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"""Matrix equation solver routines"""
# Author: Jeffrey Armstrong <jeff@approximatrix.com>
# February 24, 2012
# Modified: Chad Fulton <ChadFulton@gmail.com>
# June 19, 2014
from __future__ import division, print_function, absolute_import
import numpy as np
from numpy.linalg import inv, LinAlgError
from .basic import solve
from .lapack import get_lapack_funcs
from .decomp_schur import schur
from .special_matrices import kron
__all__ = ['solve_sylvester', 'solve_lyapunov', 'solve_discrete_lyapunov',
'solve_continuous_are', 'solve_discrete_are']
def solve_sylvester(a,b,q):
"""
Computes a solution (X) to the Sylvester equation (AX + XB = Q).
Parameters
----------
a : (M, M) array_like
Leading matrix of the Sylvester equation
b : (N, N) array_like
Trailing matrix of the Sylvester equation
q : (M, N) array_like
Right-hand side
Returns
-------
x : (M, N) ndarray
The solution to the Sylvester equation.
Raises
------
LinAlgError
If solution was not found
Notes
-----
Computes a solution to the Sylvester matrix equation via the Bartels-
Stewart algorithm. The A and B matrices first undergo Schur
decompositions. The resulting matrices are used to construct an
alternative Sylvester equation (``RY + YS^T = F``) where the R and S
matrices are in quasi-triangular form (or, when R, S or F are complex,
triangular form). The simplified equation is then solved using
``*TRSYL`` from LAPACK directly.
.. versionadded:: 0.11.0
"""
# Compute the Schur decomp form of a
r,u = schur(a, output='real')
# Compute the Schur decomp of b
s,v = schur(b.conj().transpose(), output='real')
# Construct f = u'*q*v
f = np.dot(np.dot(u.conj().transpose(), q), v)
# Call the Sylvester equation solver
trsyl, = get_lapack_funcs(('trsyl',), (r,s,f))
if trsyl is None:
raise RuntimeError('LAPACK implementation does not contain a proper Sylvester equation solver (TRSYL)')
y, scale, info = trsyl(r, s, f, tranb='C')
y = scale*y
if info < 0:
raise LinAlgError("Illegal value encountered in the %d term" % (-info,))
return np.dot(np.dot(u, y), v.conj().transpose())
def solve_lyapunov(a, q):
"""
Solves the continuous Lyapunov equation (AX + XA^H = Q) given the values
of A and Q using the Bartels-Stewart algorithm.
Parameters
----------
a : array_like
A square matrix
q : array_like
Right-hand side square matrix
Returns
-------
x : array_like
Solution to the continuous Lyapunov equation
See Also
--------
solve_sylvester : computes the solution to the Sylvester equation
Notes
-----
Because the continuous Lyapunov equation is just a special form of the
Sylvester equation, this solver relies entirely on solve_sylvester for a
solution.
.. versionadded:: 0.11.0
"""
return solve_sylvester(a, a.conj().transpose(), q)
def _solve_discrete_lyapunov_direct(a, q):
"""
Solves the discrete Lyapunov equation directly.
This function is called by the `solve_discrete_lyapunov` function with
`method=direct`. It is not supposed to be called directly.
"""
lhs = kron(a, a.conj())
lhs = np.eye(lhs.shape[0]) - lhs
x = solve(lhs, q.flatten())
return np.reshape(x, q.shape)
def _solve_discrete_lyapunov_bilinear(a, q):
"""
Solves the discrete Lyapunov equation using a bilinear transformation.
This function is called by the `solve_discrete_lyapunov` function with
`method=bilinear`. It is not supposed to be called directly.
"""
eye = np.eye(a.shape[0])
aH = a.conj().transpose()
aHI_inv = inv(aH + eye)
b = np.dot(aH - eye, aHI_inv)
c = 2*np.dot(np.dot(inv(a + eye), q), aHI_inv)
return solve_lyapunov(b.conj().transpose(), -c)
def solve_discrete_lyapunov(a, q, method=None):
"""
Solves the discrete Lyapunov equation :math:`(A'XA-X=-Q)`.
Parameters
----------
a : (M, M) array_like
A square matrix
q : (M, M) array_like
Right-hand side square matrix
method : {'direct', 'bilinear'}, optional
Type of solver.
If not given, chosen to be ``direct`` if ``M`` is less than 10 and
``bilinear`` otherwise.
Returns
-------
x : ndarray
Solution to the discrete Lyapunov equation
See Also
--------
solve_lyapunov : computes the solution to the continuous Lyapunov equation
Notes
-----
This section describes the available solvers that can be selected by the
'method' parameter. The default method is *direct* if ``M`` is less than 10
and ``bilinear`` otherwise.
Method *direct* uses a direct analytical solution to the discrete Lyapunov
equation. The algorithm is given in, for example, [1]_. However it requires
the linear solution of a system with dimension :math:`M^2` so that
performance degrades rapidly for even moderately sized matrices.
Method *bilinear* uses a bilinear transformation to convert the discrete
Lyapunov equation to a continuous Lyapunov equation :math:`(B'X+XB=-C)`
where :math:`B=(A-I)(A+I)^{-1}` and
:math:`C=2(A' + I)^{-1} Q (A + I)^{-1}`. The continuous equation can be
efficiently solved since it is a special case of a Sylvester equation.
The transformation algorithm is from Popov (1964) as described in [2]_.
.. versionadded:: 0.11.0
References
----------
.. [1] Hamilton, James D. Time Series Analysis, Princeton: Princeton
University Press, 1994. 265. Print.
http://www.scribd.com/doc/20577138/Hamilton-1994-Time-Series-Analysis
.. [2] Gajic, Z., and M.T.J. Qureshi. 2008.
Lyapunov Matrix Equation in System Stability and Control.
Dover Books on Engineering Series. Dover Publications.
"""
a = np.asarray(a)
q = np.asarray(q)
if method is None:
# Select automatically based on size of matrices
if a.shape[0] >= 10:
method = 'bilinear'
else:
method = 'direct'
meth = method.lower()
if meth == 'direct':
x = _solve_discrete_lyapunov_direct(a, q)
elif meth == 'bilinear':
x = _solve_discrete_lyapunov_bilinear(a, q)
else:
raise ValueError('Unknown solver %s' % method)
return x
def solve_continuous_are(a, b, q, r):
"""
Solves the continuous algebraic Riccati equation, or CARE, defined
as (A'X + XA - XBR^-1B'X+Q=0) directly using a Schur decomposition
method.
Parameters
----------
a : (M, M) array_like
Input
b : (M, N) array_like
Input
q : (M, M) array_like
Input
r : (N, N) array_like
Non-singular, square matrix
Returns
-------
x : (M, M) ndarray
Solution to the continuous algebraic Riccati equation
See Also
--------
solve_discrete_are : Solves the discrete algebraic Riccati equation
Notes
-----
Method taken from:
Laub, "A Schur Method for Solving Algebraic Riccati Equations."
U.S. Energy Research and Development Agency under contract
ERDA-E(49-18)-2087.
http://dspace.mit.edu/bitstream/handle/1721.1/1301/R-0859-05666488.pdf
.. versionadded:: 0.11.0
"""
try:
g = inv(r)
except LinAlgError:
raise ValueError('Matrix R in the algebraic Riccati equation solver is ill-conditioned')
g = np.dot(np.dot(b, g), b.conj().transpose())
z11 = a
z12 = -1.0*g
z21 = -1.0*q
z22 = -1.0*a.conj().transpose()
z = np.vstack((np.hstack((z11, z12)), np.hstack((z21, z22))))
# Note: we need to sort the upper left of s to have negative real parts,
# while the lower right is positive real components (Laub, p. 7)
[s, u, sorted] = schur(z, sort='lhp')
(m, n) = u.shape
u11 = u[0:m//2, 0:n//2]
u21 = u[m//2:m, 0:n//2]
u11i = inv(u11)
return np.dot(u21, u11i)
def solve_discrete_are(a, b, q, r):
"""
Solves the disctrete algebraic Riccati equation, or DARE, defined as
(X = A'XA-(A'XB)(R+B'XB)^-1(B'XA)+Q), directly using a Schur decomposition
method.
Parameters
----------
a : (M, M) array_like
Non-singular, square matrix
b : (M, N) array_like
Input
q : (M, M) array_like
Input
r : (N, N) array_like
Non-singular, square matrix
Returns
-------
x : ndarray
Solution to the continuous Lyapunov equation
See Also
--------
solve_continuous_are : Solves the continuous algebraic Riccati equation
Notes
-----
Method taken from:
Laub, "A Schur Method for Solving Algebraic Riccati Equations."
U.S. Energy Research and Development Agency under contract
ERDA-E(49-18)-2087.
http://dspace.mit.edu/bitstream/handle/1721.1/1301/R-0859-05666488.pdf
.. versionadded:: 0.11.0
"""
try:
g = inv(r)
except LinAlgError:
raise ValueError('Matrix R in the algebraic Riccati equation solver is ill-conditioned')
g = np.dot(np.dot(b, g), b.conj().transpose())
try:
ait = inv(a).conj().transpose() # ait is "A inverse transpose"
except LinAlgError:
raise ValueError('Matrix A in the algebraic Riccati equation solver is ill-conditioned')
z11 = a+np.dot(np.dot(g, ait), q)
z12 = -1.0*np.dot(g, ait)
z21 = -1.0*np.dot(ait, q)
z22 = ait
z = np.vstack((np.hstack((z11, z12)), np.hstack((z21, z22))))
# Note: we need to sort the upper left of s to lie within the unit circle,
# while the lower right is outside (Laub, p. 7)
[s, u, sorted] = schur(z, sort='iuc')
(m,n) = u.shape
u11 = u[0:m//2, 0:n//2]
u21 = u[m//2:m, 0:n//2]
u11i = inv(u11)
return np.dot(u21, u11i)
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