def trace(A):
"""
Computes the sum of the diagonal elements
of a square matrix A.
"""
# grab the diagonal elements
diag = diagonal(A)
# instantiate kahan summer
summer = KahanSum()
# compute sum of list
for d in diag:
summer.add(d)
return summer.cur_sum()
def inverse(A):
"""
Computes the inverse of a square matrix A.
Concretely, solves the linear system Ax = I
where x is a square matrix rather than a vector.
The system is solved using LU decomposition with
partial pivoting.
Params
------
- A: a numpy array of shape (N, N).
Returns
-------
- a numpy array of shape (N, N).
"""
N = A.shape[0]
P, L, U = LU(A, pivoting='partial').decompose()
# transpose P since LU returns A = PLU
P = P.T
# solve Ly = P for y
y = np.zeros_like(L)
for i in range(N):
for j in range(N):
summer = KahanSum()
for k in range(i):
summer.add(L[i, k] * y[k, j])
sum = summer.cur_sum()
y[i, j] = (P[i, j] - sum) / (L[i, i])
# solve Ux = y for x
x = np.zeros_like(U)
for i in range(N - 1, -1, -1):
for j in range(N):
summer = KahanSum()
for k in range(N - 1, i, -1):
summer.add(U[i, k] * x[k, j])
sum = summer.cur_sum()
x[i, j] = (y[i, j] - sum) / (U[i, i])
return x
def norm(x, p):
"""
Returns the p norm of a vector.
"""
v = np.array(x).flatten()
error_msg = "x must be 1D"
assert v.ndim == 1, error_msg
error_msg = "p must be >= 1"
assert p >= 1, error_msg
N = v.shape[0]
summer = KahanSum()
for i in range(N):
summer.add(np.power(np.abs(v[i]), p))
return np.power(summer.cur_sum(), 1. / p)
def _forward(self):
"""
Solves the lower triangular system Ly = b
for y by forward substitution.
"""
if self.b.ndim > 1:
num_iters = self.b.shape[1]
N = self.b.shape[0]
else:
num_iters = 1
N = self.b.shape[0]
self.y = np.zeros([N, num_iters])
for k in range(num_iters):
for i in range(N):
acc = KahanSum()
for j in range(i):
acc.add(self.L[i, j]*self.y[j, k])
if self.b.ndim > 1:
self.y[i, k] = \
(self.b[i, k] - acc.cur_sum()) / (self.L[i, i])
else:
self.y[i, k] = \
(self.b[i] - acc.cur_sum()) / (self.L[i, i])
def decompose(self, ret=True):
N = len(self.R)
self.L = np.zeros_like(self.R)
if self.crout:
for i in range(N):
for j in range(i+1):
summer = KahanSum()
for k in range(j):
summer.add(self.L[i, k] * self.L[j, k])
sum = summer.cur_sum()
if (i == j):
self.L[j, j] = np.sqrt(self.R[j, j] - sum)
else:
self.L[i, j] = (self.R[i, j] - sum) / (self.L[j, j])
if ret:
return self.L
else:
for i in range(N):
self.pivot = self.R[i, i]
# eliminate subsequent rows
for j in range(i+1, N):
for k in range(j, N):
self.R[j, k] -= self.R[i, k] * \
(self.R[i, j] / self.pivot)
# scale the current row
for k in range(i, N):
self.R[i, k] /= np.sqrt(self.pivot)
if ret:
return self.R.T
def _backward(self):
"""
Solve the upper triangular system L^Tx = y
for x by back substitution.
"""
if self.b.ndim > 1:
num_iters = self.b.shape[1]
N = self.b.shape[0]
else:
num_iters = 1
N = self.b.shape[0]
self.x = np.zeros([N, num_iters])
for k in range(num_iters):
for i in range(N-1, -1, -1):
acc = KahanSum()
for j in range(N-1, i, -1):
acc.add(self.L.T[i, j]*self.x[j, k])
self.x[i, k] = \
(self.y[i, k] - acc.cur_sum()) / (self.L.T[i, i])
if self.b.ndim == 1:
self.x = self.x.squeeze()
def _forward(self):
"""
Solves the lower triangular system Ly = b
for y by forward substitution.
If partial pivoting is used, solves the system
Ly = Pb and if full pivoting is used, solves
the system Ly = PbQ.
"""
if self.b.ndim > 1:
num_iters = self.b.shape[1]
N = self.b.shape[0]
else:
num_iters = 1
N = self.b.shape[0]
self.y = np.zeros([N, num_iters])
if self.pivoting is None:
right_hand = self.b
elif self.pivoting == "partial":
right_hand = np.dot(self.P, self.b)
else:
right_hand = np.dot(self.P, self.b)
right_hand = np.dot(right_hand[:, np.newaxis].T, self.Q)
right_hand = right_hand.squeeze().T
for k in range(num_iters):
for i in range(N):
acc = KahanSum()
for j in range(i):
acc.add(self.L[i, j]*self.y[j, k])
if self.b.ndim > 1:
self.y[i, k] = right_hand[i, k] - acc.cur_sum()
else:
self.y[i, k] = right_hand[i] - acc.cur_sum()