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# TODO: don't use round
from __future__ import division
import pytest
from mpmath import *
xrange = libmp.backend.xrange
# XXX: these shouldn't be visible(?)
LU_decomp = mp.LU_decomp
L_solve = mp.L_solve
U_solve = mp.U_solve
householder = mp.householder
improve_solution = mp.improve_solution
A1 = matrix([[3, 1, 6],
[2, 1, 3],
[1, 1, 1]])
b1 = [2, 7, 4]
A2 = matrix([[ 2, -1, -1, 2],
[ 6, -2, 3, -1],
[-4, 2, 3, -2],
[ 2, 0, 4, -3]])
b2 = [3, -3, -2, -1]
A3 = matrix([[ 1, 0, -1, -1, 0],
[ 0, 1, 1, 0, -1],
[ 4, -5, 2, 0, 0],
[ 0, 0, -2, 9,-12],
[ 0, 5, 0, 0, 12]])
b3 = [0, 0, 0, 0, 50]
A4 = matrix([[10.235, -4.56, 0., -0.035, 5.67],
[-2.463, 1.27, 3.97, -8.63, 1.08],
[-6.58, 0.86, -0.257, 9.32, -43.6 ],
[ 9.83, 7.39, -17.25, 0.036, 24.86],
[-9.31, 34.9, 78.56, 1.07, 65.8 ]])
b4 = [8.95, 20.54, 7.42, 5.60, 58.43]
A5 = matrix([[ 1, 2, -4],
[-2, -3, 5],
[ 3, 5, -8]])
A6 = matrix([[ 1.377360, 2.481400, 5.359190],
[ 2.679280, -1.229560, 25.560210],
[-1.225280+1.e6, 9.910180, -35.049900-1.e6]])
b6 = [23.500000, -15.760000, 2.340000]
A7 = matrix([[1, -0.5],
[2, 1],
[-2, 6]])
b7 = [3, 2, -4]
A8 = matrix([[1, 2, 3],
[-1, 0, 1],
[-1, -2, -1],
[1, 0, -1]])
b8 = [1, 2, 3, 4]
A9 = matrix([[ 4, 2, -2],
[ 2, 5, -4],
[-2, -4, 5.5]])
b9 = [10, 16, -15.5]
A10 = matrix([[1.0 + 1.0j, 2.0, 2.0],
[4.0, 5.0, 6.0],
[7.0, 8.0, 9.0]])
b10 = [1.0, 1.0 + 1.0j, 1.0]
def test_LU_decomp():
A = A3.copy()
b = b3
A, p = LU_decomp(A)
y = L_solve(A, b, p)
x = U_solve(A, y)
assert p == [2, 1, 2, 3]
assert [round(i, 14) for i in x] == [3.78953107960742, 2.9989094874591098,
-0.081788440567070006, 3.8713195201744801, 2.9171210468920399]
A = A4.copy()
b = b4
A, p = LU_decomp(A)
y = L_solve(A, b, p)
x = U_solve(A, y)
assert p == [0, 3, 4, 3]
assert [round(i, 14) for i in x] == [2.6383625899619201, 2.6643834462368399,
0.79208015947958998, -2.5088376454101899, -1.0567657691375001]
A = randmatrix(3)
bak = A.copy()
LU_decomp(A, overwrite=1)
assert A != bak
def test_inverse():
for A in [A1, A2, A5]:
inv = inverse(A)
assert mnorm(A*inv - eye(A.rows), 1) < 1.e-14
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 range(1, 13):
A.append([1. / (i + j - 1) for j in range(1, n + 1)])
return matrix(A)
residuals = []
refres = []
for n in range(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 hilbert_cmplx(n):
# Complexified Hilbert matrix
A = hilbert(2*n,n)
v = randmatrix(2*n, 2, min=-1, max=1)
v = v.apply(lambda x: exp(1J*pi()*x))
A = diag(v[:,0])*A*diag(v[:n,1])
return A
residuals_cmplx = []
refres_cmplx = []
for n in range(2, 10):
A = hilbert_cmplx(n)
H, p, x, r = householder(A.copy())
residuals_cmplx.append(norm(r, 2))
refres_cmplx.append(norm(residual(A[:,:n-1], x, A[:,n-1]), 2))
assert norm(matrix(residuals_cmplx) - matrix(refres_cmplx), inf) < 1.e-13
def test_factorization():
A = randmatrix(5)
P, L, U = lu(A)
assert mnorm(P*A - L*U, 1) < 1.e-15
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_overdet_complex():
A = matrix([[1, 2j], [3, 4j], [5, 6]])
b = matrix([1 + j, 2, -j])
assert norm(residual(A, lu_solve(A, b), b)) < 1.0208
def test_singular():
mp.dps = 15
A = [[5.6, 1.2], [7./15, .1]]
B = repr(zeros(2))
b = [1, 2]
for i in ['lu_solve(%s, %s)' % (A, b), 'lu_solve(%s, %s)' % (B, b),
'qr_solve(%s, %s)' % (A, b), 'qr_solve(%s, %s)' % (B, b)]:
pytest.raises((ZeroDivisionError, ValueError), lambda: eval(i))
def test_cholesky():
assert fp.cholesky(fp.matrix(A9)) == fp.matrix([[2, 0, 0], [1, 2, 0], [-1, -3/2, 3/2]])
x = fp.cholesky_solve(A9, b9)
assert fp.norm(fp.residual(A9, x, b9), fp.inf) == 0
def test_det():
assert det(A1) == 1
assert round(det(A2), 14) == 8
assert round(det(A3)) == 1834
assert round(det(A4)) == 4443376
assert det(A5) == 1
assert round(det(A6)) == 78356463
assert det(zeros(3)) == 0
def test_cond():
mp.dps = 15
A = matrix([[1.2969, 0.8648], [0.2161, 0.1441]])
assert cond(A, lambda x: mnorm(x,1)) == mpf('327065209.73817754')
assert cond(A, lambda x: mnorm(x,inf)) == mpf('327065209.73817754')
assert cond(A, lambda x: mnorm(x,'F')) == mpf('249729266.80008656')
@extradps(50)
def test_precision():
A = randmatrix(10, 10)
assert mnorm(inverse(inverse(A)) - A, 1) < 1.e-45
def test_interval_matrix():
mp.dps = 15
iv.dps = 15
a = iv.matrix([['0.1','0.3','1.0'],['7.1','5.5','4.8'],['3.2','4.4','5.6']])
b = iv.matrix(['4','0.6','0.5'])
c = iv.lu_solve(a, b)
assert c[0].delta < 1e-13
assert c[1].delta < 1e-13
assert c[2].delta < 1e-13
assert 5.25823271130625686059275 in c[0]
assert -13.155049396267837541163 in c[1]
assert 7.42069154774972557628979 in c[2]
def test_LU_cache():
A = randmatrix(3)
LU = LU_decomp(A)
assert A._LU == LU_decomp(A)
A[0,0] = -1000
assert A._LU is None
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_exp_pade():
for i in range(3):
dps = 15
extra = 15
mp.dps = dps + extra
dm = 0
N = 3
dg = range(1,N+1)
a = diag(dg)
expa = diag([exp(x) for x in dg])
# choose a random matrix not close to be singular
# to avoid adding too much extra precision in computing
# m**-1 * M * m
while abs(dm) < 0.01:
m = randmatrix(N)
dm = det(m)
m = m/dm
a1 = m**-1 * a * m
e2 = m**-1 * expa * m
mp.dps = dps
e1 = expm(a1, method='pade')
mp.dps = dps + extra
d = e2 - e1
#print d
mp.dps = dps
assert norm(d, inf).ae(0)
mp.dps = 15
def test_qr():
mp.dps = 15 # used default value for dps
lowlimit = -9 # lower limit of matrix element value
uplimit = 9 # uppter limit of matrix element value
maxm = 4 # max matrix size
flg = False # toggle to create real vs complex matrix
zero = mpf('0.0')
for k in xrange(0,10):
exdps = 0
mode = 'full'
flg = bool(k % 2)
# generate arbitrary matrix size (2 to maxm)
num1 = nint(maxm*rand())
num2 = nint(maxm*rand())
m = int(max(num1, num2))
n = int(min(num1, num2))
# create matrix
A = mp.matrix(m,n)
# populate matrix values with arbitrary integers
if flg:
flg = False
dtype = 'complex'
for j in xrange(0,n):
for i in xrange(0,m):
val = nint(lowlimit + (uplimit-lowlimit)*rand())
val2 = nint(lowlimit + (uplimit-lowlimit)*rand())
A[i,j] = mpc(val, val2)
else:
flg = True
dtype = 'real'
for j in xrange(0,n):
for i in xrange(0,m):
val = nint(lowlimit + (uplimit-lowlimit)*rand())
A[i,j] = mpf(val)
# perform A -> QR decomposition
Q, R = qr(A, mode, edps = exdps)
#print('\n\n A = \n', nstr(A, 4))
#print('\n Q = \n', nstr(Q, 4))
#print('\n R = \n', nstr(R, 4))
#print('\n Q*R = \n', nstr(Q*R, 4))
maxnorm = mpf('1.0E-11')
n1 = norm(A - Q * R)
#print '\n Norm of A - Q * R = ', n1
assert n1 <= maxnorm
if dtype == 'real':
n1 = norm(eye(m) - Q.T * Q)
#print ' Norm of I - Q.T * Q = ', n1
assert n1 <= maxnorm
n1 = norm(eye(m) - Q * Q.T)
#print ' Norm of I - Q * Q.T = ', n1
assert n1 <= maxnorm
if dtype == 'complex':
n1 = norm(eye(m) - Q.T * Q.conjugate())
#print ' Norm of I - Q.T * Q.conjugate() = ', n1
assert n1 <= maxnorm
n1 = norm(eye(m) - Q.conjugate() * Q.T)
#print ' Norm of I - Q.conjugate() * Q.T = ', n1
assert n1 <= maxnorm
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