\]
'''
answer = int(det(A))
return { 'answer' : answer, 'text' : text}
def Sistema4x4(i):
A = randMatrix(4, 4, -5, 5, prng=prng)
P, L, D, U = A.LUdecompositionFF()
text = fr'''
Calcule o determinante da matriz \[ {latex(A)} .\]
'''
answer = int(det(A))
return {'answer': answer, 'text': text}
def _compare_tensorflow_matrix_scalar(variables, expr):
f = lambdify(variables, expr, 'tensorflow')
random_matrices = [
randMatrix(v.rows, v.cols).evalf() / 100 for v in variables]
graph = tf.Graph()
r = None
with graph.as_default():
random_variables = [eval(tensorflow_code(i)) for i in random_matrices]
session = tf.compat.v1.Session(graph=graph)
r = session.run(f(*random_variables))
e = expr.subs({k: v for k, v in zip(variables, random_matrices)})
e = e.doit()
assert abs(r-e) < 10**-6
# Find the induced metric w.r.t. the E1 E2 basis
xE = xu.dot(xu)
xF = xu.dot(xv)
xG = xv.dot(xv)
xI = Matrix([[xE, xF], [xF, xG]])
xI_pt = xI.subs(uvpt) # Same as XI_pt!?
# Find the shape operator in uv space
# Any vector in uv space can be transformed into {E1, E2} space,
# multipled by the shape operator, then transformed back into uv space
S_pt_uv = (xJ_pt.inv() * S_pt * xJ_pt).evalf()
# S is self-adjoint w.r.t. the induced metric XI
print 'Verify that S is self-adjoint w.r.t. the induced metrix I'
print 'i.e. g(SX,Y) = g(X,SY) for all vectors X,Y'
randX = randMatrix(2,1)
randY = randMatrix(2,1)
gSXY = (S_pt_uv * randX).transpose() * xI_pt * randY
gXSY = randX.transpose() * xI_pt * (S_pt_uv * randY)
print '{} = g(SX,Y) = g(X,SY) = {}'.format(gSXY, gXSY)
# ============
# Plot results
# ============
import itertools
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
import numpy as np
# Set limits
xMin = -2
def generate_Z(alpha, d):
Z = randMatrix(alpha, d, 0, 1)
while Z.rank() != alpha:
Z = randMatrix(alpha, d, 0, 1)
return Z
def generate_b_col_vector(alpha):
b = randMatrix(alpha, 1, 0, 1)
while b.rank() == 0:
b = randMatrix(alpha, 1, 0, 1)
return b
def generate_b_row_vector(alpha):
b = randMatrix(1, alpha, 0, 1)
while b.rank() == 0:
b = randMatrix(1, alpha, 0, 1)
return b
