def jacobian(functions, variables):
"""
Return the Jacobian matrix, which is the matrix of partial
derivatives in which the i,j entry of the Jacobian matrix is the
partial derivative diff(functions[i], variables[j]).
EXAMPLES::
sage: x,y = var('x,y')
sage: g=x^2-2*x*y
sage: jacobian(g, (x,y))
[2*x - 2*y -2*x]
The Jacobian of the Jacobian should give us the "second derivative", which is the Hessian matrix::
sage: jacobian(jacobian(g, (x,y)), (x,y))
[ 2 -2]
[-2 0]
sage: g.hessian()
[ 2 -2]
[-2 0]
sage: f=(x^3*sin(y), cos(x)*sin(y), exp(x))
sage: jacobian(f, (x,y))
[ 3*x^2*sin(y) x^3*cos(y)]
[-sin(x)*sin(y) cos(x)*cos(y)]
[ e^x 0]
sage: jacobian(f, (y,x))
[ x^3*cos(y) 3*x^2*sin(y)]
[ cos(x)*cos(y) -sin(x)*sin(y)]
[ 0 e^x]
"""
if is_Matrix(functions) and (functions.nrows() == 1
or functions.ncols() == 1):
functions = functions.list()
elif not (isinstance(functions, (tuple, list)) or is_Vector(functions)):
functions = [functions]
if not isinstance(variables, (tuple, list)) and not is_Vector(variables):
variables = [variables]
return matrix([[diff(f, v) for v in variables] for f in functions])
def jacobian(functions, variables):
"""
Return the Jacobian matrix, which is the matrix of partial
derivatives in which the i,j entry of the Jacobian matrix is the
partial derivative diff(functions[i], variables[j]).
EXAMPLES::
sage: x,y = var('x,y')
sage: g=x^2-2*x*y
sage: jacobian(g, (x,y))
[2*x - 2*y -2*x]
The Jacobian of the Jacobian should give us the "second derivative", which is the Hessian matrix::
sage: jacobian(jacobian(g, (x,y)), (x,y))
[ 2 -2]
[-2 0]
sage: g.hessian()
[ 2 -2]
[-2 0]
sage: f=(x^3*sin(y), cos(x)*sin(y), exp(x))
sage: jacobian(f, (x,y))
[ 3*x^2*sin(y) x^3*cos(y)]
[-sin(x)*sin(y) cos(x)*cos(y)]
[ e^x 0]
sage: jacobian(f, (y,x))
[ x^3*cos(y) 3*x^2*sin(y)]
[ cos(x)*cos(y) -sin(x)*sin(y)]
[ 0 e^x]
"""
if is_Matrix(functions) and (functions.nrows()==1 or functions.ncols()==1):
functions = functions.list()
elif not (isinstance(functions, (tuple, list)) or is_Vector(functions)):
functions = [functions]
if not isinstance(variables, (tuple, list)) and not is_Vector(variables):
variables = [variables]
return matrix([[diff(f, v) for v in variables] for f in functions])
def wronskian(*args):
"""
Returns the Wronskian of the provided functions, differentiating with
respect to the given variable. If no variable is provided,
diff(f) is called for each function f.
wronskian(f1,...,fn, x) returns the Wronskian of f1,...,fn, with
derivatives taken with respect to x.
wronskian(f1,...,fn) returns the Wronskian of f1,...,fn where
k'th derivatives are computed by doing `.derivative(k)' on each
function.
The Wronskian of a list of functions is a determinant of derivatives.
The nth row (starting from 0) is a list of the nth derivatives of the
given functions.
For two functions::
| f g |
W(f, g) = det| | = f*g' - g*f'.
| f' g' |
EXAMPLES::
sage: wronskian(e^x, x^2)
-x^2*e^x + 2*x*e^x
sage: x,y = var('x, y')
sage: wronskian(x*y, log(x), x)
-y*log(x) + y
If your functions are in a list, you can use `*' to turn them into
arguments to :func:`wronskian`::
sage: wronskian(*[x^k for k in range(1, 5)])
12*x^4
If you want to use 'x' as one of the functions in the Wronskian,
you can't put it last or it will be interpreted as the variable
with respect to which we differentiate. There are several ways to
get around this.
Two-by-two Wronskian of sin(x) and e^x::
sage: wronskian(sin(x), e^x, x)
e^x*sin(x) - e^x*cos(x)
Or don't put x last::
sage: wronskian(x, sin(x), e^x)
(e^x*sin(x) + e^x*cos(x))*x - 2*e^x*sin(x)
Example where one of the functions is constant::
sage: wronskian(1, e^(-x), e^(2*x))
-6*e^x
NOTES:
- http://en.wikipedia.org/wiki/Wronskian
- http://planetmath.org/encyclopedia/WronskianDeterminant.html
AUTHORS:
- Dan Drake (2008-03-12)
"""
if len(args) == 0:
raise TypeError('wronskian() takes at least one argument (0 given)')
elif len(args) == 1:
# a 1x1 Wronskian is just its argument
return args[0]
else:
if is_SymbolicVariable(args[-1]):
# if last argument is a variable, peel it off and
# differentiate the other args
v = args[-1]
fs = args[0:-1]
row = lambda n: map(lambda f: diff(f, v, n), fs)
else:
# if the last argument isn't a variable, just run
# .derivative on everything
fs = args
row = lambda n: map(lambda f: diff(f, n), fs)
# NOTE: I rewrote the below as two lines to avoid a possible subtle
# memory management problem on some platforms (only VMware as far
# as we know?). See trac #2990.
# There may still be a real problem that this is just hiding for now.
A = matrix(map(row, range(len(fs))))
return A.determinant()
def wronskian(*args):
"""
Returns the Wronskian of the provided functions, differentiating with
respect to the given variable. If no variable is provided,
diff(f) is called for each function f.
wronskian(f1,...,fn, x) returns the Wronskian of f1,...,fn, with
derivatives taken with respect to x.
wronskian(f1,...,fn) returns the Wronskian of f1,...,fn where
k'th derivatives are computed by doing ``.derivative(k)`` on each
function.
The Wronskian of a list of functions is a determinant of derivatives.
The nth row (starting from 0) is a list of the nth derivatives of the
given functions.
For two functions::
| f g |
W(f, g) = det| | = f*g' - g*f'.
| f' g' |
EXAMPLES::
sage: wronskian(e^x, x^2)
-x^2*e^x + 2*x*e^x
sage: x,y = var('x, y')
sage: wronskian(x*y, log(x), x)
-y*log(x) + y
If your functions are in a list, you can use `*' to turn them into
arguments to :func:`wronskian`::
sage: wronskian(*[x^k for k in range(1, 5)])
12*x^4
If you want to use 'x' as one of the functions in the Wronskian,
you can't put it last or it will be interpreted as the variable
with respect to which we differentiate. There are several ways to
get around this.
Two-by-two Wronskian of sin(x) and e^x::
sage: wronskian(sin(x), e^x, x)
-cos(x)*e^x + e^x*sin(x)
Or don't put x last::
sage: wronskian(x, sin(x), e^x)
(cos(x)*e^x + e^x*sin(x))*x - 2*e^x*sin(x)
Example where one of the functions is constant::
sage: wronskian(1, e^(-x), e^(2*x))
-6*e^x
NOTES:
- http://en.wikipedia.org/wiki/Wronskian
- http://planetmath.org/encyclopedia/WronskianDeterminant.html
AUTHORS:
- Dan Drake (2008-03-12)
"""
if len(args) == 0:
raise TypeError('wronskian() takes at least one argument (0 given)')
elif len(args) == 1:
# a 1x1 Wronskian is just its argument
return args[0]
else:
if is_SymbolicVariable(args[-1]):
# if last argument is a variable, peel it off and
# differentiate the other args
v = args[-1]
fs = args[0:-1]
row = lambda n: map(lambda f: diff(f, v, n), fs)
else:
# if the last argument isn't a variable, just run
# .derivative on everything
fs = args
row = lambda n: map(lambda f: diff(f, n), fs)
# NOTE: I rewrote the below as two lines to avoid a possible subtle
# memory management problem on some platforms (only VMware as far
# as we know?). See trac #2990.
# There may still be a real problem that this is just hiding for now.
A = matrix(map(row, range(len(fs))))
return A.determinant()
