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Mini Shell
Mini Shell
"""
Some examples have been taken from:
http://www.math.uwaterloo.ca/~hwolkowi//matrixcookbook.pdf
"""
from sympy import (MatrixSymbol, Inverse, symbols, Determinant, Trace,
sin, exp, cos, tan, log, S, sqrt,
hadamard_product, DiagMatrix, OneMatrix,
HadamardProduct, HadamardPower, KroneckerDelta, Sum,
Rational)
from sympy import MatAdd, Identity, MatMul, ZeroMatrix
from sympy.tensor.array.array_derivatives import ArrayDerivative
from sympy.matrices.expressions import hadamard_power
k = symbols("k")
i, j = symbols("i j")
m, n = symbols("m n")
X = MatrixSymbol("X", k, k)
x = MatrixSymbol("x", k, 1)
y = MatrixSymbol("y", k, 1)
A = MatrixSymbol("A", k, k)
B = MatrixSymbol("B", k, k)
C = MatrixSymbol("C", k, k)
D = MatrixSymbol("D", k, k)
a = MatrixSymbol("a", k, 1)
b = MatrixSymbol("b", k, 1)
c = MatrixSymbol("c", k, 1)
d = MatrixSymbol("d", k, 1)
KDelta = lambda i, j: KroneckerDelta(i, j, (0, k-1))
def _check_derivative_with_explicit_matrix(expr, x, diffexpr, dim=2):
# TODO: this is commented because it slows down the tests.
return
expr = expr.xreplace({k: dim})
x = x.xreplace({k: dim})
diffexpr = diffexpr.xreplace({k: dim})
expr = expr.as_explicit()
x = x.as_explicit()
diffexpr = diffexpr.as_explicit()
assert expr.diff(x).reshape(*diffexpr.shape).tomatrix() == diffexpr
def test_matrix_derivative_by_scalar():
assert A.diff(i) == ZeroMatrix(k, k)
assert (A*(X + B)*c).diff(i) == ZeroMatrix(k, 1)
assert x.diff(i) == ZeroMatrix(k, 1)
assert (x.T*y).diff(i) == ZeroMatrix(1, 1)
assert (x*x.T).diff(i) == ZeroMatrix(k, k)
assert (x + y).diff(i) == ZeroMatrix(k, 1)
assert hadamard_power(x, 2).diff(i) == ZeroMatrix(k, 1)
assert hadamard_power(x, i).diff(i).dummy_eq(
HadamardProduct(x.applyfunc(log), HadamardPower(x, i)))
assert hadamard_product(x, y).diff(i) == ZeroMatrix(k, 1)
assert hadamard_product(i*OneMatrix(k, 1), x, y).diff(i) == hadamard_product(x, y)
assert (i*x).diff(i) == x
assert (sin(i)*A*B*x).diff(i) == cos(i)*A*B*x
assert x.applyfunc(sin).diff(i) == ZeroMatrix(k, 1)
assert Trace(i**2*X).diff(i) == 2*i*Trace(X)
mu = symbols("mu")
expr = (2*mu*x)
assert expr.diff(x) == 2*mu*Identity(k)
def test_matrix_derivative_non_matrix_result():
# This is a 4-dimensional array:
assert A.diff(A) == ArrayDerivative(A, A)
assert A.T.diff(A) == ArrayDerivative(A.T, A)
assert (2*A).diff(A) == ArrayDerivative(2*A, A)
assert MatAdd(A, A).diff(A) == ArrayDerivative(MatAdd(A, A), A)
assert (A + B).diff(A) == ArrayDerivative(A + B, A) # TODO: `B` can be removed.
def test_matrix_derivative_trivial_cases():
# Cookbook example 33:
# TODO: find a way to represent a four-dimensional zero-array:
assert X.diff(A) == ArrayDerivative(X, A)
def test_matrix_derivative_with_inverse():
# Cookbook example 61:
expr = a.T*Inverse(X)*b
assert expr.diff(X) == -Inverse(X).T*a*b.T*Inverse(X).T
# Cookbook example 62:
expr = Determinant(Inverse(X))
# Not implemented yet:
# assert expr.diff(X) == -Determinant(X.inv())*(X.inv()).T
# Cookbook example 63:
expr = Trace(A*Inverse(X)*B)
assert expr.diff(X) == -(X**(-1)*B*A*X**(-1)).T
# Cookbook example 64:
expr = Trace(Inverse(X + A))
assert expr.diff(X) == -(Inverse(X + A)).T**2
def test_matrix_derivative_vectors_and_scalars():
assert x.diff(x) == Identity(k)
assert x[i, 0].diff(x[m, 0]).doit() == KDelta(m, i)
assert x.T.diff(x) == Identity(k)
# Cookbook example 69:
expr = x.T*a
assert expr.diff(x) == a
assert expr[0, 0].diff(x[m, 0]).doit() == a[m, 0]
expr = a.T*x
assert expr.diff(x) == a
# Cookbook example 70:
expr = a.T*X*b
assert expr.diff(X) == a*b.T
# Cookbook example 71:
expr = a.T*X.T*b
assert expr.diff(X) == b*a.T
# Cookbook example 72:
expr = a.T*X*a
assert expr.diff(X) == a*a.T
expr = a.T*X.T*a
assert expr.diff(X) == a*a.T
# Cookbook example 77:
expr = b.T*X.T*X*c
assert expr.diff(X) == X*b*c.T + X*c*b.T
# Cookbook example 78:
expr = (B*x + b).T*C*(D*x + d)
assert expr.diff(x) == B.T*C*(D*x + d) + D.T*C.T*(B*x + b)
# Cookbook example 81:
expr = x.T*B*x
assert expr.diff(x) == B*x + B.T*x
# Cookbook example 82:
expr = b.T*X.T*D*X*c
assert expr.diff(X) == D.T*X*b*c.T + D*X*c*b.T
# Cookbook example 83:
expr = (X*b + c).T*D*(X*b + c)
assert expr.diff(X) == D*(X*b + c)*b.T + D.T*(X*b + c)*b.T
assert str(expr[0, 0].diff(X[m, n]).doit()) == \
'b[n, 0]*Sum((c[_i_1, 0] + Sum(X[_i_1, _i_3]*b[_i_3, 0], (_i_3, 0, k - 1)))*D[_i_1, m], (_i_1, 0, k - 1)) + Sum((c[_i_2, 0] + Sum(X[_i_2, _i_4]*b[_i_4, 0], (_i_4, 0, k - 1)))*D[m, _i_2]*b[n, 0], (_i_2, 0, k - 1))'
def test_matrix_derivatives_of_traces():
expr = Trace(A)*A
assert expr.diff(A) == ArrayDerivative(Trace(A)*A, A)
assert expr[i, j].diff(A[m, n]).doit() == (
KDelta(i, m)*KDelta(j, n)*Trace(A) +
KDelta(m, n)*A[i, j]
)
## First order:
# Cookbook example 99:
expr = Trace(X)
assert expr.diff(X) == Identity(k)
assert expr.rewrite(Sum).diff(X[m, n]).doit() == KDelta(m, n)
# Cookbook example 100:
expr = Trace(X*A)
assert expr.diff(X) == A.T
assert expr.rewrite(Sum).diff(X[m, n]).doit() == A[n, m]
# Cookbook example 101:
expr = Trace(A*X*B)
assert expr.diff(X) == A.T*B.T
assert expr.rewrite(Sum).diff(X[m, n]).doit().dummy_eq((A.T*B.T)[m, n])
# Cookbook example 102:
expr = Trace(A*X.T*B)
assert expr.diff(X) == B*A
# Cookbook example 103:
expr = Trace(X.T*A)
assert expr.diff(X) == A
# Cookbook example 104:
expr = Trace(A*X.T)
assert expr.diff(X) == A
# Cookbook example 105:
# TODO: TensorProduct is not supported
#expr = Trace(TensorProduct(A, X))
#assert expr.diff(X) == Trace(A)*Identity(k)
## Second order:
# Cookbook example 106:
expr = Trace(X**2)
assert expr.diff(X) == 2*X.T
# Cookbook example 107:
expr = Trace(X**2*B)
assert expr.diff(X) == (X*B + B*X).T
expr = Trace(MatMul(X, X, B))
assert expr.diff(X) == (X*B + B*X).T
# Cookbook example 108:
expr = Trace(X.T*B*X)
assert expr.diff(X) == B*X + B.T*X
# Cookbook example 109:
expr = Trace(B*X*X.T)
assert expr.diff(X) == B*X + B.T*X
# Cookbook example 110:
expr = Trace(X*X.T*B)
assert expr.diff(X) == B*X + B.T*X
# Cookbook example 111:
expr = Trace(X*B*X.T)
assert expr.diff(X) == X*B.T + X*B
# Cookbook example 112:
expr = Trace(B*X.T*X)
assert expr.diff(X) == X*B.T + X*B
# Cookbook example 113:
expr = Trace(X.T*X*B)
assert expr.diff(X) == X*B.T + X*B
# Cookbook example 114:
expr = Trace(A*X*B*X)
assert expr.diff(X) == A.T*X.T*B.T + B.T*X.T*A.T
# Cookbook example 115:
expr = Trace(X.T*X)
assert expr.diff(X) == 2*X
expr = Trace(X*X.T)
assert expr.diff(X) == 2*X
# Cookbook example 116:
expr = Trace(B.T*X.T*C*X*B)
assert expr.diff(X) == C.T*X*B*B.T + C*X*B*B.T
# Cookbook example 117:
expr = Trace(X.T*B*X*C)
assert expr.diff(X) == B*X*C + B.T*X*C.T
# Cookbook example 118:
expr = Trace(A*X*B*X.T*C)
assert expr.diff(X) == A.T*C.T*X*B.T + C*A*X*B
# Cookbook example 119:
expr = Trace((A*X*B + C)*(A*X*B + C).T)
assert expr.diff(X) == 2*A.T*(A*X*B + C)*B.T
# Cookbook example 120:
# TODO: no support for TensorProduct.
# expr = Trace(TensorProduct(X, X))
# expr = Trace(X)*Trace(X)
# expr.diff(X) == 2*Trace(X)*Identity(k)
# Higher Order
# Cookbook example 121:
expr = Trace(X**k)
#assert expr.diff(X) == k*(X**(k-1)).T
# Cookbook example 122:
expr = Trace(A*X**k)
#assert expr.diff(X) == # Needs indices
# Cookbook example 123:
expr = Trace(B.T*X.T*C*X*X.T*C*X*B)
assert expr.diff(X) == C*X*X.T*C*X*B*B.T + C.T*X*B*B.T*X.T*C.T*X + C*X*B*B.T*X.T*C*X + C.T*X*X.T*C.T*X*B*B.T
# Other
# Cookbook example 124:
expr = Trace(A*X**(-1)*B)
assert expr.diff(X) == -Inverse(X).T*A.T*B.T*Inverse(X).T
# Cookbook example 125:
expr = Trace(Inverse(X.T*C*X)*A)
# Warning: result in the cookbook is equivalent if B and C are symmetric:
assert expr.diff(X) == - X.inv().T*A.T*X.inv()*C.inv().T*X.inv().T - X.inv().T*A*X.inv()*C.inv()*X.inv().T
# Cookbook example 126:
expr = Trace((X.T*C*X).inv()*(X.T*B*X))
assert expr.diff(X) == -2*C*X*(X.T*C*X).inv()*X.T*B*X*(X.T*C*X).inv() + 2*B*X*(X.T*C*X).inv()
# Cookbook example 127:
expr = Trace((A + X.T*C*X).inv()*(X.T*B*X))
# Warning: result in the cookbook is equivalent if B and C are symmetric:
assert expr.diff(X) == B*X*Inverse(A + X.T*C*X) - C*X*Inverse(A + X.T*C*X)*X.T*B*X*Inverse(A + X.T*C*X) - C.T*X*Inverse(A.T + (C*X).T*X)*X.T*B.T*X*Inverse(A.T + (C*X).T*X) + B.T*X*Inverse(A.T + (C*X).T*X)
def test_derivatives_of_complicated_matrix_expr():
expr = a.T*(A*X*(X.T*B + X*A) + B.T*X.T*(a*b.T*(X*D*X.T + X*(X.T*B + A*X)*D*B - X.T*C.T*A)*B + B*(X*D.T + B*A*X*A.T - 3*X*D))*B + 42*X*B*X.T*A.T*(X + X.T))*b
result = (B*(B*A*X*A.T - 3*X*D + X*D.T) + a*b.T*(X*(A*X + X.T*B)*D*B + X*D*X.T - X.T*C.T*A)*B)*B*b*a.T*B.T + B**2*b*a.T*B.T*X.T*a*b.T*X*D + 42*A*X*B.T*X.T*a*b.T + B*D*B**3*b*a.T*B.T*X.T*a*b.T*X + B*b*a.T*A*X + 42*a*b.T*(X + X.T)*A*X*B.T + b*a.T*X*B*a*b.T*B.T**2*X*D.T + b*a.T*X*B*a*b.T*B.T**3*D.T*(B.T*X + X.T*A.T) + 42*b*a.T*X*B*X.T*A.T + 42*A.T*(X + X.T)*b*a.T*X*B + A.T*B.T**2*X*B*a*b.T*B.T*A + A.T*a*b.T*(A.T*X.T + B.T*X) + A.T*X.T*b*a.T*X*B*a*b.T*B.T**3*D.T + B.T*X*B*a*b.T*B.T*D - 3*B.T*X*B*a*b.T*B.T*D.T - C.T*A*B**2*b*a.T*B.T*X.T*a*b.T + X.T*A.T*a*b.T*A.T
assert expr.diff(X) == result
def test_mixed_deriv_mixed_expressions():
expr = 3*Trace(A)
assert expr.diff(A) == 3*Identity(k)
expr = k
deriv = expr.diff(A)
assert isinstance(deriv, ZeroMatrix)
assert deriv == ZeroMatrix(k, k)
expr = Trace(A)**2
assert expr.diff(A) == (2*Trace(A))*Identity(k)
expr = Trace(A)*A
# TODO: this is not yet supported:
assert expr.diff(A) == ArrayDerivative(expr, A)
expr = Trace(Trace(A)*A)
assert expr.diff(A) == (2*Trace(A))*Identity(k)
expr = Trace(Trace(Trace(A)*A)*A)
assert expr.diff(A) == (3*Trace(A)**2)*Identity(k)
def test_derivatives_matrix_norms():
expr = x.T*y
assert expr.diff(x) == y
assert expr[0, 0].diff(x[m, 0]).doit() == y[m, 0]
expr = (x.T*y)**S.Half
assert expr.diff(x) == y/(2*sqrt(x.T*y))
expr = (x.T*x)**S.Half
assert expr.diff(x) == x*(x.T*x)**Rational(-1, 2)
expr = (c.T*a*x.T*b)**S.Half
assert expr.diff(x) == b/(2*sqrt(c.T*a*x.T*b))*c.T*a
expr = (c.T*a*x.T*b)**Rational(1, 3)
assert expr.diff(x) == b*(c.T*a*x.T*b)**Rational(-2, 3)*c.T*a/3
expr = (a.T*X*b)**S.Half
assert expr.diff(X) == a/(2*sqrt(a.T*X*b))*b.T
expr = d.T*x*(a.T*X*b)**S.Half*y.T*c
assert expr.diff(X) == a*d.T*x/(2*sqrt(a.T*X*b))*y.T*c*b.T
def test_derivatives_elementwise_applyfunc():
from sympy.matrices.expressions.diagonal import DiagMatrix
expr = x.applyfunc(tan)
assert expr.diff(x).dummy_eq(
DiagMatrix(x.applyfunc(lambda x: tan(x)**2 + 1)))
assert expr[i, 0].diff(x[m, 0]).doit() == (tan(x[i, 0])**2 + 1)*KDelta(i, m)
_check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
expr = (i**2*x).applyfunc(sin)
assert expr.diff(i).dummy_eq(
HadamardProduct((2*i)*x, (i**2*x).applyfunc(cos)))
assert expr[i, 0].diff(i).doit() == 2*i*x[i, 0]*cos(i**2*x[i, 0])
_check_derivative_with_explicit_matrix(expr, i, expr.diff(i))
expr = (log(i)*A*B).applyfunc(sin)
assert expr.diff(i).dummy_eq(
HadamardProduct(A*B/i, (log(i)*A*B).applyfunc(cos)))
_check_derivative_with_explicit_matrix(expr, i, expr.diff(i))
expr = A*x.applyfunc(exp)
# TODO: restore this result (currently returning the transpose):
# assert expr.diff(x).dummy_eq(DiagMatrix(x.applyfunc(exp))*A.T)
_check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
expr = x.T*A*x + k*y.applyfunc(sin).T*x
assert expr.diff(x).dummy_eq(A.T*x + A*x + k*y.applyfunc(sin))
_check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
expr = x.applyfunc(sin).T*y
# TODO: restore (currently returning the traspose):
# assert expr.diff(x).dummy_eq(DiagMatrix(x.applyfunc(cos))*y)
_check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
expr = (a.T * X * b).applyfunc(sin)
assert expr.diff(X).dummy_eq(a*(a.T*X*b).applyfunc(cos)*b.T)
_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
expr = a.T * X.applyfunc(sin) * b
assert expr.diff(X).dummy_eq(
DiagMatrix(a)*X.applyfunc(cos)*DiagMatrix(b))
_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
expr = a.T * (A*X*B).applyfunc(sin) * b
assert expr.diff(X).dummy_eq(
A.T*DiagMatrix(a)*(A*X*B).applyfunc(cos)*DiagMatrix(b)*B.T)
_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
expr = a.T * (A*X*b).applyfunc(sin) * b.T
# TODO: not implemented
#assert expr.diff(X) == ...
#_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
expr = a.T*A*X.applyfunc(sin)*B*b
assert expr.diff(X).dummy_eq(
DiagMatrix(A.T*a)*X.applyfunc(cos)*DiagMatrix(B*b))
expr = a.T * (A*X.applyfunc(sin)*B).applyfunc(log) * b
# TODO: wrong
# assert expr.diff(X) == A.T*DiagMatrix(a)*(A*X.applyfunc(sin)*B).applyfunc(Lambda(k, 1/k))*DiagMatrix(b)*B.T
expr = a.T * (X.applyfunc(sin)).applyfunc(log) * b
# TODO: wrong
# assert expr.diff(X) == DiagMatrix(a)*X.applyfunc(sin).applyfunc(Lambda(k, 1/k))*DiagMatrix(b)
def test_derivatives_of_hadamard_expressions():
# Hadamard Product
expr = hadamard_product(a, x, b)
assert expr.diff(x) == DiagMatrix(hadamard_product(b, a))
expr = a.T*hadamard_product(A, X, B)*b
assert expr.diff(X) == DiagMatrix(a)*hadamard_product(B, A)*DiagMatrix(b)
# Hadamard Power
expr = hadamard_power(x, 2)
assert expr.diff(x).doit() == 2*DiagMatrix(x)
expr = hadamard_power(x.T, 2)
assert expr.diff(x).doit() == 2*DiagMatrix(x)
expr = hadamard_power(x, S.Half)
assert expr.diff(x) == S.Half*DiagMatrix(hadamard_power(x, Rational(-1, 2)))
expr = hadamard_power(a.T*X*b, 2)
assert expr.diff(X) == 2*a*a.T*X*b*b.T
expr = hadamard_power(a.T*X*b, S.Half)
assert expr.diff(X) == a/2*hadamard_power(a.T*X*b, Rational(-1, 2))*b.T
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