def test_householder():
mp.dps = 15
A, b = A8, b8
H, p, x, r = householder(extend(A, b))
assert H == matrix([[mpf('3.0'), mpf('-2.0'),
mpf('-1.0'), 0],
[
-1.0,
mpf('3.333333333333333'),
mpf('-2.9999999999999991'),
mpf('2.0')
],
[
-1.0,
mpf('-0.66666666666666674'),
mpf('2.8142135623730948'),
mpf('-2.8284271247461898')
],
[
1.0,
mpf('-1.3333333333333333'),
mpf('-0.20000000000000018'),
mpf('4.2426406871192857')
]])
assert p == [-2, -2, mpf('-1.4142135623730949')]
assert round(norm(r, 2), 10) == 4.2426406870999998
y = [
102.102, 58.344, 36.463, 24.310, 17.017, 12.376, 9.282, 7.140, 5.610,
4.488, 3.6465, 3.003
]
def coeff(n):
# similiar to Hilbert matrix
A = []
for i in xrange(1, 13):
A.append([1. / (i + j - 1) for j in xrange(1, n + 1)])
return matrix(A)
residuals = []
refres = []
for n in xrange(2, 7):
A = coeff(n)
H, p, x, r = householder(extend(A, y))
x = matrix(x)
y = matrix(y)
residuals.append(norm(r, 2))
refres.append(norm(residual(A, x, y), 2))
assert [round(res, 10) for res in residuals] == [
15.1733888877, 0.82378073210000002, 0.302645887, 0.0260109244,
0.00058653999999999998
]
assert norm(matrix(residuals) - matrix(refres), inf) < 1.e-13
def test_norms():
# matrix norms
A = matrix([[1, -2], [-3, -1], [2, 1]])
assert mnorm(A,1) == 6
assert mnorm(A,inf) == 4
assert mnorm(A,'F') == sqrt(20)
# vector norms
assert norm(-3) == 3
x = [1, -2, 7, -12]
assert norm(x, 1) == 22
assert round(norm(x, 2), 10) == 14.0712472795
assert round(norm(x, 10), 10) == 12.0054633727
assert norm(x, inf) == 12
def test_solve():
assert norm(residual(A6, lu_solve(A6, b6), b6), inf) < 1.e-10
assert norm(residual(A7, lu_solve(A7, b7), b7), inf) < 1.5
assert norm(residual(A8, lu_solve(A8, b8), b8), inf) <= 3 + 1.e-10
assert norm(residual(A6, qr_solve(A6, b6)[0], b6), inf) < 1.e-10
assert norm(residual(A7, qr_solve(A7, b7)[0], b7), inf) < 1.5
assert norm(residual(A8, qr_solve(A8, b8)[0], b8), 2) <= 4.3
assert norm(residual(A10, lu_solve(A10, b10), b10), 2) < 1.e-10
assert norm(residual(A10, qr_solve(A10, b10)[0], b10), 2) < 1.e-10
def test_solve():
assert norm(residual(A6, lu_solve(A6, b6), b6), inf) < 1.e-10
assert norm(residual(A7, lu_solve(A7, b7), b7), inf) < 1.5
assert norm(residual(A8, lu_solve(A8, b8), b8), inf) <= 3 + 1.e-10
assert norm(residual(A6, qr_solve(A6, b6)[0], b6), inf) < 1.e-10
assert norm(residual(A7, qr_solve(A7, b7)[0], b7), inf) < 1.5
assert norm(residual(A8, qr_solve(A8, b8)[0], b8), 2) <= 4.3
assert norm(residual(A10, lu_solve(A10, b10), b10), 2) < 1.e-10
assert norm(residual(A10, qr_solve(A10, b10)[0], b10), 2) < 1.e-10
def test_householder():
mp.dps = 15
A, b = A8, b8
H, p, x, r = householder(extend(A, b))
assert H == matrix(
[[mpf('3.0'), mpf('-2.0'), mpf('-1.0'), 0],
[-1.0,mpf('3.333333333333333'),mpf('-2.9999999999999991'),mpf('2.0')],
[-1.0, mpf('-0.66666666666666674'),mpf('2.8142135623730948'),
mpf('-2.8284271247461898')],
[1.0, mpf('-1.3333333333333333'),mpf('-0.20000000000000018'),
mpf('4.2426406871192857')]])
assert p == [-2, -2, mpf('-1.4142135623730949')]
assert round(norm(r, 2), 10) == 4.2426406870999998
y = [102.102, 58.344, 36.463, 24.310, 17.017, 12.376, 9.282, 7.140, 5.610,
4.488, 3.6465, 3.003]
def coeff(n):
# similiar to Hilbert matrix
A = []
for i in xrange(1, 13):
A.append([1. / (i + j - 1) for j in xrange(1, n + 1)])
return matrix(A)
residuals = []
refres = []
for n in xrange(2, 7):
A = coeff(n)
H, p, x, r = householder(extend(A, y))
x = matrix(x)
y = matrix(y)
residuals.append(norm(r, 2))
refres.append(norm(residual(A, x, y), 2))
assert [round(res, 10) for res in residuals] == [15.1733888877,
0.82378073210000002, 0.302645887, 0.0260109244,
0.00058653999999999998]
assert norm(matrix(residuals) - matrix(refres), inf) < 1.e-13
def test_exp_pade():
for i in range(3):
dps = 15
extra = 5
mp.dps = dps + extra
dm = 0
while not dm:
m = randmatrix(3)
dm = det(m)
m = m / dm
a = diag([1, 2, 3])
a1 = m**-1 * a * m
mp.dps = dps
e1 = exp_pade(a1)
mp.dps = dps + extra
e2 = m * a1 * m**-1
d = e2 - a
#print d
mp.dps = dps
assert norm(d, inf).ae(0)
mp.dps = 15
def test_exp_pade():
for i in range(3):
dps = 15
extra = 5
mp.dps = dps + extra
dm = 0
while not dm:
m = randmatrix(3)
dm = det(m)
m = m/dm
a = diag([1,2,3])
a1 = m**-1 * a * m
mp.dps = dps
e1 = exp_pade(a1)
mp.dps = dps + extra
e2 = m * a1 * m**-1
d = e2 - a
#print d
mp.dps = dps
assert norm(d, inf).ae(0)
mp.dps = 15
def test_improve_solution():
A = randmatrix(5, min=1e-20, max=1e20)
b = randmatrix(5, 1, min=-1000, max=1000)
x1 = lu_solve(A, b) + randmatrix(5, 1, min=-1e-5, max=1.e-5)
x2 = improve_solution(A, x1, b)
assert norm(residual(A, x2, b), 2) < norm(residual(A, x1, b), 2)
def test_cholesky():
A9.force_type = float
assert cholesky(A9) == matrix([[2, 0, 0], [1, 2, 0], [-1, -3 / 2, 3 / 2]])
x = cholesky_solve(A9, b9)
assert norm(residual(A9, x, b9), inf) == 0
def test_improve_solution():
A = randmatrix(5, min=1e-20, max=1e20)
b = randmatrix(5, 1, min=-1000, max=1000)
x1 = lu_solve(A, b) + randmatrix(5, 1, min=-1e-5, max=1.e-5)
x2 = improve_solution(A, x1, b)
assert norm(residual(A, x2, b), 2) < norm(residual(A, x1, b), 2)
def test_cholesky():
A9.force_type = float
assert cholesky(A9) == matrix([[2, 0, 0], [1, 2, 0], [-1, -3/2, 3/2]])
x = cholesky_solve(A9, b9)
assert norm(residual(A9, x, b9), inf) == 0
