vectors_in_intervals.utility

Utility functions

Functions

solve_left_for_roots(A, b)	Find a solution for x*A = b that works for matrices whose entries are roots.
solve_without_division(A, b)	Solve a linear system of equations without division.

Classes

CombinationsIncluding(mset, k[, elements])

Combinatorial object of all combinations that include given elements

class vectors_in_intervals.utility.CombinationsIncluding(mset, k, elements=None)

Combinatorial object of all combinations that include given elements

EXAMPLES:

We generate all subsets of range(4) with 2 elements that include the element 2:

sage: from vectors_in_intervals.utility import CombinationsIncluding
sage: C = CombinationsIncluding(4, 2, [2])
sage: list(C)
[[0, 2], [1, 2], [2, 3]]
sage: list(reversed(C))
[[2, 3], [1, 2], [0, 2]]

category(self)

File: sage/structure/sage_object.pyx (starting at line 484)

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 self to a string s, which can later be reconstituted as self using loads(s).

There is an optional boolean argument compress which defaults to True.

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

parent(self)

File: sage/structure/sage_object.pyx (starting at line 518)

Return the type of self to 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_name attribute.

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

vectors_in_intervals.utility.solve_left_for_roots(A, b)

Find a solution for x*A = b that works for matrices whose entries are roots.

INPUT:

• A – a matrix

• b – a vector

NOTE:

The built in method ``solve_left`` for matrices fails occasionally.

vectors_in_intervals.utility.solve_without_division(A, b)

Solve a linear system of equations without division.

The system is A x = c b where c is a positive constant. Uses an elementary vector.

EXAMPLES:

sage: from vectors_in_intervals.utility import solve_without_division
sage: A = matrix([[1, 2], [0, 1], [1, -1]])
sage: b = vector([1, 0, 1])
sage: solve_without_division(A, b)
(1, 0)
sage: A = matrix([[1, 4], [0, 2], [1, -2]])
sage: b = vector([6, 2, 0])
sage: solve_without_division(A, b)
(4, 2)
sage: A.solve_right(b)
(2, 1)
sage: A = matrix([[1, 1, 1], [0, 1, 2]])
sage: b = vector([2, 3])
sage: solve_without_division(A, b)
(0, 1, 1)