Path: public / 2015-11-05 The Coercion Framework of SageMath.sagews

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Multiplying Apples and Oranges: Transparent Arithmetic with different Data Types in SageMath

Or: An Introduction to SageMath's Coercion Framework

Daniel Krenn, AAU Klagenfurt, Austria

@Purdue University, 5.11.2015

R.<a,b,c> = ZZ[[]]
R

Multivariate Power Series Ring in a, b, c over Integer Ring

a

a.parent()

Multivariate Power Series Ring in a, b, c over Integer Ring

1.parent()

Integer Ring

x.parent()

Symbolic Ring

1 / (1-a)

1 + a + a^2 + a^3 + a^4 + a^5 + a^6 + a^7 + a^8 + a^9 + a^10 + a^11 + O(a, b, c)^12

What is all this about?

11 + 5 / 2015

4434/403

11.parent()

Integer Ring

(5 / 2015).parent()

Rational Field

What is the Goal of Coercions

from SageMath's documentation:

"The primary goal [...] is to be able to transparently do
arithmetic, comparisons, etc. between elements of distinct
sets."

A short Look behind the Scenes

QQ.coerce_map_from(ZZ)  # seeing what's going on

Natural morphism:
  From: Integer Ring
  To:   Rational Field

An Exercise

P.<t> = ZZ[]
P

Univariate Polynomial Ring in t over Integer Ring

z0 = t
z0.parent()

Univariate Polynomial Ring in t over Integer Ring

z1 = t / 2
z1.parent()

Univariate Polynomial Ring in t over Rational Field

z2 = t / 1
z2.parent()

Univariate Polynomial Ring in t over Rational Field

z3 = 1 / t
z3.parent()

Fraction Field of Univariate Polynomial Ring in t over Integer Ring

(t / (1+I)).parent()

Symbolic Ring

I.parent()

Symbolic Ring

I^2

-1

K = QQ.extension(x^2+1, 'i'); i = K.gen()
K

Number Field in i with defining polynomial x^2 + 1

i^2

-1

(t/(1+i)).parent()

Univariate Polynomial Ring in t over Number Field in i with defining polynomial x^2 + 1

Coercions vs. Conversions

ZZ(2/1)

2

ZZ(3/2)

Error in lines 1-1
Traceback (most recent call last):
  File "/usr/local/lib/python2.7/dist-packages/smc_sagews/sage_server.py", line 905, in execute
    exec compile(block+'\n', '', 'single') in namespace, locals
  File "", line 1, in <module>
  File "sage/structure/parent.pyx", line 1098, in sage.structure.parent.Parent.__call__ (/projects/53b77207-8614-4086-a032-432af4b4cdbd/sage-dev-images/sage-6.10.beta4/src/build/cythonized/sage/structure/parent.c:9820)
    return mor._call_(x)
  File "sage/rings/rational.pyx", line 3745, in sage.rings.rational.Q_to_Z._call_ (/projects/53b77207-8614-4086-a032-432af4b4cdbd/sage-dev-images/sage-6.10.beta4/src/build/cythonized/sage/rings/rational.c:30593)
    raise TypeError, "no conversion of this rational to integer"
TypeError: no conversion of this rational to integer

Another Example: Real Numbers

RR

Real Field with 53 bits of precision

a = RR(pi)  # a conversion occurs here
a

3.14159265358979

pi.parent()

Symbolic Ring

R2 = RealField(2)
R2

Real Field with 2 bits of precision

b = R2(3)
b

3.0

R2(7)

8.0

R2(15)

16.

c = a + b
c, c.parent()

(6.0, Real Field with 2 bits of precision)

Comparisions

a == b

True

A more Challenging Example

P  # above: P.<t> = ZZ[]

Univariate Polynomial Ring in t over Integer Ring

d = (t^2 + 11*t) + 5/2015
d

t^2 + 11*t + 1/403

Q = d.parent()

P.coerce_map_from(QQ) is None, QQ.coerce_map_from(P) is None

(True, True)

Looking behind the Scene: Now really...

from sage.structure.element import get_coercion_model
cm = get_coercion_model()
cm  # What a strange object!

<sage.structure.coerce.CoercionModel_cache_maps object at 0x7f0c260f6738>

cm.common_parent(P, QQ)

Univariate Polynomial Ring in t over Rational Field

cm.explain(P, QQ)

Action discovered.
    Right scalar multiplication by Rational Field on Univariate Polynomial Ring in t over Integer Ring
Result lives in Univariate Polynomial Ring in t over Rational Field
Univariate Polynomial Ring in t over Rational Field

Discovering new Parents

M.<m, n> = ZZ[]
M

Multivariate Polynomial Ring in m, n over Integer Ring

alpha = m^2 * n + 42 * n^2
alpha

m^2*n + 42*n^2

N.<n, o> = QQ[]
N

Multivariate Polynomial Ring in n, o over Rational Field

beta = n^2 / 3 + o
beta

1/3*n^2 + o

gamma = alpha + beta
gamma, gamma.parent()

(m^2*n + 127/3*n^2 + o, Multivariate Polynomial Ring in m, n, o over Rational Field)

cm.explain(M, N)

Coercion on left operand via
    Conversion map:
      From: Multivariate Polynomial Ring in m, n over Integer Ring
      To:   Multivariate Polynomial Ring in m, n, o over Rational Field
Coercion on right operand via
    Conversion map:
      From: Multivariate Polynomial Ring in n, o over Rational Field
      To:   Multivariate Polynomial Ring in m, n, o over Rational Field
Arithmetic performed after coercions.
Result lives in Multivariate Polynomial Ring in m, n, o over Rational Field
Multivariate Polynomial Ring in m, n, o over Rational Field

Constructions for Discovering new Parents

M.construction()

(MPoly[m,n], Integer Ring)

N.construction()

(MPoly[n,o], Rational Field)

QQ.construction()

(FractionField, Integer Ring)

cm.common_parent(M, N).construction()

(MPoly[m,n,o], Rational Field)

Properties of Coercions / Axioms

A coercion is defined on all elements of a parent.
Coercions are structure preserving.
There is at most one coercion from one parent to another.
Coercions can be composed.
The identity is a coercion