"""
Utility functions for working with CVXOPT matrices (instances of the
-``cvxopt.base.matrix`` class).
+class:`cvxopt.base.matrix` class).
"""
from math import sqrt
from cvxopt import matrix
-from cvxopt.lapack import syev
+from cvxopt.lapack import gees, syevr
+
+import options
+
def append_col(left, right):
"""
- Append the matrix ``right`` to the right side of the matrix ``left``.
+ Append two matrices side-by-side.
+
+ Parameters
+ ----------
+
+ left, right : matrix
+ The two matrices to append to one another.
+
+ Returns
+ -------
+
+ matrix
+ A new matrix consisting of ``right`` appended to the right
+ of ``left``.
- EXAMPLES:
+ Examples
+ --------
>>> A = matrix([1,2,3,4], (2,2))
>>> B = matrix([5,6,7,8,9,10], (2,3))
+ >>> print(A)
+ [ 1 3]
+ [ 2 4]
+ <BLANKLINE>
+ >>> print(B)
+ [ 5 7 9]
+ [ 6 8 10]
+ <BLANKLINE>
>>> print(append_col(A,B))
[ 1 3 5 7 9]
[ 2 4 6 8 10]
"""
return matrix([left.trans(), right.trans()]).trans()
+
def append_row(top, bottom):
"""
- Append the matrix ``bottom`` to the bottom of the matrix ``top``.
+ Append two matrices top-to-bottom.
+
+ Parameters
+ ----------
+
+ top, bottom : matrix
+ The two matrices to append to one another.
+
+ Returns
+ -------
+
+ matrix
+ A new matrix consisting of ``bottom`` appended below ``top``.
- EXAMPLES:
+ Examples
+ --------
>>> A = matrix([1,2,3,4], (2,2))
>>> B = matrix([5,6,7,8,9,10], (3,2))
+ >>> print(A)
+ [ 1 3]
+ [ 2 4]
+ <BLANKLINE>
+ >>> print(B)
+ [ 5 8]
+ [ 6 9]
+ [ 7 10]
+ <BLANKLINE>
>>> print(append_row(A,B))
[ 1 3]
[ 2 4]
return matrix([top, bottom])
-def eigenvalues(real_matrix):
+def eigenvalues(symmat):
"""
- Return the eigenvalues of the given ``real_matrix``.
+ Return the eigenvalues of the given symmetric real matrix.
+
+ On the surface, this appears redundant to the :func:`eigenvalues_re`
+ function. However, if we know in advance that our input is
+ symmetric, a better algorithm can be used.
+
+ Parameters
+ ----------
+
+ symmat : matrix
+ The real symmetric matrix whose eigenvalues you want.
- EXAMPLES:
+ Returns
+ -------
+
+ list of float
+ A list of the eigenvalues (in no particular order) of ``symmat``.
+
+ Raises
+ ------
+
+ TypeError
+ If the input matrix is not symmetric.
+
+ Examples
+ --------
>>> A = matrix([[2,1],[1,2]], tc='d')
>>> eigenvalues(A)
[1.0, 3.0]
+ If the input matrix is not symmetric, it may not have real
+ eigenvalues, and we don't know what to do::
+
+ >>> A = matrix([[1,2],[3,4]])
+ >>> eigenvalues(A)
+ Traceback (most recent call last):
+ ...
+ TypeError: input must be a symmetric real matrix
+
"""
- domain_dim = real_matrix.size[0] # Assume ``real_matrix`` is square.
+ if not norm(symmat.trans() - symmat) < options.ABS_TOL:
+ # Ensure that ``symmat`` is symmetric (and thus square).
+ raise TypeError('input must be a symmetric real matrix')
+
+ domain_dim = symmat.size[0]
eigs = matrix(0, (domain_dim, 1), tc='d')
- syev(real_matrix, eigs)
+ syevr(symmat, eigs)
return list(eigs)
+def eigenvalues_re(anymat):
+ """
+ Return the real parts of the eigenvalues of the given square matrix.
+
+ Parameters
+ ----------
+
+ anymat : matrix
+ The square matrix whose eigenvalues you want.
+
+ Returns
+ -------
+
+ list of float
+ A list of the real parts (in no particular order) of the
+ eigenvalues of ``anymat``.
+
+ Raises
+ ------
+
+ TypeError
+ If the input matrix is not square.
+
+ Examples
+ --------
+
+ This is symmetric and has two real eigenvalues:
+
+ >>> A = matrix([[2,1],[1,2]], tc='d')
+ >>> sorted(eigenvalues_re(A))
+ [1.0, 3.0]
+
+ But this rotation matrix has eigenvalues `i` and `-i`, both of whose
+ real parts are zero:
+
+ >>> A = matrix([[0,-1],[1,0]])
+ >>> eigenvalues_re(A)
+ [0.0, 0.0]
+
+ If the input matrix is not square, it doesn't have eigenvalues::
+
+ >>> A = matrix([[1,2],[3,4],[5,6]])
+ >>> eigenvalues_re(A)
+ Traceback (most recent call last):
+ ...
+ TypeError: input matrix must be square
+
+ """
+ if not anymat.size[0] == anymat.size[1]:
+ raise TypeError('input matrix must be square')
+
+ domain_dim = anymat.size[0]
+ eigs = matrix(0, (domain_dim, 1), tc='z')
+
+ # Create a copy of ``anymat`` here for two reasons:
+ #
+ # 1. ``gees`` clobbers its input.
+ # 2. We need to ensure that the type code of ``dummy`` is 'd' or 'z'.
+ #
+ dummy = matrix(anymat, anymat.size, tc='d')
+
+ gees(dummy, eigs)
+ return [eig.real for eig in eigs]
+
+
def identity(domain_dim):
"""
- Return a ``domain_dim``-by-``domain_dim`` dense integer identity
- matrix.
+ Create an identity matrix of the given dimensions.
- EXAMPLES:
+ Parameters
+ ----------
+
+ domain_dim : int
+ The dimension of the vector space on which the identity will act.
+
+ Returns
+ -------
+
+ matrix
+ A ``domain_dim``-by-``domain_dim`` dense integer identity matrix.
+
+ Raises
+ ------
+
+ ValueError
+ If you ask for the identity on zero or fewer dimensions.
+
+ Examples
+ --------
>>> print(identity(3))
[ 1 0 0]
def inner_product(vec1, vec2):
"""
- Compute the (Euclidean) inner product of the two vectors ``vec1``
- and ``vec2``.
+ Compute the Euclidean inner product of two vectors.
+
+ Parameters
+ ----------
+
+ vec1, vec2 : matrix
+ The two vectors whose inner product you want.
+
+ Returns
+ -------
+
+ float
+ The inner product of ``vec1`` and ``vec2``.
+
+ Raises
+ ------
- EXAMPLES:
+ TypeError
+ If the lengths of ``vec1`` and ``vec2`` differ.
+
+ Examples
+ --------
>>> x = [1,2,3]
>>> y = [3,4,1]
if not len(vec1) == len(vec2):
raise TypeError('the lengths of vec1 and vec2 must match')
- return sum([x*y for (x,y) in zip(vec1,vec2)])
+ return sum([x*y for (x, y) in zip(vec1, vec2)])
def norm(matrix_or_vector):
"""
- Return the Frobenius norm of ``matrix_or_vector``, which is the same
- thing as its Euclidean norm when it's a vector (when one of its
- dimensions is unity).
+ Return the Frobenius norm of a matrix or vector.
+
+ When the input is a vector, its matrix-Frobenius norm is the same
+ thing as its vector-Euclidean norm.
+
+ Parameters
+ ----------
+
+ matrix_or_vector : matrix
+ The matrix or vector whose norm you want.
+
+ Returns
+ -------
- EXAMPLES:
+ float
+ The norm of ``matrix_or_vector``.
+
+ Examples
+ --------
>>> v = matrix([1,1])
>>> print('{:.5f}'.format(norm(v)))
2.0
"""
- return sqrt(inner_product(matrix_or_vector,matrix_or_vector))
+ return sqrt(inner_product(matrix_or_vector, matrix_or_vector))
-def vec(real_matrix):
+def vec(mat):
"""
- Create a long vector in column-major order from ``real_matrix``.
+ Create a long vector in column-major order from ``mat``.
+
+ Parameters
+ ----------
+
+ mat : matrix
+ Any sort of real matrix that you want written as a long vector.
+
+ Returns
+ -------
+
+ matrix
+ An ``len(mat)``-by-``1`` long column vector containign the
+ entries of ``mat`` in column major order.
- EXAMPLES:
+ Examples
+ --------
>>> A = matrix([[1,2],[3,4]])
>>> print(A)
[ 4]
<BLANKLINE>
- Note that if ``real_matrix`` is a vector, this function is a no-op:
+ Note that if ``mat`` is a vector, this function is a no-op:
>>> v = matrix([1,2,3,4], (4,1))
>>> print(v)
<BLANKLINE>
"""
- return matrix(real_matrix, (len(real_matrix), 1))
+ return matrix(mat, (len(mat), 1))
projects / dunshire.git / blobdiff