def regular_completion(matrix):
m,n = matrix.shape
r = matrix.rank()
#~ IPS()
assert m!=n, "There is no regular completion of a square matrix."
if m<n:
assert r==m, "Matrix does not have full row rank."
A, B, V_pi = reshape_matrix_columns(matrix)
zeros = sp.zeros(n-m,m)
ones = sp.eye(n-m)
S = st.col_stack(zeros,ones)
completion = S*V_pi.T
regular_matrix = st.row_stack(matrix,completion)
assert st.rnd_number_rank(regular_matrix)==n, "Regular completion seems to be wrong."
elif n<m:
assert r==n, "Matrix does not have full column rank."
A, B, V_pi = reshape_matrix_columns(matrix.T)
zeros = sp.zeros(m-n,n)
ones = sp.eye(m-n)
S = st.col_stack(zeros,ones)
completion = V_pi*S.T
regular_matrix = st.col_stack(completion,matrix)
assert st.rnd_number_rank(regular_matrix)==m, "Regular completion seems to be wrong."
return completion
def regular_completion(matrix):
m, n = matrix.shape
r = matrix.rank()
#~ IPS()
assert m != n, "There is no regular completion of a square matrix."
if m < n:
assert r == m, "Matrix does not have full row rank."
A, B, V_pi = reshape_matrix_columns(matrix)
zeros = sp.zeros(n - m, m)
ones = sp.eye(n - m)
S = st.col_stack(zeros, ones)
completion = S * V_pi.T
regular_matrix = st.row_stack(matrix, completion)
assert st.rnd_number_rank(
regular_matrix) == n, "Regular completion seems to be wrong."
elif n < m:
assert r == n, "Matrix does not have full column rank."
A, B, V_pi = reshape_matrix_columns(matrix.T)
zeros = sp.zeros(m - n, n)
ones = sp.eye(m - n)
S = st.col_stack(zeros, ones)
completion = V_pi * S.T
regular_matrix = st.col_stack(completion, matrix)
assert st.rnd_number_rank(
regular_matrix) == m, "Regular completion seems to be wrong."
return completion
def is_linearly_independent(matrix, column_vector):
m, n = matrix.shape
rank1 = st.rnd_number_rank(matrix)
tmp = st.concat_cols(matrix, column_vector)
rank2 = st.rnd_number_rank(tmp)
assert rank2 >= rank1
return rank2 > rank1
def is_linearly_independent(matrix, column_vector):
m, n = matrix.shape
rank1 = st.rnd_number_rank(matrix)
tmp = st.concat_cols(matrix, column_vector)
rank2 = st.rnd_number_rank(tmp)
assert rank2 >= rank1
return rank2 > rank1
def left_pseudo_inverse(matrix):
m, n = matrix.shape
r = st.rnd_number_rank(matrix)
#assert r==n, "Matrix does not have full column rank!"
assert not is_zero_matrix(matrix), "Matrix is zero matrix!"
transposed_rpinv = right_pseudo_inverse(matrix.T)
matrix_lpinv = transposed_rpinv.T
assert is_unit_matrix(matrix_lpinv*matrix), "Leftpseudoinverse is not correct."
return matrix_lpinv
def left_pseudo_inverse(matrix):
m, n = matrix.shape
r = st.rnd_number_rank(matrix)
#assert r==n, "Matrix does not have full column rank!"
assert not is_zero_matrix(matrix), "Matrix is zero matrix!"
transposed_rpinv = right_pseudo_inverse(matrix.T)
matrix_lpinv = transposed_rpinv.T
assert is_unit_matrix(matrix_lpinv *
matrix), "Leftpseudoinverse is not correct."
return matrix_lpinv
def has_left_ortho_complement(matrix):
r = st.rnd_number_rank(matrix)
m, n = matrix.shape
if(n>=m):
return True if (r<m) else False
return True
def is_regular_matrix(matrix):
assert is_square_matrix(matrix), "Matrix is not a square matrix."
m, n = matrix.shape
r = st.rnd_number_rank(matrix)
return True if (r == m) else False
def has_full_row_rank(matrix):
m, n = matrix.shape
r = st.rnd_number_rank(matrix)
return True if (m==r) else False
def reshape_matrix_columns(P):
m0, n0 = P.shape
# pick m0 of the simplest lin. independent columns of P:
P_new = remove_zero_columns(P)
m1, n1 = P_new.shape
list_of_cols = matrix_to_vectorlist(P_new)
# sort by "complexity"
if pinv_optimization == "free_symbols":
cols_sorted_by_atoms = sorted(list_of_cols,
key=lambda x: x.free_symbols,
reverse=False)
elif pinv_optimization == "count_ops":
cols_sorted_by_atoms = sorted(list_of_cols,
key=lambda x: nr_of_ops(x),
reverse=False)
elif pinv_optimization == "none":
cols_sorted_by_atoms = list_of_cols
# sort by number of zero entries
colvecs_sorted = sorted(cols_sorted_by_atoms,
key=lambda x: count_zero_entries(x),
reverse=True)
# pick m suitable column vectors and add to new matrix A: ----------
A = colvecs_sorted[0]
for j in range(len(colvecs_sorted) - 1):
column = colvecs_sorted[j + 1]
if is_linearly_independent(A, column):
A = st.concat_cols(A, column)
if st.rnd_number_rank(A) == m1:
break
assert A.is_square
# calculate transformation matrix R: -------------------------------
#R_tilde = sp.Matrix([])
used_cols = []
R = sp.Matrix([])
for k in range(m1):
new_column_index = k
old_column_index = get_column_index_of_matrix(P, A.col(k))
used_cols.append(old_column_index)
tmp = sp.zeros(n0, 1)
tmp[old_column_index] = 1
#R_tilde = st.concat_cols(R_tilde,tmp)
R = st.concat_cols(R, tmp)
#R=R_tilde
# remainder columns of R matrix
#m2,n2 = R_tilde.shape
m2, n2 = R.shape
for l in range(n0):
if l not in used_cols:
R_col = sp.zeros(n0, 1)
R_col[l] = 1
R = st.concat_cols(R, R_col)
m3, n3 = A.shape
r2 = st.rnd_number_rank(A)
assert m3 == r2, "A problem occured in reshaping the matrix."
# calculate B matrix: ----------------------------------------------
B = sp.Matrix([])
tmp = P * R
for i in range(n0 - m0):
B = st.concat_cols(B, tmp.col(m0 + i))
assert is_zero_matrix(
(P * R) -
st.concat_cols(A, B)), "A problem occured in reshaping the matrix."
return A, B, R
def reshape_matrix_columns(P):
m0, n0 = P.shape
# pick m0 of the simplest lin. independent columns of P:
P_new = remove_zero_columns(P)
m1, n1 = P_new.shape
list_of_cols = matrix_to_vectorlist(P_new)
# sort by "complexity"
if pinv_optimization=="free_symbols":
cols_sorted_by_atoms = sorted( list_of_cols, key=lambda x: x.free_symbols, reverse=False)
elif pinv_optimization=="count_ops":
cols_sorted_by_atoms = sorted( list_of_cols, key=lambda x: nr_of_ops(x), reverse=False)
elif pinv_optimization=="none":
cols_sorted_by_atoms = list_of_cols
# sort by number of zero entries
colvecs_sorted = sorted( cols_sorted_by_atoms, key=lambda x: count_zero_entries(x), reverse=True)
# pick m suitable column vectors and add to new matrix A: ----------
A = colvecs_sorted[0]
for j in xrange(len(colvecs_sorted)-1):
column = colvecs_sorted[j+1]
if is_linearly_independent(A,column):
A = st.concat_cols(A,column)
if st.rnd_number_rank(A)==m1:
break
assert A.is_square
# calculate transformation matrix R: -------------------------------
#R_tilde = sp.Matrix([])
used_cols = []
R = sp.Matrix([])
for k in xrange(m1):
new_column_index = k
old_column_index = get_column_index_of_matrix(P,A.col(k))
used_cols.append(old_column_index)
tmp = sp.zeros(n0,1)
tmp[old_column_index] = 1
#R_tilde = st.concat_cols(R_tilde,tmp)
R = st.concat_cols(R,tmp)
#R=R_tilde
# remainder columns of R matrix
#m2,n2 = R_tilde.shape
m2,n2 = R.shape
for l in xrange(n0):
if l not in used_cols:
R_col = sp.zeros(n0,1)
R_col[l] = 1
R = st.concat_cols(R,R_col)
m3, n3 = A.shape
r2 = st.rnd_number_rank(A)
assert m3==r2, "A problem occured in reshaping the matrix."
# calculate B matrix: ----------------------------------------------
B = sp.Matrix([])
tmp = P*R
for i in xrange(n0-m0):
B = st.concat_cols(B,tmp.col(m0+i))
assert is_zero_matrix( (P*R) - st.concat_cols(A,B)), "A problem occured in reshaping the matrix."
return A, B, R
def is_special_case(Bi):
""" Checks for special case
"""
n, p = Bi.shape
return True if (st.rnd_number_rank(Bi) < p) else False
def is_regular_matrix(matrix):
assert is_square_matrix(matrix), "Matrix is not a square matrix."
m, n = matrix.shape
r = st.rnd_number_rank(matrix)
return True if (r == m) else False
def is_special_case(Bi):
""" Checks for special case
"""
n, p = Bi.shape
return True if (st.rnd_number_rank(Bi) < p) else False
def has_right_ortho_complement(matrix):
r = st.rnd_number_rank(matrix)
m, n = matrix.shape
if(m>=n):
return True if (r<n) else False
return True
def has_right_ortho_complement(matrix):
r = st.rnd_number_rank(matrix)
m, n = matrix.shape
if (m >= n):
return True if (r < n) else False
return True
def has_left_ortho_complement(matrix):
r = st.rnd_number_rank(matrix)
m, n = matrix.shape
if (n >= m):
return True if (r < m) else False
return True
def has_full_row_rank(matrix):
m, n = matrix.shape
r = st.rnd_number_rank(matrix)
return True if (m == r) else False
def end_condition(Bi):
""" The algorithm ends if Bi has full row rank.
"""
n, p = Bi.shape
return True if (n == st.rnd_number_rank(Bi)) else False
def end_condition(Bi):
""" The algorithm ends if Bi has full row rank.
"""
n, p = Bi.shape
return True if (n == st.rnd_number_rank(Bi)) else False