vectors_in_intervals.linear_inequality_systems¶
Linear inequality systems¶
EXAMPLES:
sage: from vectors_in_intervals import *
sage: A = matrix([[1, 2], [0, 1]])
sage: B = matrix([[2, 3]])
sage: C = matrix([[-1, 0]])
sage: S = HomogeneousSystem(A, B, C)
sage: S.intervals
[(0, +oo), (0, +oo), [0, +oo), {0}]
sage: S.solve()
(0, 1)
sage: S.certify_existence()
(2, 1, 3, 0)
sage: S.certify_nonexistence()
Traceback (most recent call last):
...
ValueError: A solution exists!
sage: S.certify()
(True, (2, 1, 3, 0))
sage: S.certify(reverse=True)
(True, (2, 1, 3, 0))
sage: S.certify_parallel()
(True, (2, 1, 3, 0))
sage: S.certify_parallel(reverse=True)
(True, (2, 1, 3, 0))
sage: S.certify_parallel(random=True)
(True, (2, 1, 3, 0))
We consider another system:
sage: M = matrix([[1, 0], [0, 1], [1, 1], [0, 1]])
sage: lower_bounds = [2, 5, 0, -oo]
sage: upper_bounds = [5, oo, 8, 5]
sage: lower_bounds_closed = [True, True, False, False]
sage: upper_bounds_closed = [False, False, False, True]
sage: I = Intervals.from_bounds(lower_bounds, upper_bounds, lower_bounds_closed, upper_bounds_closed)
sage: S = LinearInequalitySystem(M, I)
sage: S.solve()
(5/2, 5)
sage: S.certify_existence()
(5, 15, 1, 2, 1, 0, 0)
sage: S.certify()
(True, (5, 15, 1, 2, 1, 0, 0))
sage: S.certify(reverse=True)
(True, (3, 7, 1, 1, 0, 0, 0))
sage: # S.certify_parallel() # TODO SignalError: Segmentation Fault
(True, (3, 7, 1, 1, 0, 0, 0))
We consider yet another system:
sage: A = matrix([[1, 0], [1, 1]])
sage: B = matrix([[-1, -1]])
sage: b = vector([1, 0])
sage: c = vector([0])
sage: S = InhomogeneousSystem(A, B, b, c)
sage: S.certify()
(False, (0, 1, 1))
sage: S.certify_parallel()
(False, (0, 1, 1))
sage: S.certify_parallel(random=True)
(False, (0, 1, 1))
In the case of homogeneous systems, we can use cocircuits to certify:
sage: A = matrix([[1, 2], [0, 1]])
sage: B = matrix([[2, 3]])
sage: C = matrix([[-1, 0]])
sage: S = HomogeneousSystemCocircuits(A, B, C) # TODO: not implemented
sage: S.solve() # TODO: not implemented
Traceback (most recent call last):
...
ValueError: Can't solve using cocircuits!
sage: S.certify_existence() # TODO: not implemented
(+++0)
sage: # S.certify() # TODO
Now, we consider the example:
sage: A = matrix([[1, 0], [0, 1]])
sage: B = matrix([[2, -3]])
sage: C = matrix([[-1, -1]])
sage: S = HomogeneousSystemCocircuits(A, B, C) # TODO: not implemented
sage: S.certify_nonexistence() # TODO: not implemented
(++0+)
Functions
|
Convert a general system to a homogeneous system. |
|
Convert an inhomogeneous system to a homogeneous system. |
|
Translate a general system into an inhomogeneous system. |
Classes
|
A class for homogeneous linear inequality systems |
|
A class for inhomogeneous linear inequality systems |
|
A class for linear inequality systems given by a matrix and intervals |
- class vectors_in_intervals.linear_inequality_systems.HomogeneousSystem(A: sage.matrix.constructor.matrix, B: sage.matrix.constructor.matrix, C: sage.matrix.constructor.matrix, result: Optional[bool] = None)¶
A class for homogeneous linear inequality systems
A x > 0,B x >= 0,C x = 0TESTS:
sage: from vectors_in_intervals import * sage: A = matrix([[0, 1], [0, 1], [0, 1]]) sage: B = zero_matrix(0, 2) sage: C = matrix([[1, 1], [0, 0]]) sage: S = HomogeneousSystem(A, B, C) sage: S.certify_existence() (1, 1, 1, 0, 0)
- candidate_generator(dual: bool = True, reverse: bool = False, random: bool = False) collections.abc.Generator¶
Return a generator of elementary vectors.
- category(self)¶
File: sage/structure/sage_object.pyx (starting at line 484)
- certify(reverse: bool = False) tuple¶
Return a boolean and a certificate for solvability.
- certify_existence(reverse: bool = False, random: bool = False)¶
Certify existence of a solution if one exists.
Otherwise, a
ValueErroris raised.Note
If a solution exists and
randomis set to true, this method will never finish.
- certify_nonexistence(reverse: bool = False, random: bool = False)¶
Certify nonexistence of solutions.
Otherwise, a
ValueErroris raised.Note
If a solution exists and
randomis set to true, this method will never finish.
- certify_parallel(reverse: bool = False, random: bool = False) tuple¶
Return a boolean and a certificate for solvability.
Attempts to find a solution and certify nonexistence in parallel.
- dump(self, filename, compress=True)¶
File: sage/structure/sage_object.pyx (starting at line 445)
Same as self.save(filename, compress)
- dumps(self, compress=True)¶
File: sage/structure/sage_object.pyx (starting at line 451)
Dump
selfto a strings, which can later be reconstituted asselfusingloads(s).There is an optional boolean argument
compresswhich defaults toTrue.EXAMPLES:
sage: from sage.misc.persist import comp sage: O = SageObject() sage: p_comp = O.dumps() sage: p_uncomp = O.dumps(compress=False) sage: comp.decompress(p_comp) == p_uncomp True sage: import pickletools sage: pickletools.dis(p_uncomp) 0: \x80 PROTO 2 2: c GLOBAL 'sage.structure.sage_object SageObject' 41: q BINPUT ... 43: ) EMPTY_TUPLE 44: \x81 NEWOBJ 45: q BINPUT ... 47: . STOP highest protocol among opcodes = 2
- exists_orthogonal_vector(v) bool¶
Check if an orthogonal vector exists in the intervals.
- property intervals: vectors_in_intervals.intervals.Intervals¶
Return the corresponding intervals.
- property matrix: sage.matrix.constructor.matrix¶
Return the corresponding matrix.
- parent(self)¶
File: sage/structure/sage_object.pyx (starting at line 518)
Return the type of
selfto support the coercion framework.EXAMPLES:
sage: t = log(sqrt(2) - 1) + log(sqrt(2) + 1); t log(sqrt(2) + 1) + log(sqrt(2) - 1) sage: u = t.maxima_methods() sage: u.parent() <class 'sage.symbolic.maxima_wrapper.MaximaWrapper'>
- rename(self, x=None)¶
File: sage/structure/sage_object.pyx (starting at line 68)
Change self so it prints as x, where x is a string.
Note
This is only supported for Python classes that derive from SageObject.
EXAMPLES:
sage: x = PolynomialRing(QQ, 'x', sparse=True).gen() sage: g = x^3 + x - 5 sage: g x^3 + x - 5 sage: g.rename('a polynomial') sage: g a polynomial sage: g + x x^3 + 2*x - 5 sage: h = g^100 sage: str(h)[:20] 'x^300 + 100*x^298 - ' sage: h.rename('x^300 + ...') sage: h x^300 + ...
Real numbers are not Python classes, so rename is not supported:
sage: a = 3.14 sage: type(a) <... 'sage.rings.real_mpfr.RealLiteral'> sage: a.rename('pi') Traceback (most recent call last): ... NotImplementedError: object does not support renaming: 3.14000000000000
Note
The reason C-extension types are not supported by default is if they were then every single one would have to carry around an extra attribute, which would be slower and waste a lot of memory.
To support them for a specific class, add a
cdef public __custom_nameattribute.
- reset_name(self)¶
File: sage/structure/sage_object.pyx (starting at line 125)
Remove the custom name of an object.
EXAMPLES:
sage: P.<x> = QQ[] sage: P Univariate Polynomial Ring in x over Rational Field sage: P.rename('A polynomial ring') sage: P A polynomial ring sage: P.reset_name() sage: P Univariate Polynomial Ring in x over Rational Field
- save(self, filename=None, compress=True)¶
File: sage/structure/sage_object.pyx (starting at line 420)
Save self to the given filename.
EXAMPLES:
sage: f = x^3 + 5 sage: f.save(os.path.join(SAGE_TMP, 'file')) sage: load(os.path.join(SAGE_TMP, 'file.sobj')) x^3 + 5
- solve(reverse: bool = False, random: bool = False)¶
Compute a solution if existent.
This approach sums up positive elementary vectors in the row space. It doesn’t use division.
Note
If no solution exists, and
randomis true, this method will never finish.
- to_homogeneous() vectors_in_intervals.linear_inequality_systems.HomogeneousSystem¶
Return the equivalent homogeneous system.
- class vectors_in_intervals.linear_inequality_systems.InhomogeneousSystem(A: sage.matrix.constructor.matrix, B: sage.matrix.constructor.matrix, b: sage.modules.free_module_element.vector, c: sage.modules.free_module_element.vector, result: Optional[bool] = None)¶
A class for inhomogeneous linear inequality systems
A x <= b,B x <= c- candidate_generator(dual: bool = True, reverse: bool = False, random: bool = False) collections.abc.Generator¶
Return a generator of elementary vectors.
- category(self)¶
File: sage/structure/sage_object.pyx (starting at line 484)
- certify(reverse: bool = False) tuple¶
Return a boolean and a certificate for solvability.
- certify_existence(reverse: bool = False, random: bool = False)¶
Certify existence of a solution if one exists.
Otherwise, a
ValueErroris raised.Note
If a solution exists and
randomis set to true, this method will never finish.
- certify_nonexistence(reverse: bool = False, random: bool = False)¶
Certify nonexistence of solutions.
Otherwise, a
ValueErroris raised.Note
If a solution exists and
randomis set to true, this method will never finish.
- certify_parallel(reverse: bool = False, random: bool = False) tuple¶
Return a boolean and a certificate for solvability.
Attempts to find a solution and certify nonexistence in parallel.
- dump(self, filename, compress=True)¶
File: sage/structure/sage_object.pyx (starting at line 445)
Same as self.save(filename, compress)
- dumps(self, compress=True)¶
File: sage/structure/sage_object.pyx (starting at line 451)
Dump
selfto a strings, which can later be reconstituted asselfusingloads(s).There is an optional boolean argument
compresswhich defaults toTrue.EXAMPLES:
sage: from sage.misc.persist import comp sage: O = SageObject() sage: p_comp = O.dumps() sage: p_uncomp = O.dumps(compress=False) sage: comp.decompress(p_comp) == p_uncomp True sage: import pickletools sage: pickletools.dis(p_uncomp) 0: \x80 PROTO 2 2: c GLOBAL 'sage.structure.sage_object SageObject' 41: q BINPUT ... 43: ) EMPTY_TUPLE 44: \x81 NEWOBJ 45: q BINPUT ... 47: . STOP highest protocol among opcodes = 2
- exists_orthogonal_vector(v) bool¶
Check if an orthogonal vector exists in the intervals.
- property intervals: vectors_in_intervals.intervals.Intervals¶
Return the corresponding intervals.
- property matrix: sage.matrix.constructor.matrix¶
Return the corresponding matrix.
- parent(self)¶
File: sage/structure/sage_object.pyx (starting at line 518)
Return the type of
selfto support the coercion framework.EXAMPLES:
sage: t = log(sqrt(2) - 1) + log(sqrt(2) + 1); t log(sqrt(2) + 1) + log(sqrt(2) - 1) sage: u = t.maxima_methods() sage: u.parent() <class 'sage.symbolic.maxima_wrapper.MaximaWrapper'>
- rename(self, x=None)¶
File: sage/structure/sage_object.pyx (starting at line 68)
Change self so it prints as x, where x is a string.
Note
This is only supported for Python classes that derive from SageObject.
EXAMPLES:
sage: x = PolynomialRing(QQ, 'x', sparse=True).gen() sage: g = x^3 + x - 5 sage: g x^3 + x - 5 sage: g.rename('a polynomial') sage: g a polynomial sage: g + x x^3 + 2*x - 5 sage: h = g^100 sage: str(h)[:20] 'x^300 + 100*x^298 - ' sage: h.rename('x^300 + ...') sage: h x^300 + ...
Real numbers are not Python classes, so rename is not supported:
sage: a = 3.14 sage: type(a) <... 'sage.rings.real_mpfr.RealLiteral'> sage: a.rename('pi') Traceback (most recent call last): ... NotImplementedError: object does not support renaming: 3.14000000000000
Note
The reason C-extension types are not supported by default is if they were then every single one would have to carry around an extra attribute, which would be slower and waste a lot of memory.
To support them for a specific class, add a
cdef public __custom_nameattribute.
- reset_name(self)¶
File: sage/structure/sage_object.pyx (starting at line 125)
Remove the custom name of an object.
EXAMPLES:
sage: P.<x> = QQ[] sage: P Univariate Polynomial Ring in x over Rational Field sage: P.rename('A polynomial ring') sage: P A polynomial ring sage: P.reset_name() sage: P Univariate Polynomial Ring in x over Rational Field
- save(self, filename=None, compress=True)¶
File: sage/structure/sage_object.pyx (starting at line 420)
Save self to the given filename.
EXAMPLES:
sage: f = x^3 + 5 sage: f.save(os.path.join(SAGE_TMP, 'file')) sage: load(os.path.join(SAGE_TMP, 'file.sobj')) x^3 + 5
- solve(reverse: bool = False, random: bool = False)¶
Compute a solution for this linear inequality system.
If no solution exists, a
ValueErroris raised.
- to_homogeneous() vectors_in_intervals.linear_inequality_systems.HomogeneousSystem¶
Return the equivalent homogeneous system.
- class vectors_in_intervals.linear_inequality_systems.LinearInequalitySystem(matrix: sage.matrix.constructor.matrix, intervals: Optional[vectors_in_intervals.intervals.Intervals] = None, result: Optional[bool] = None)¶
A class for linear inequality systems given by a matrix and intervals
- candidate_generator(dual: bool = True, reverse: bool = False, random: bool = False) collections.abc.Generator¶
Return a generator of elementary vectors.
- category(self)¶
File: sage/structure/sage_object.pyx (starting at line 484)
- certify(reverse: bool = False) tuple¶
Return a boolean and a certificate for solvability.
- certify_existence(reverse: bool = False, random: bool = False)¶
Certify existence of a solution if one exists.
Otherwise, a
ValueErroris raised.Note
If a solution exists and
randomis set to true, this method will never finish.
- certify_nonexistence(reverse: bool = False, random: bool = False)¶
Certify nonexistence of solutions.
Otherwise, a
ValueErroris raised.Note
If a solution exists and
randomis set to true, this method will never finish.
- certify_parallel(reverse: bool = False, random: bool = False) tuple¶
Return a boolean and a certificate for solvability.
Attempts to find a solution and certify nonexistence in parallel.
- dump(self, filename, compress=True)¶
File: sage/structure/sage_object.pyx (starting at line 445)
Same as self.save(filename, compress)
- dumps(self, compress=True)¶
File: sage/structure/sage_object.pyx (starting at line 451)
Dump
selfto a strings, which can later be reconstituted asselfusingloads(s).There is an optional boolean argument
compresswhich defaults toTrue.EXAMPLES:
sage: from sage.misc.persist import comp sage: O = SageObject() sage: p_comp = O.dumps() sage: p_uncomp = O.dumps(compress=False) sage: comp.decompress(p_comp) == p_uncomp True sage: import pickletools sage: pickletools.dis(p_uncomp) 0: \x80 PROTO 2 2: c GLOBAL 'sage.structure.sage_object SageObject' 41: q BINPUT ... 43: ) EMPTY_TUPLE 44: \x81 NEWOBJ 45: q BINPUT ... 47: . STOP highest protocol among opcodes = 2
- exists_orthogonal_vector(v) bool¶
Check if an orthogonal vector exists in the intervals.
- property intervals: vectors_in_intervals.intervals.Intervals¶
Return the corresponding intervals.
- property matrix: sage.matrix.constructor.matrix¶
Return the corresponding matrix.
- parent(self)¶
File: sage/structure/sage_object.pyx (starting at line 518)
Return the type of
selfto support the coercion framework.EXAMPLES:
sage: t = log(sqrt(2) - 1) + log(sqrt(2) + 1); t log(sqrt(2) + 1) + log(sqrt(2) - 1) sage: u = t.maxima_methods() sage: u.parent() <class 'sage.symbolic.maxima_wrapper.MaximaWrapper'>
- rename(self, x=None)¶
File: sage/structure/sage_object.pyx (starting at line 68)
Change self so it prints as x, where x is a string.
Note
This is only supported for Python classes that derive from SageObject.
EXAMPLES:
sage: x = PolynomialRing(QQ, 'x', sparse=True).gen() sage: g = x^3 + x - 5 sage: g x^3 + x - 5 sage: g.rename('a polynomial') sage: g a polynomial sage: g + x x^3 + 2*x - 5 sage: h = g^100 sage: str(h)[:20] 'x^300 + 100*x^298 - ' sage: h.rename('x^300 + ...') sage: h x^300 + ...
Real numbers are not Python classes, so rename is not supported:
sage: a = 3.14 sage: type(a) <... 'sage.rings.real_mpfr.RealLiteral'> sage: a.rename('pi') Traceback (most recent call last): ... NotImplementedError: object does not support renaming: 3.14000000000000
Note
The reason C-extension types are not supported by default is if they were then every single one would have to carry around an extra attribute, which would be slower and waste a lot of memory.
To support them for a specific class, add a
cdef public __custom_nameattribute.
- reset_name(self)¶
File: sage/structure/sage_object.pyx (starting at line 125)
Remove the custom name of an object.
EXAMPLES:
sage: P.<x> = QQ[] sage: P Univariate Polynomial Ring in x over Rational Field sage: P.rename('A polynomial ring') sage: P A polynomial ring sage: P.reset_name() sage: P Univariate Polynomial Ring in x over Rational Field
- save(self, filename=None, compress=True)¶
File: sage/structure/sage_object.pyx (starting at line 420)
Save self to the given filename.
EXAMPLES:
sage: f = x^3 + 5 sage: f.save(os.path.join(SAGE_TMP, 'file')) sage: load(os.path.join(SAGE_TMP, 'file.sobj')) x^3 + 5
- solve(reverse: bool = False, random: bool = False)¶
Compute a solution for this linear inequality system.
If no solution exists, a
ValueErroris raised.
- to_homogeneous() vectors_in_intervals.linear_inequality_systems.HomogeneousSystem¶
Return the equivalent homogeneous system.
- vectors_in_intervals.linear_inequality_systems.homogeneous_from_general(system: vectors_in_intervals.linear_inequality_systems.LinearInequalitySystem) vectors_in_intervals.linear_inequality_systems.HomogeneousSystem¶
Convert a general system to a homogeneous system.
EXAMPLE:
sage: from vectors_in_intervals import * sage: M = matrix([[1, 0], [0, 1], [1, 1]]) sage: lower_bounds = [2, 5, 0] sage: upper_bounds = [5, oo, 0] sage: lower_bounds_closed = [True, True, True] sage: upper_bounds_closed = [False, False, True] sage: I = Intervals.from_bounds(lower_bounds, upper_bounds, lower_bounds_closed, upper_bounds_closed) sage: S = LinearInequalitySystem(M, I) sage: homogeneous_from_general(S) [ 1 0 -5] [ 0 0 -1] [--------] [-1 0 2] [ 0 -1 5] [--------] [ 1 1 0] x in [(0, +oo), (0, +oo), [0, +oo), [0, +oo), {0}]
- vectors_in_intervals.linear_inequality_systems.homogeneous_from_inhomogeneous(system: vectors_in_intervals.linear_inequality_systems.InhomogeneousSystem) vectors_in_intervals.linear_inequality_systems.HomogeneousSystem¶
Convert an inhomogeneous system to a homogeneous system.
- vectors_in_intervals.linear_inequality_systems.inhomogeneous_from_general(system: vectors_in_intervals.linear_inequality_systems.LinearInequalitySystem) vectors_in_intervals.linear_inequality_systems.InhomogeneousSystem¶
Translate a general system into an inhomogeneous system.