def run_eig(A, verbose=0):
if verbose > 1:
print("original matrix (eig):\n", A)
n = A.rows
E, EL, ER = mp.eig(A, left=True, right=True)
if verbose > 1:
print("E:\n", E)
print("EL:\n", EL)
print("ER:\n", ER)
eps = mp.exp(0.8 * mp.log(mp.eps))
err0 = 0
for i in xrange(n):
B = A * ER[:, i] - E[i] * ER[:, i]
err0 = max(err0, mp.mnorm(B))
B = EL[i, :] * A - EL[i, :] * E[i]
err0 = max(err0, mp.mnorm(B))
err0 /= n * n
if verbose > 0:
print("difference (E):", err0)
assert err0 < eps
def run_eig(A, verbose = 0):
if verbose > 1:
print("original matrix (eig):\n", A)
n = A.rows
E, EL, ER = mp.eig(A, left = True, right = True)
if verbose > 1:
print("E:\n", E)
print("EL:\n", EL)
print("ER:\n", ER)
eps = mp.exp(0.8 * mp.log(mp.eps))
err0 = 0
for i in xrange(n):
B = A * ER[:,i] - E[i] * ER[:,i]
err0 = max(err0, mp.mnorm(B))
B = EL[i,:] * A - EL[i,:] * E[i]
err0 = max(err0, mp.mnorm(B))
err0 /= n * n
if verbose > 0:
print("difference (E):", err0)
assert err0 < eps
def run_eigsy(A, verbose = False):
if verbose:
print("original matrix:\n", str(A))
D, Q = mp.eigsy(A)
B = Q * mp.diag(D) * Q.transpose()
C = A - B
E = Q * Q.transpose() - mp.eye(A.rows)
if verbose:
print("eigenvalues:\n", D)
print("eigenvectors:\n", Q)
NC = mp.mnorm(C)
NE = mp.mnorm(E)
if verbose:
print("difference:", NC, "\n", C, "\n")
print("difference:", NE, "\n", E, "\n")
eps = mp.exp( 0.8 * mp.log(mp.eps))
assert NC < eps
assert NE < eps
return NC
def run_eigsy(A, verbose=False):
if verbose:
print("original matrix:\n", str(A))
D, Q = mp.eigsy(A)
B = Q * mp.diag(D) * Q.transpose()
C = A - B
E = Q * Q.transpose() - mp.eye(A.rows)
if verbose:
print("eigenvalues:\n", D)
print("eigenvectors:\n", Q)
NC = mp.mnorm(C)
NE = mp.mnorm(E)
if verbose:
print("difference:", NC, "\n", C, "\n")
print("difference:", NE, "\n", E, "\n")
eps = mp.exp(0.8 * mp.log(mp.eps))
assert NC < eps
assert NE < eps
return NC
def test_svd_test_case():
# a test case from Golub and Reinsch
# (see wilkinson/reinsch: handbook for auto. comp., vol ii-linear algebra, 134-151(1971).)
eps = mp.exp(0.8 * mp.log(mp.eps))
a = [[22, 10, 2, 3, 7],
[14, 7, 10, 0, 8],
[-1, 13, -1, -11, 3],
[-3, -2, 13, -2, 4],
[ 9, 8, 1, -2, 4],
[ 9, 1, -7, 5, -1],
[ 2, -6, 6, 5, 1],
[ 4, 5, 0, -2, 2]]
a = mp.matrix(a)
b = mp.matrix([mp.sqrt(1248), 20, mp.sqrt(384), 0, 0])
S = mp.svd_r(a, compute_uv = False)
S -= b
assert mp.mnorm(S) < eps
S = mp.svd_c(a, compute_uv = False)
S -= b
assert mp.mnorm(S) < eps
def run_gauss(qtype, a, b):
eps = 1e-5
d, e = mp.gauss_quadrature(len(a), qtype)
d -= mp.matrix(a)
e -= mp.matrix(b)
assert mp.mnorm(d) < eps
assert mp.mnorm(e) < eps
def run_gauss(qtype, a, b):
eps = 1e-5
d, e = mp.gauss_quadrature(len(a), qtype)
d -= mp.matrix(a)
e -= mp.matrix(b)
assert mp.mnorm(d) < eps
assert mp.mnorm(e) < eps
def run_svd_r(A, full_matrices=False, verbose=True):
m, n = A.rows, A.cols
eps = mp.exp(0.8 * mp.log(mp.eps))
if verbose:
print("original matrix:\n", str(A))
print("full", full_matrices)
U, S0, V = mp.svd_r(A, full_matrices=full_matrices)
S = mp.zeros(U.cols, V.rows)
for j in xrange(min(m, n)):
S[j, j] = S0[j]
if verbose:
print("U:\n", str(U))
print("S:\n", str(S0))
print("V:\n", str(V))
C = U * S * V - A
err = mp.mnorm(C)
if verbose:
print("C\n", str(C), "\n", err)
assert err < eps
D = V * V.transpose() - mp.eye(V.rows)
err = mp.mnorm(D)
if verbose:
print("D:\n", str(D), "\n", err)
assert err < eps
E = U.transpose() * U - mp.eye(U.cols)
err = mp.mnorm(E)
if verbose:
print("E:\n", str(E), "\n", err)
assert err < eps
def run_svd_r(A, full_matrices = False, verbose = True):
m, n = A.rows, A.cols
eps = mp.exp(0.8 * mp.log(mp.eps))
if verbose:
print("original matrix:\n", str(A))
print("full", full_matrices)
U, S0, V = mp.svd_r(A, full_matrices = full_matrices)
S = mp.zeros(U.cols, V.rows)
for j in xrange(min(m, n)):
S[j,j] = S0[j]
if verbose:
print("U:\n", str(U))
print("S:\n", str(S0))
print("V:\n", str(V))
C = U * S * V - A
err = mp.mnorm(C)
if verbose:
print("C\n", str(C), "\n", err)
assert err < eps
D = V * V.transpose() - mp.eye(V.rows)
err = mp.mnorm(D)
if verbose:
print("D:\n", str(D), "\n", err)
assert err < eps
E = U.transpose() * U - mp.eye(U.cols)
err = mp.mnorm(E)
if verbose:
print("E:\n", str(E), "\n", err)
assert err < eps
