def test_matrix_numpy():
try:
import numpy
except ImportError:
return
l = [[1, 2], [3, 4], [5, 6]]
a = numpy.matrix(l)
assert matrix(l) == matrix(a)
def test_matrix_transform():
A = matrix([[1, 2], [3, 4], [5, 6]])
assert A.T == A.transpose() == matrix([[1, 3, 5], [2, 4, 6]])
swap_row(A, 1, 2)
assert A == matrix([[1, 2], [5, 6], [3, 4]])
l = [1, 2]
swap_row(l, 0, 1)
assert l == [2, 1]
assert extend(eye(3), [1,2,3]) == matrix([[1,0,0,1],[0,1,0,2],[0,0,1,3]])
def test_matrix_transform():
A = matrix([[1, 2], [3, 4], [5, 6]])
assert A.T == A.transpose() == matrix([[1, 3, 5], [2, 4, 6]])
swap_row(A, 1, 2)
assert A == matrix([[1, 2], [5, 6], [3, 4]])
l = [1, 2]
swap_row(l, 0, 1)
assert l == [2, 1]
assert extend(eye(3), [1, 2, 3]) == matrix([[1, 0, 0, 1], [0, 1, 0, 2],
[0, 0, 1, 3]])
def test_vector():
x = matrix([0, 1, 2, 3, 4])
assert x == matrix([[0], [1], [2], [3], [4]])
assert x[3] == 3
assert len(x._matrix__data) == 4
assert list(x) == range(5)
x[0] = -10
x[4] = 0
assert x[0] == -10
assert len(x) == len(x.T) == 5
assert x.T*x == matrix([[114]])
def test_vector():
x = matrix([0, 1, 2, 3, 4])
assert x == matrix([[0], [1], [2], [3], [4]])
assert x[3] == 3
assert len(x._matrix__data) == 4
assert list(x) == range(5)
x[0] = -10
x[4] = 0
assert x[0] == -10
assert len(x) == len(x.T) == 5
assert x.T * x == matrix([[114]])
def test_interval_matrix():
a = matrix([['0.1','0.3','1.0'],['7.1','5.5','4.8'],['3.2','4.4','5.6']],
force_type=mpi)
b = matrix(['4','0.6','0.5'], force_type=mpi)
c = 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_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_interval_matrix():
a = matrix(
[['0.1', '0.3', '1.0'], ['7.1', '5.5', '4.8'], ['3.2', '4.4', '5.6']],
force_type=mpi)
b = matrix(['4', '0.6', '0.5'], force_type=mpi)
c = 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_matrix_creation():
assert diag([1, 2, 3]) == matrix([[1, 0, 0], [0, 2, 0], [0, 0, 3]])
A1 = ones(2, 3)
assert A1.rows == 2 and A1.cols == 3
for a in A1:
assert a == 1
A2 = zeros(3, 2)
assert A2.rows == 3 and A2.cols == 2
for a in A2:
assert a == 0
assert randmatrix(10) != randmatrix(10)
one = mpf(1)
assert hilbert(3) == matrix([[one, one / 2, one / 3], [one / 2, one / 3, one / 4], [one / 3, one / 4, one / 5]])
def test_matrix_creation():
assert diag([1, 2, 3]) == matrix([[1, 0, 0], [0, 2, 0], [0, 0, 3]])
A1 = ones(2, 3)
assert A1.rows == 2 and A1.cols == 3
for a in A1:
assert a == 1
A2 = zeros(3, 2)
assert A2.rows == 3 and A2.cols == 2
for a in A2:
assert a == 0
assert randmatrix(10) != randmatrix(10)
def test_matrix_creation():
assert diag([1, 2, 3]) == matrix([[1, 0, 0], [0, 2, 0], [0, 0, 3]])
A1 = ones(2, 3)
assert A1.rows == 2 and A1.cols == 3
for a in A1:
assert a == 1
A2 = zeros(3, 2)
assert A2.rows == 3 and A2.cols == 2
for a in A2:
assert a == 0
assert randmatrix(10) != randmatrix(10)
def test_norms():
# matrix norms
A = matrix([[1, -2], [-3, -1], [2, 1]])
assert mnorm_1(A) == 6
assert mnorm_oo(A) == 4
assert mnorm_F(A) == sqrt(20)
# vector norms
x = [1, -2, 7, -12]
assert norm_p(x, 1) == 22
assert round(norm_p(x, 2), 10) == 14.0712472795
assert round(norm_p(x, 10), 10) == 12.0054633727
assert norm_p(x, inf) == 12
def test_norms():
# matrix norms
A = matrix([[1, -2], [-3, -1], [2, 1]])
assert mnorm_1(A) == 6
assert mnorm_oo(A) == 4
assert mnorm_F(A) == sqrt(20)
# vector norms
x = [1, -2, 7, -12]
assert norm_p(x, 1) == 22
assert round(norm_p(x, 2), 10) == 14.0712472795
assert round(norm_p(x, 10), 10) == 12.0054633727
assert norm_p(x, inf) == 12
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
# TODO: don't use round
from __future__ import division
from sympy.mpmath.matrices import matrix, norm_p, mnorm_1, mnorm_oo, mnorm_F, \
randmatrix, eye, zeros
from sympy.mpmath.linalg import * # TODO: absolute imports
from sympy.mpmath.mptypes import *
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],
def test_matrix_power():
A = matrix([[1, 2], [3, 4]])
assert A**2 == A*A
assert A**3 == A*A*A
assert A**-1 == inverse(A)
assert A**-2 == inverse(A*A)
def test_matrix_basic():
A1 = matrix(3)
for i in xrange(3):
A1[i, i] = 1
assert A1 == eye(3)
assert A1 == matrix(A1)
A2 = matrix(3, 2)
assert not A2._matrix__data
A3 = matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
assert list(A3) == range(1, 10)
A3[1, 1] = 0
assert not (1, 1) in A3._matrix__data
A4 = matrix([[1, 2, 3], [4, 5, 6]])
A5 = matrix([[6, -1], [3, 2], [0, -3]])
assert A4 * A5 == matrix([[12, -6], [39, -12]])
assert A1 * A3 == A3 * A1 == A3
try:
A2 * A2
assert False
except ValueError:
pass
l = [[10, 20, 30], [40, 0, 60], [70, 80, 90]]
A6 = matrix(l)
assert A6.tolist() == l
assert A6 == eval(repr(A6))
A6 = matrix(A6, force_type=float)
assert A6 == eval(repr(A6))
assert A3 * 10 == 10 * A3 == A6
assert A2.rows == 3
assert A2.cols == 2
A3.rows = 2
A3.cols = 2
assert len(A3._matrix__data) == 3
assert A4 + A4 == 2 * A4
try:
A4 + A2
except ValueError:
pass
assert sum(A1 - A1) == 0
A7 = matrix([[1, 2], [3, 4], [5, 6], [7, 8]])
x = matrix([10, -10])
assert A7 * x == matrix([-10, -10, -10, -10])
A8 = ones(5)
assert sum((A8 + 1) - (2 - zeros(5))) == 0
assert (1 + ones(4)) / 2 - 1 == zeros(4)
assert eye(3)**10 == eye(3)
try:
A7**2
assert False
except ValueError:
pass
A9 = randmatrix(3)
A10 = matrix(A9)
A9[0, 0] = -100
assert A9 != A10
A11 = matrix(randmatrix(2, 3), force_type=mpi)
for a in A11:
assert isinstance(a, mpi)
def test_matrix_numpy():
l = [[1, 2], [3, 4], [5, 6]]
a = numpy.matrix(l)
assert matrix(l) == matrix(a)
def test_matrix_basic():
A1 = matrix(3)
for i in xrange(3):
A1[i,i] = 1
assert A1 == eye(3)
assert A1 == matrix(A1)
A2 = matrix(3, 2)
assert not A2._matrix__data
A3 = matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
assert list(A3) == range(1, 10)
A3[1,1] = 0
assert not (1, 1) in A3._matrix__data
A4 = matrix([[1, 2, 3], [4, 5, 6]])
A5 = matrix([[6, -1], [3, 2], [0, -3]])
assert A4 * A5 == matrix([[12, -6], [39, -12]])
assert A1 * A3 == A3 * A1 == A3
try:
A2 * A2
assert False
except ValueError:
pass
l = [[10, 20, 30], [40, 0, 60], [70, 80, 90]]
A6 = matrix(l)
assert A6.tolist() == l
assert A6 == eval(repr(A6))
A6 = matrix(A6, force_type=float)
assert A6 == eval(repr(A6))
assert A6*1j == eval(repr(A6*1j))
assert A3 * 10 == 10 * A3 == A6
assert A2.rows == 3
assert A2.cols == 2
A3.rows = 2
A3.cols = 2
assert len(A3._matrix__data) == 3
assert A4 + A4 == 2*A4
try:
A4 + A2
except ValueError:
pass
assert sum(A1 - A1) == 0
A7 = matrix([[1, 2], [3, 4], [5, 6], [7, 8]])
x = matrix([10, -10])
assert A7*x == matrix([-10, -10, -10, -10])
A8 = ones(5)
assert sum((A8 + 1) - (2 - zeros(5))) == 0
assert (1 + ones(4)) / 2 - 1 == zeros(4)
assert eye(3)**10 == eye(3)
try:
A7**2
assert False
except ValueError:
pass
A9 = randmatrix(3)
A10 = matrix(A9)
A9[0,0] = -100
assert A9 != A10
A11 = matrix(randmatrix(2, 3), force_type=mpi)
for a in A11:
assert isinstance(a, mpi)
assert nstr(A9)
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')
def test_matrix_power():
A = matrix([[1, 2], [3, 4]])
assert A**2 == A * A
assert A**3 == A * A * A
assert A**-1 == inverse(A)
assert A**-2 == inverse(A * A)
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)
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')
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_matrix_numpy():
l = [[1, 2], [3, 4], [5, 6]]
a = numpy.matrix(l)
assert matrix(l) == matrix(a)
def test_cond():
A = matrix([[1.2969, 0.8648], [0.2161, 0.1441]])
assert cond(A, mnorm_1) == mpf('327065209.73817754')
assert cond(A, mnorm_oo) == mpf('327065209.73817748')
assert cond(A, mnorm_F) == mpf('249729266.80008656')
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_p(residual(A9, x, b9), inf) == 0
# TODO: don't use round
from __future__ import division
from sympy.mpmath.matrices import matrix, norm, mnorm, randmatrix, eye, zeros, diag
from sympy.mpmath.linalg import * # TODO: absolute imports
from sympy.mpmath.mptypes import *
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]
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)
def test_cond():
A = matrix([[1.2969, 0.8648], [0.2161, 0.1441]])
assert cond(A, mnorm_1) == mpf('327065209.73817754')
assert cond(A, mnorm_oo) == mpf('327065209.73817748')
assert cond(A, mnorm_F) == mpf('249729266.80008656')