565 lines
20 KiB
Python
565 lines
20 KiB
Python
|
from sympy.core.function import expand_mul
|
||
|
|
from sympy.core.numbers import (I, Rational)
|
||
|
|
from sympy.core.singleton import S
|
||
|
|
from sympy.core.symbol import (Symbol, symbols)
|
||
|
|
from sympy.core.sympify import sympify
|
||
|
|
from sympy.simplify.simplify import simplify
|
||
|
|
from sympy.matrices.matrices import (ShapeError, NonSquareMatrixError)
|
||
|
|
from sympy.matrices import (
|
||
|
|
ImmutableMatrix, Matrix, eye, ones, ImmutableDenseMatrix, dotprodsimp)
|
||
|
|
from sympy.testing.pytest import raises
|
||
|
|
from sympy.matrices.common import NonInvertibleMatrixError
|
||
|
|
from sympy.solvers.solveset import linsolve
|
||
|
|
from sympy.abc import x, y
|
||
|
|
|
||
|
|
def test_issue_17247_expression_blowup_29():
|
||
|
|
M = Matrix(S('''[
|
||
|
|
[ -3/4, 45/32 - 37*I/16, 0, 0],
|
||
|
|
[-149/64 + 49*I/32, -177/128 - 1369*I/128, 0, -2063/256 + 541*I/128],
|
||
|
|
[ 0, 9/4 + 55*I/16, 2473/256 + 137*I/64, 0],
|
||
|
|
[ 0, 0, 0, -177/128 - 1369*I/128]]'''))
|
||
|
|
with dotprodsimp(True):
|
||
|
|
assert M.gauss_jordan_solve(ones(4, 1)) == (Matrix(S('''[
|
||
|
|
[ -32549314808672/3306971225785 - 17397006745216*I/3306971225785],
|
||
|
|
[ 67439348256/3306971225785 - 9167503335872*I/3306971225785],
|
||
|
|
[-15091965363354518272/21217636514687010905 + 16890163109293858304*I/21217636514687010905],
|
||
|
|
[ -11328/952745 + 87616*I/952745]]''')), Matrix(0, 1, []))
|
||
|
|
|
||
|
|
def test_issue_17247_expression_blowup_30():
|
||
|
|
M = Matrix(S('''[
|
||
|
|
[ -3/4, 45/32 - 37*I/16, 0, 0],
|
||
|
|
[-149/64 + 49*I/32, -177/128 - 1369*I/128, 0, -2063/256 + 541*I/128],
|
||
|
|
[ 0, 9/4 + 55*I/16, 2473/256 + 137*I/64, 0],
|
||
|
|
[ 0, 0, 0, -177/128 - 1369*I/128]]'''))
|
||
|
|
with dotprodsimp(True):
|
||
|
|
assert M.cholesky_solve(ones(4, 1)) == Matrix(S('''[
|
||
|
|
[ -32549314808672/3306971225785 - 17397006745216*I/3306971225785],
|
||
|
|
[ 67439348256/3306971225785 - 9167503335872*I/3306971225785],
|
||
|
|
[-15091965363354518272/21217636514687010905 + 16890163109293858304*I/21217636514687010905],
|
||
|
|
[ -11328/952745 + 87616*I/952745]]'''))
|
||
|
|
|
||
|
|
# @XFAIL # This calculation hangs with dotprodsimp.
|
||
|
|
# def test_issue_17247_expression_blowup_31():
|
||
|
|
# M = Matrix([
|
||
|
|
# [x + 1, 1 - x, 0, 0],
|
||
|
|
# [1 - x, x + 1, 0, x + 1],
|
||
|
|
# [ 0, 1 - x, x + 1, 0],
|
||
|
|
# [ 0, 0, 0, x + 1]])
|
||
|
|
# with dotprodsimp(True):
|
||
|
|
# assert M.LDLsolve(ones(4, 1)) == Matrix([
|
||
|
|
# [(x + 1)/(4*x)],
|
||
|
|
# [(x - 1)/(4*x)],
|
||
|
|
# [(x + 1)/(4*x)],
|
||
|
|
# [ 1/(x + 1)]])
|
||
|
|
|
||
|
|
def test_issue_17247_expression_blowup_32():
|
||
|
|
M = Matrix([
|
||
|
|
[x + 1, 1 - x, 0, 0],
|
||
|
|
[1 - x, x + 1, 0, x + 1],
|
||
|
|
[ 0, 1 - x, x + 1, 0],
|
||
|
|
[ 0, 0, 0, x + 1]])
|
||
|
|
with dotprodsimp(True):
|
||
|
|
assert M.LUsolve(ones(4, 1)) == Matrix([
|
||
|
|
[(x + 1)/(4*x)],
|
||
|
|
[(x - 1)/(4*x)],
|
||
|
|
[(x + 1)/(4*x)],
|
||
|
|
[ 1/(x + 1)]])
|
||
|
|
|
||
|
|
def test_LUsolve():
|
||
|
|
A = Matrix([[2, 3, 5],
|
||
|
|
[3, 6, 2],
|
||
|
|
[8, 3, 6]])
|
||
|
|
x = Matrix(3, 1, [3, 7, 5])
|
||
|
|
b = A*x
|
||
|
|
soln = A.LUsolve(b)
|
||
|
|
assert soln == x
|
||
|
|
A = Matrix([[0, -1, 2],
|
||
|
|
[5, 10, 7],
|
||
|
|
[8, 3, 4]])
|
||
|
|
x = Matrix(3, 1, [-1, 2, 5])
|
||
|
|
b = A*x
|
||
|
|
soln = A.LUsolve(b)
|
||
|
|
assert soln == x
|
||
|
|
A = Matrix([[2, 1], [1, 0], [1, 0]]) # issue 14548
|
||
|
|
b = Matrix([3, 1, 1])
|
||
|
|
assert A.LUsolve(b) == Matrix([1, 1])
|
||
|
|
b = Matrix([3, 1, 2]) # inconsistent
|
||
|
|
raises(ValueError, lambda: A.LUsolve(b))
|
||
|
|
A = Matrix([[0, -1, 2],
|
||
|
|
[5, 10, 7],
|
||
|
|
[8, 3, 4],
|
||
|
|
[2, 3, 5],
|
||
|
|
[3, 6, 2],
|
||
|
|
[8, 3, 6]])
|
||
|
|
x = Matrix([2, 1, -4])
|
||
|
|
b = A*x
|
||
|
|
soln = A.LUsolve(b)
|
||
|
|
assert soln == x
|
||
|
|
A = Matrix([[0, -1, 2], [5, 10, 7]]) # underdetermined
|
||
|
|
x = Matrix([-1, 2, 0])
|
||
|
|
b = A*x
|
||
|
|
raises(NotImplementedError, lambda: A.LUsolve(b))
|
||
|
|
|
||
|
|
A = Matrix(4, 4, lambda i, j: 1/(i+j+1) if i != 3 else 0)
|
||
|
|
b = Matrix.zeros(4, 1)
|
||
|
|
raises(NonInvertibleMatrixError, lambda: A.LUsolve(b))
|
||
|
|
|
||
|
|
|
||
|
|
def test_QRsolve():
|
||
|
|
A = Matrix([[2, 3, 5],
|
||
|
|
[3, 6, 2],
|
||
|
|
[8, 3, 6]])
|
||
|
|
x = Matrix(3, 1, [3, 7, 5])
|
||
|
|
b = A*x
|
||
|
|
soln = A.QRsolve(b)
|
||
|
|
assert soln == x
|
||
|
|
x = Matrix([[1, 2], [3, 4], [5, 6]])
|
||
|
|
b = A*x
|
||
|
|
soln = A.QRsolve(b)
|
||
|
|
assert soln == x
|
||
|
|
|
||
|
|
A = Matrix([[0, -1, 2],
|
||
|
|
[5, 10, 7],
|
||
|
|
[8, 3, 4]])
|
||
|
|
x = Matrix(3, 1, [-1, 2, 5])
|
||
|
|
b = A*x
|
||
|
|
soln = A.QRsolve(b)
|
||
|
|
assert soln == x
|
||
|
|
x = Matrix([[7, 8], [9, 10], [11, 12]])
|
||
|
|
b = A*x
|
||
|
|
soln = A.QRsolve(b)
|
||
|
|
assert soln == x
|
||
|
|
|
||
|
|
def test_errors():
|
||
|
|
raises(ShapeError, lambda: Matrix([1]).LUsolve(Matrix([[1, 2], [3, 4]])))
|
||
|
|
|
||
|
|
def test_cholesky_solve():
|
||
|
|
A = Matrix([[2, 3, 5],
|
||
|
|
[3, 6, 2],
|
||
|
|
[8, 3, 6]])
|
||
|
|
x = Matrix(3, 1, [3, 7, 5])
|
||
|
|
b = A*x
|
||
|
|
soln = A.cholesky_solve(b)
|
||
|
|
assert soln == x
|
||
|
|
A = Matrix([[0, -1, 2],
|
||
|
|
[5, 10, 7],
|
||
|
|
[8, 3, 4]])
|
||
|
|
x = Matrix(3, 1, [-1, 2, 5])
|
||
|
|
b = A*x
|
||
|
|
soln = A.cholesky_solve(b)
|
||
|
|
assert soln == x
|
||
|
|
A = Matrix(((1, 5), (5, 1)))
|
||
|
|
x = Matrix((4, -3))
|
||
|
|
b = A*x
|
||
|
|
soln = A.cholesky_solve(b)
|
||
|
|
assert soln == x
|
||
|
|
A = Matrix(((9, 3*I), (-3*I, 5)))
|
||
|
|
x = Matrix((-2, 1))
|
||
|
|
b = A*x
|
||
|
|
soln = A.cholesky_solve(b)
|
||
|
|
assert expand_mul(soln) == x
|
||
|
|
A = Matrix(((9*I, 3), (-3 + I, 5)))
|
||
|
|
x = Matrix((2 + 3*I, -1))
|
||
|
|
b = A*x
|
||
|
|
soln = A.cholesky_solve(b)
|
||
|
|
assert expand_mul(soln) == x
|
||
|
|
a00, a01, a11, b0, b1 = symbols('a00, a01, a11, b0, b1')
|
||
|
|
A = Matrix(((a00, a01), (a01, a11)))
|
||
|
|
b = Matrix((b0, b1))
|
||
|
|
x = A.cholesky_solve(b)
|
||
|
|
assert simplify(A*x) == b
|
||
|
|
|
||
|
|
|
||
|
|
def test_LDLsolve():
|
||
|
|
A = Matrix([[2, 3, 5],
|
||
|
|
[3, 6, 2],
|
||
|
|
[8, 3, 6]])
|
||
|
|
x = Matrix(3, 1, [3, 7, 5])
|
||
|
|
b = A*x
|
||
|
|
soln = A.LDLsolve(b)
|
||
|
|
assert soln == x
|
||
|
|
|
||
|
|
A = Matrix([[0, -1, 2],
|
||
|
|
[5, 10, 7],
|
||
|
|
[8, 3, 4]])
|
||
|
|
x = Matrix(3, 1, [-1, 2, 5])
|
||
|
|
b = A*x
|
||
|
|
soln = A.LDLsolve(b)
|
||
|
|
assert soln == x
|
||
|
|
|
||
|
|
A = Matrix(((9, 3*I), (-3*I, 5)))
|
||
|
|
x = Matrix((-2, 1))
|
||
|
|
b = A*x
|
||
|
|
soln = A.LDLsolve(b)
|
||
|
|
assert expand_mul(soln) == x
|
||
|
|
|
||
|
|
A = Matrix(((9*I, 3), (-3 + I, 5)))
|
||
|
|
x = Matrix((2 + 3*I, -1))
|
||
|
|
b = A*x
|
||
|
|
soln = A.LDLsolve(b)
|
||
|
|
assert expand_mul(soln) == x
|
||
|
|
|
||
|
|
A = Matrix(((9, 3), (3, 9)))
|
||
|
|
x = Matrix((1, 1))
|
||
|
|
b = A * x
|
||
|
|
soln = A.LDLsolve(b)
|
||
|
|
assert expand_mul(soln) == x
|
||
|
|
|
||
|
|
A = Matrix([[-5, -3, -4], [-3, -7, 7]])
|
||
|
|
x = Matrix([[8], [7], [-2]])
|
||
|
|
b = A * x
|
||
|
|
raises(NotImplementedError, lambda: A.LDLsolve(b))
|
||
|
|
|
||
|
|
|
||
|
|
def test_lower_triangular_solve():
|
||
|
|
|
||
|
|
raises(NonSquareMatrixError,
|
||
|
|
lambda: Matrix([1, 0]).lower_triangular_solve(Matrix([0, 1])))
|
||
|
|
raises(ShapeError,
|
||
|
|
lambda: Matrix([[1, 0], [0, 1]]).lower_triangular_solve(Matrix([1])))
|
||
|
|
raises(ValueError,
|
||
|
|
lambda: Matrix([[2, 1], [1, 2]]).lower_triangular_solve(
|
||
|
|
Matrix([[1, 0], [0, 1]])))
|
||
|
|
|
||
|
|
A = Matrix([[1, 0], [0, 1]])
|
||
|
|
B = Matrix([[x, y], [y, x]])
|
||
|
|
C = Matrix([[4, 8], [2, 9]])
|
||
|
|
|
||
|
|
assert A.lower_triangular_solve(B) == B
|
||
|
|
assert A.lower_triangular_solve(C) == C
|
||
|
|
|
||
|
|
|
||
|
|
def test_upper_triangular_solve():
|
||
|
|
|
||
|
|
raises(NonSquareMatrixError,
|
||
|
|
lambda: Matrix([1, 0]).upper_triangular_solve(Matrix([0, 1])))
|
||
|
|
raises(ShapeError,
|
||
|
|
lambda: Matrix([[1, 0], [0, 1]]).upper_triangular_solve(Matrix([1])))
|
||
|
|
raises(TypeError,
|
||
|
|
lambda: Matrix([[2, 1], [1, 2]]).upper_triangular_solve(
|
||
|
|
Matrix([[1, 0], [0, 1]])))
|
||
|
|
|
||
|
|
A = Matrix([[1, 0], [0, 1]])
|
||
|
|
B = Matrix([[x, y], [y, x]])
|
||
|
|
C = Matrix([[2, 4], [3, 8]])
|
||
|
|
|
||
|
|
assert A.upper_triangular_solve(B) == B
|
||
|
|
assert A.upper_triangular_solve(C) == C
|
||
|
|
|
||
|
|
|
||
|
|
def test_diagonal_solve():
|
||
|
|
raises(TypeError, lambda: Matrix([1, 1]).diagonal_solve(Matrix([1])))
|
||
|
|
A = Matrix([[1, 0], [0, 1]])*2
|
||
|
|
B = Matrix([[x, y], [y, x]])
|
||
|
|
assert A.diagonal_solve(B) == B/2
|
||
|
|
|
||
|
|
A = Matrix([[1, 0], [1, 2]])
|
||
|
|
raises(TypeError, lambda: A.diagonal_solve(B))
|
||
|
|
|
||
|
|
def test_pinv_solve():
|
||
|
|
# Fully determined system (unique result, identical to other solvers).
|
||
|
|
A = Matrix([[1, 5], [7, 9]])
|
||
|
|
B = Matrix([12, 13])
|
||
|
|
assert A.pinv_solve(B) == A.cholesky_solve(B)
|
||
|
|
assert A.pinv_solve(B) == A.LDLsolve(B)
|
||
|
|
assert A.pinv_solve(B) == Matrix([sympify('-43/26'), sympify('71/26')])
|
||
|
|
assert A * A.pinv() * B == B
|
||
|
|
# Fully determined, with two-dimensional B matrix.
|
||
|
|
B = Matrix([[12, 13, 14], [15, 16, 17]])
|
||
|
|
assert A.pinv_solve(B) == A.cholesky_solve(B)
|
||
|
|
assert A.pinv_solve(B) == A.LDLsolve(B)
|
||
|
|
assert A.pinv_solve(B) == Matrix([[-33, -37, -41], [69, 75, 81]]) / 26
|
||
|
|
assert A * A.pinv() * B == B
|
||
|
|
# Underdetermined system (infinite results).
|
||
|
|
A = Matrix([[1, 0, 1], [0, 1, 1]])
|
||
|
|
B = Matrix([5, 7])
|
||
|
|
solution = A.pinv_solve(B)
|
||
|
|
w = {}
|
||
|
|
for s in solution.atoms(Symbol):
|
||
|
|
# Extract dummy symbols used in the solution.
|
||
|
|
w[s.name] = s
|
||
|
|
assert solution == Matrix([[w['w0_0']/3 + w['w1_0']/3 - w['w2_0']/3 + 1],
|
||
|
|
[w['w0_0']/3 + w['w1_0']/3 - w['w2_0']/3 + 3],
|
||
|
|
[-w['w0_0']/3 - w['w1_0']/3 + w['w2_0']/3 + 4]])
|
||
|
|
assert A * A.pinv() * B == B
|
||
|
|
# Overdetermined system (least squares results).
|
||
|
|
A = Matrix([[1, 0], [0, 0], [0, 1]])
|
||
|
|
B = Matrix([3, 2, 1])
|
||
|
|
assert A.pinv_solve(B) == Matrix([3, 1])
|
||
|
|
# Proof the solution is not exact.
|
||
|
|
assert A * A.pinv() * B != B
|
||
|
|
|
||
|
|
def test_pinv_rank_deficient():
|
||
|
|
# Test the four properties of the pseudoinverse for various matrices.
|
||
|
|
As = [Matrix([[1, 1, 1], [2, 2, 2]]),
|
||
|
|
Matrix([[1, 0], [0, 0]]),
|
||
|
|
Matrix([[1, 2], [2, 4], [3, 6]])]
|
||
|
|
|
||
|
|
for A in As:
|
||
|
|
A_pinv = A.pinv(method="RD")
|
||
|
|
AAp = A * A_pinv
|
||
|
|
ApA = A_pinv * A
|
||
|
|
assert simplify(AAp * A) == A
|
||
|
|
assert simplify(ApA * A_pinv) == A_pinv
|
||
|
|
assert AAp.H == AAp
|
||
|
|
assert ApA.H == ApA
|
||
|
|
|
||
|
|
for A in As:
|
||
|
|
A_pinv = A.pinv(method="ED")
|
||
|
|
AAp = A * A_pinv
|
||
|
|
ApA = A_pinv * A
|
||
|
|
assert simplify(AAp * A) == A
|
||
|
|
assert simplify(ApA * A_pinv) == A_pinv
|
||
|
|
assert AAp.H == AAp
|
||
|
|
assert ApA.H == ApA
|
||
|
|
|
||
|
|
# Test solving with rank-deficient matrices.
|
||
|
|
A = Matrix([[1, 0], [0, 0]])
|
||
|
|
# Exact, non-unique solution.
|
||
|
|
B = Matrix([3, 0])
|
||
|
|
solution = A.pinv_solve(B)
|
||
|
|
w1 = solution.atoms(Symbol).pop()
|
||
|
|
assert w1.name == 'w1_0'
|
||
|
|
assert solution == Matrix([3, w1])
|
||
|
|
assert A * A.pinv() * B == B
|
||
|
|
# Least squares, non-unique solution.
|
||
|
|
B = Matrix([3, 1])
|
||
|
|
solution = A.pinv_solve(B)
|
||
|
|
w1 = solution.atoms(Symbol).pop()
|
||
|
|
assert w1.name == 'w1_0'
|
||
|
|
assert solution == Matrix([3, w1])
|
||
|
|
assert A * A.pinv() * B != B
|
||
|
|
|
||
|
|
def test_gauss_jordan_solve():
|
||
|
|
|
||
|
|
# Square, full rank, unique solution
|
||
|
|
A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 10]])
|
||
|
|
b = Matrix([3, 6, 9])
|
||
|
|
sol, params = A.gauss_jordan_solve(b)
|
||
|
|
assert sol == Matrix([[-1], [2], [0]])
|
||
|
|
assert params == Matrix(0, 1, [])
|
||
|
|
|
||
|
|
# Square, full rank, unique solution, B has more columns than rows
|
||
|
|
A = eye(3)
|
||
|
|
B = Matrix([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]])
|
||
|
|
sol, params = A.gauss_jordan_solve(B)
|
||
|
|
assert sol == B
|
||
|
|
assert params == Matrix(0, 4, [])
|
||
|
|
|
||
|
|
# Square, reduced rank, parametrized solution
|
||
|
|
A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
|
||
|
|
b = Matrix([3, 6, 9])
|
||
|
|
sol, params, freevar = A.gauss_jordan_solve(b, freevar=True)
|
||
|
|
w = {}
|
||
|
|
for s in sol.atoms(Symbol):
|
||
|
|
# Extract dummy symbols used in the solution.
|
||
|
|
w[s.name] = s
|
||
|
|
assert sol == Matrix([[w['tau0'] - 1], [-2*w['tau0'] + 2], [w['tau0']]])
|
||
|
|
assert params == Matrix([[w['tau0']]])
|
||
|
|
assert freevar == [2]
|
||
|
|
|
||
|
|
# Square, reduced rank, parametrized solution, B has two columns
|
||
|
|
A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
|
||
|
|
B = Matrix([[3, 4], [6, 8], [9, 12]])
|
||
|
|
sol, params, freevar = A.gauss_jordan_solve(B, freevar=True)
|
||
|
|
w = {}
|
||
|
|
for s in sol.atoms(Symbol):
|
||
|
|
# Extract dummy symbols used in the solution.
|
||
|
|
w[s.name] = s
|
||
|
|
assert sol == Matrix([[w['tau0'] - 1, w['tau1'] - Rational(4, 3)],
|
||
|
|
[-2*w['tau0'] + 2, -2*w['tau1'] + Rational(8, 3)],
|
||
|
|
[w['tau0'], w['tau1']],])
|
||
|
|
assert params == Matrix([[w['tau0'], w['tau1']]])
|
||
|
|
assert freevar == [2]
|
||
|
|
|
||
|
|
# Square, reduced rank, parametrized solution
|
||
|
|
A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
|
||
|
|
b = Matrix([0, 0, 0])
|
||
|
|
sol, params = A.gauss_jordan_solve(b)
|
||
|
|
w = {}
|
||
|
|
for s in sol.atoms(Symbol):
|
||
|
|
w[s.name] = s
|
||
|
|
assert sol == Matrix([[-2*w['tau0'] - 3*w['tau1']],
|
||
|
|
[w['tau0']], [w['tau1']]])
|
||
|
|
assert params == Matrix([[w['tau0']], [w['tau1']]])
|
||
|
|
|
||
|
|
# Square, reduced rank, parametrized solution
|
||
|
|
A = Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0]])
|
||
|
|
b = Matrix([0, 0, 0])
|
||
|
|
sol, params = A.gauss_jordan_solve(b)
|
||
|
|
w = {}
|
||
|
|
for s in sol.atoms(Symbol):
|
||
|
|
w[s.name] = s
|
||
|
|
assert sol == Matrix([[w['tau0']], [w['tau1']], [w['tau2']]])
|
||
|
|
assert params == Matrix([[w['tau0']], [w['tau1']], [w['tau2']]])
|
||
|
|
|
||
|
|
# Square, reduced rank, no solution
|
||
|
|
A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
|
||
|
|
b = Matrix([0, 0, 1])
|
||
|
|
raises(ValueError, lambda: A.gauss_jordan_solve(b))
|
||
|
|
|
||
|
|
# Rectangular, tall, full rank, unique solution
|
||
|
|
A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
|
||
|
|
b = Matrix([0, 0, 1, 0])
|
||
|
|
sol, params = A.gauss_jordan_solve(b)
|
||
|
|
assert sol == Matrix([[Rational(-1, 2)], [0], [Rational(1, 6)]])
|
||
|
|
assert params == Matrix(0, 1, [])
|
||
|
|
|
||
|
|
# Rectangular, tall, full rank, unique solution, B has less columns than rows
|
||
|
|
A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
|
||
|
|
B = Matrix([[0,0], [0, 0], [1, 2], [0, 0]])
|
||
|
|
sol, params = A.gauss_jordan_solve(B)
|
||
|
|
assert sol == Matrix([[Rational(-1, 2), Rational(-2, 2)], [0, 0], [Rational(1, 6), Rational(2, 6)]])
|
||
|
|
assert params == Matrix(0, 2, [])
|
||
|
|
|
||
|
|
# Rectangular, tall, full rank, no solution
|
||
|
|
A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
|
||
|
|
b = Matrix([0, 0, 0, 1])
|
||
|
|
raises(ValueError, lambda: A.gauss_jordan_solve(b))
|
||
|
|
|
||
|
|
# Rectangular, tall, full rank, no solution, B has two columns (2nd has no solution)
|
||
|
|
A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
|
||
|
|
B = Matrix([[0,0], [0, 0], [1, 0], [0, 1]])
|
||
|
|
raises(ValueError, lambda: A.gauss_jordan_solve(B))
|
||
|
|
|
||
|
|
# Rectangular, tall, full rank, no solution, B has two columns (1st has no solution)
|
||
|
|
A = Matrix([[1, 5, 3], [2, 1, 6], [1, 7, 9], [1, 4, 3]])
|
||
|
|
B = Matrix([[0,0], [0, 0], [0, 1], [1, 0]])
|
||
|
|
raises(ValueError, lambda: A.gauss_jordan_solve(B))
|
||
|
|
|
||
|
|
# Rectangular, tall, reduced rank, parametrized solution
|
||
|
|
A = Matrix([[1, 5, 3], [2, 10, 6], [3, 15, 9], [1, 4, 3]])
|
||
|
|
b = Matrix([0, 0, 0, 1])
|
||
|
|
sol, params = A.gauss_jordan_solve(b)
|
||
|
|
w = {}
|
||
|
|
for s in sol.atoms(Symbol):
|
||
|
|
w[s.name] = s
|
||
|
|
assert sol == Matrix([[-3*w['tau0'] + 5], [-1], [w['tau0']]])
|
||
|
|
assert params == Matrix([[w['tau0']]])
|
||
|
|
|
||
|
|
# Rectangular, tall, reduced rank, no solution
|
||
|
|
A = Matrix([[1, 5, 3], [2, 10, 6], [3, 15, 9], [1, 4, 3]])
|
||
|
|
b = Matrix([0, 0, 1, 1])
|
||
|
|
raises(ValueError, lambda: A.gauss_jordan_solve(b))
|
||
|
|
|
||
|
|
# Rectangular, wide, full rank, parametrized solution
|
||
|
|
A = Matrix([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 1, 12]])
|
||
|
|
b = Matrix([1, 1, 1])
|
||
|
|
sol, params = A.gauss_jordan_solve(b)
|
||
|
|
w = {}
|
||
|
|
for s in sol.atoms(Symbol):
|
||
|
|
w[s.name] = s
|
||
|
|
assert sol == Matrix([[2*w['tau0'] - 1], [-3*w['tau0'] + 1], [0],
|
||
|
|
[w['tau0']]])
|
||
|
|
assert params == Matrix([[w['tau0']]])
|
||
|
|
|
||
|
|
# Rectangular, wide, reduced rank, parametrized solution
|
||
|
|
A = Matrix([[1, 2, 3, 4], [5, 6, 7, 8], [2, 4, 6, 8]])
|
||
|
|
b = Matrix([0, 1, 0])
|
||
|
|
sol, params = A.gauss_jordan_solve(b)
|
||
|
|
w = {}
|
||
|
|
for s in sol.atoms(Symbol):
|
||
|
|
w[s.name] = s
|
||
|
|
assert sol == Matrix([[w['tau0'] + 2*w['tau1'] + S.Half],
|
||
|
|
[-2*w['tau0'] - 3*w['tau1'] - Rational(1, 4)],
|
||
|
|
[w['tau0']], [w['tau1']]])
|
||
|
|
assert params == Matrix([[w['tau0']], [w['tau1']]])
|
||
|
|
# watch out for clashing symbols
|
||
|
|
x0, x1, x2, _x0 = symbols('_tau0 _tau1 _tau2 tau1')
|
||
|
|
M = Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
|
||
|
|
A = M[:, :-1]
|
||
|
|
b = M[:, -1:]
|
||
|
|
sol, params = A.gauss_jordan_solve(b)
|
||
|
|
assert params == Matrix(3, 1, [x0, x1, x2])
|
||
|
|
assert sol == Matrix(5, 1, [x0, 0, x1, _x0, x2])
|
||
|
|
|
||
|
|
# Rectangular, wide, reduced rank, no solution
|
||
|
|
A = Matrix([[1, 2, 3, 4], [5, 6, 7, 8], [2, 4, 6, 8]])
|
||
|
|
b = Matrix([1, 1, 1])
|
||
|
|
raises(ValueError, lambda: A.gauss_jordan_solve(b))
|
||
|
|
|
||
|
|
# Test for immutable matrix
|
||
|
|
A = ImmutableMatrix([[1, 0], [0, 1]])
|
||
|
|
B = ImmutableMatrix([1, 2])
|
||
|
|
sol, params = A.gauss_jordan_solve(B)
|
||
|
|
assert sol == ImmutableMatrix([1, 2])
|
||
|
|
assert params == ImmutableMatrix(0, 1, [])
|
||
|
|
assert sol.__class__ == ImmutableDenseMatrix
|
||
|
|
assert params.__class__ == ImmutableDenseMatrix
|
||
|
|
|
||
|
|
# Test placement of free variables
|
||
|
|
A = Matrix([[1, 0, 0, 0], [0, 0, 0, 1]])
|
||
|
|
b = Matrix([1, 1])
|
||
|
|
sol, params = A.gauss_jordan_solve(b)
|
||
|
|
w = {}
|
||
|
|
for s in sol.atoms(Symbol):
|
||
|
|
w[s.name] = s
|
||
|
|
assert sol == Matrix([[1], [w['tau0']], [w['tau1']], [1]])
|
||
|
|
assert params == Matrix([[w['tau0']], [w['tau1']]])
|
||
|
|
|
||
|
|
|
||
|
|
def test_linsolve_underdetermined_AND_gauss_jordan_solve():
|
||
|
|
#Test placement of free variables as per issue 19815
|
||
|
|
A = Matrix([[1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
|
||
|
|
[1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0],
|
||
|
|
[0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
|
||
|
|
[0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0],
|
||
|
|
[0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0],
|
||
|
|
[0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0],
|
||
|
|
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0],
|
||
|
|
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1]])
|
||
|
|
B = Matrix([1, 2, 1, 1, 1, 1, 1, 2])
|
||
|
|
sol, params = A.gauss_jordan_solve(B)
|
||
|
|
w = {}
|
||
|
|
for s in sol.atoms(Symbol):
|
||
|
|
w[s.name] = s
|
||
|
|
assert params == Matrix([[w['tau0']], [w['tau1']], [w['tau2']],
|
||
|
|
[w['tau3']], [w['tau4']], [w['tau5']]])
|
||
|
|
assert sol == Matrix([[1 - 1*w['tau2']],
|
||
|
|
[w['tau2']],
|
||
|
|
[1 - 1*w['tau0'] + w['tau1']],
|
||
|
|
[w['tau0']],
|
||
|
|
[w['tau3'] + w['tau4']],
|
||
|
|
[-1*w['tau3'] - 1*w['tau4'] - 1*w['tau1']],
|
||
|
|
[1 - 1*w['tau2']],
|
||
|
|
[w['tau1']],
|
||
|
|
[w['tau2']],
|
||
|
|
[w['tau3']],
|
||
|
|
[w['tau4']],
|
||
|
|
[1 - 1*w['tau5']],
|
||
|
|
[w['tau5']],
|
||
|
|
[1]])
|
||
|
|
|
||
|
|
from sympy.abc import j,f
|
||
|
|
# https://github.com/sympy/sympy/issues/20046
|
||
|
|
A = Matrix([
|
||
|
|
[1, 1, 1, 1, 1, 1, 1, 1, 1],
|
||
|
|
[0, -1, 0, -1, 0, -1, 0, -1, -j],
|
||
|
|
[0, 0, 0, 0, 1, 1, 1, 1, f]
|
||
|
|
])
|
||
|
|
|
||
|
|
sol_1=Matrix(list(linsolve(A))[0])
|
||
|
|
|
||
|
|
tau0, tau1, tau2, tau3, tau4 = symbols('tau:5')
|
||
|
|
|
||
|
|
assert sol_1 == Matrix([[-f - j - tau0 + tau2 + tau4 + 1],
|
||
|
|
[j - tau1 - tau2 - tau4],
|
||
|
|
[tau0],
|
||
|
|
[tau1],
|
||
|
|
[f - tau2 - tau3 - tau4],
|
||
|
|
[tau2],
|
||
|
|
[tau3],
|
||
|
|
[tau4]])
|
||
|
|
|
||
|
|
# https://github.com/sympy/sympy/issues/19815
|
||
|
|
sol_2 = A[:, : -1 ] * sol_1 - A[:, -1 ]
|
||
|
|
assert sol_2 == Matrix([[0], [0], [0]])
|
||
|
|
|
||
|
|
|
||
|
|
def test_solve():
|
||
|
|
A = Matrix([[1,2], [2,4]])
|
||
|
|
b = Matrix([[3], [4]])
|
||
|
|
raises(ValueError, lambda: A.solve(b)) #no solution
|
||
|
|
b = Matrix([[ 4], [8]])
|
||
|
|
raises(ValueError, lambda: A.solve(b)) #infinite solution
|