def hessian(f, varlist):
"""Compute Hessian matrix for a function f
see: http://en.wikipedia.org/wiki/Hessian_matrix
See Also
========
sympy.matrices.mutable.Matrix.jacobian
wronskian
"""
# f is the expression representing a function f, return regular matrix
if is_sequence(varlist):
m = len(varlist)
if not m:
raise ShapeError("`len(varlist)` must not be zero.")
elif isinstance(varlist, MatrixBase):
m = varlist.cols
if not m:
raise ShapeError("`varlist.cols` must not be zero.")
if varlist.rows != 1:
raise ShapeError("`varlist` must be a row vector.")
else:
raise ValueError("Improper variable list in hessian function")
if not getattr(f, 'diff'):
# check differentiability
raise ValueError("Function `f` (%s) is not differentiable" % f)
out = zeros(m)
for i in range(m):
for j in range(i, m):
out[i, j] = f.diff(varlist[i]).diff(varlist[j])
for i in range(m):
for j in range(i):
out[i, j] = out[j, i]
return out
def copyin_matrix(self, key, value):
"""Copy in values from a matrix into the given bounds.
Parameters
==========
key : slice
The section of this matrix to replace.
value : Matrix
The matrix to copy values from.
Examples
========
>>> from sympy.matrices import Matrix, eye
>>> M = Matrix([[0, 1], [2, 3], [4, 5]])
>>> I = eye(3)
>>> I[:3, :2] = M
>>> I
Matrix([
[0, 1, 0],
[2, 3, 0],
[4, 5, 1]])
>>> I[0, 1] = M
>>> I
Matrix([
[0, 0, 1],
[2, 2, 3],
[4, 4, 5]])
See Also
========
copyin_list
"""
rlo, rhi, clo, chi = self.key2bounds(key)
shape = value.shape
dr, dc = rhi - rlo, chi - clo
if shape != (dr, dc):
raise ShapeError(
filldedent("The Matrix `value` doesn't have the "
"same dimensions "
"as the in sub-Matrix given by `key`."))
for i in range(value.rows):
for j in range(value.cols):
self[i + rlo, j + clo] = value[i, j]
def __new__(cls, lhs, rhs, rel_cls, assert_symmetry=True):
lhs = sympify(lhs)
rhs = sympify(rhs)
if assert_symmetry:
if lhs.is_Matrix and hasattr(lhs, 'is_symmetric') and \
not lhs.is_symmetric():
raise NonSymmetricMatrixError('lhs matrix is not symmetric')
if rhs.is_Matrix and hasattr(rhs, 'is_symmetric') and \
not rhs.is_symmetric():
raise NonSymmetricMatrixError('rsh matrix is not symmetric')
if lhs.is_Matrix and rhs.is_Matrix:
if lhs.shape != rhs.shape:
raise ShapeError('LMI matrices have different shapes')
elif not ((lhs.is_Matrix and rhs.is_zero) or
(lhs.is_zero and rhs.is_Matrix)):
raise ValueError('LMI sides must be two matrices '
'or a matrix and a zero')
return rel_cls.__new__(cls, lhs, rhs)
def matrix_multiply_elementwise(A, B):
"""Return the Hadamard product (elementwise product) of A and B
>>> from sympy.matrices import matrix_multiply_elementwise
>>> from sympy.matrices import Matrix
>>> A = Matrix([[0, 1, 2], [3, 4, 5]])
>>> B = Matrix([[1, 10, 100], [100, 10, 1]])
>>> matrix_multiply_elementwise(A, B)
[ 0, 10, 200]
[300, 40, 5]
See Also
========
__mul__
"""
if A.shape != B.shape:
raise ShapeError()
shape = A.shape
return classof(A, B)._new(shape[0], shape[1],
lambda i, j: A[i, j] * B[i, j])
def hessian(f, varlist, constraints=[]):
"""Compute Hessian matrix for a function f wrt parameters in varlist
which may be given as a sequence or a row/column vector. A list of
constraints may optionally be given.
Examples
========
>>> from sympy import Function, hessian, pprint
>>> from sympy.abc import x, y
>>> f = Function('f')(x, y)
>>> g1 = Function('g')(x, y)
>>> g2 = x**2 + 3*y
>>> pprint(hessian(f, (x, y), [g1, g2]))
[ d d ]
[ 0 0 --(g(x, y)) --(g(x, y)) ]
[ dx dy ]
[ ]
[ 0 0 2*x 3 ]
[ ]
[ 2 2 ]
[d d d ]
[--(g(x, y)) 2*x ---(f(x, y)) -----(f(x, y))]
[dx 2 dy dx ]
[ dx ]
[ ]
[ 2 2 ]
[d d d ]
[--(g(x, y)) 3 -----(f(x, y)) ---(f(x, y)) ]
[dy dy dx 2 ]
[ dy ]
References
==========
http://en.wikipedia.org/wiki/Hessian_matrix
See Also
========
sympy.matrices.mutable.Matrix.jacobian
wronskian
"""
# f is the expression representing a function f, return regular matrix
if isinstance(varlist, MatrixBase):
if 1 not in varlist.shape:
raise ShapeError("`varlist` must be a column or row vector.")
if varlist.cols == 1:
varlist = varlist.T
varlist = varlist.tolist()[0]
if is_sequence(varlist):
n = len(varlist)
if not n:
raise ShapeError("`len(varlist)` must not be zero.")
else:
raise ValueError("Improper variable list in hessian function")
if not getattr(f, 'diff'):
# check differentiability
raise ValueError("Function `f` (%s) is not differentiable" % f)
m = len(constraints)
N = m + n
out = zeros(N)
for k, g in enumerate(constraints):
if not getattr(g, 'diff'):
# check differentiability
raise ValueError("Function `f` (%s) is not differentiable" % f)
for i in range(n):
out[k, i + m] = g.diff(varlist[i])
for i in range(n):
for j in range(i, n):
out[i + m, j + m] = f.diff(varlist[i]).diff(varlist[j])
for i in range(N):
for j in range(i + 1, N):
out[j, i] = out[i, j]
return out