Programming¶

Loading and Attaching Sage files¶

Next we illustrate how to load programs written in a separate file into Sage. Create a file called example.sage with the following content:

print("Hello World")
print(2^3)

You can read in and execute example.sage file using the load command.

Sage

sage: load("example.sage")
Hello World
8

Python

>>> from sage.all import *
>>> load("example.sage")
Hello World
8

You can also attach a Sage file to a running session using the attach command:

Sage

sage: attach("example.sage")
Hello World
8

Python

>>> from sage.all import *
>>> attach("example.sage")
Hello World
8

Now if you change example.sage and enter one blank line into Sage (i.e., hit return), then the contents of example.sage will be automatically reloaded into Sage.

In particular, attach automatically reloads a file whenever it changes, which is handy when debugging code, whereas load only loads a file once.

When Sage loads example.sage it converts it to Python, which is then executed by the Python interpreter. This conversion is minimal; it mainly involves wrapping integer literals in Integer() floating point literals in RealNumber(), replacing ^’s by **’s, and replacing e.g., R.2 by R.gen(2). The converted version of example.sage is contained in the same directory as example.sage and is called example.sage.py. This file contains the following code:

print("Hello World")
print(Integer(2)**Integer(3))

Integer literals are wrapped and the ^ is replaced by a **. (In Python ^ means “exclusive or” and ** means “exponentiation”.)

(This preparsing is implemented in sage/misc/interpreter.py.)

You can paste multi-line indented code into Sage as long as there are newlines to make new blocks (this is not necessary in files). However, the best way to enter such code into Sage is to save it to a file and use attach, as described above.

Creating Compiled Code¶

Speed is crucial in mathematical computations. Though Python is a convenient very high-level language, certain calculations can be several orders of magnitude faster than in Python if they are implemented using static types in a compiled language. Some aspects of Sage would have been too slow if it had been written entirely in Python. To deal with this, Sage supports a compiled “version” of Python called Cython ([Cyt] and [Pyr]). Cython is simultaneously similar to both Python and C. Most Python constructions, including list comprehensions, conditional expressions, code like += are allowed; you can also import code that you have written in other Python modules. Moreover, you can declare arbitrary C variables, and arbitrary C library calls can be made directly. The resulting code is converted to C and compiled using a C compiler.

In order to make your own compiled Sage code, give the file an .spyx extension (instead of .sage). If you are working with the command-line interface, you can attach and load compiled code exactly like with interpreted code (at the moment, attaching and loading Cython code is not supported with the notebook interface). The actual compilation is done “behind the scenes” without your having to do anything explicit. The compiled shared object library is stored under $HOME/.sage/temp/hostname/pid/spyx. These files are deleted when you exit Sage.

NO Sage preparsing is applied to spyx files, e.g., 1/3 will result in 0 in a spyx file instead of the rational number $1/3$. If foo is a function in the Sage library, to use it from a spyx file import sage.all and use sage.all.foo.

import sage.all
def foo(n):
    return sage.all.factorial(n)

Accessing C Functions in Separate Files¶

It is also easy to access C functions defined in separate *.c files. Here’s an example. Create files test.c and test.spyx in the same directory with contents:

The pure C code: test.c

int add_one(int n) {
  return n + 1;
}

The Cython code: test.spyx:

cdef extern from "test.c":
    int add_one(int n)

def test(n):
    return add_one(n)

Then the following works:

Sage

sage: attach("test.spyx")
Compiling (...)/test.spyx...
sage: test(10)
11

Python

>>> from sage.all import *
>>> attach("test.spyx")
Compiling (...)/test.spyx...
>>> test(Integer(10))
11

If an additional library foo is needed to compile the C code generated from a Cython file, add the line clib foo to the Cython source. Similarly, an additional C file bar can be included in the compilation with the declaration cfile bar.

Standalone Python/Sage Scripts¶

The following standalone Sage script factors integers, polynomials, etc:

#!/usr/bin/env sage

import sys

if len(sys.argv) != 2:
    print("Usage: %s <n>" % sys.argv[0])
    print("Outputs the prime factorization of n.")
    sys.exit(1)

print(factor(sage_eval(sys.argv[1])))

In order to use this script, your SAGE_ROOT must be in your PATH. If the above script is called factor, here is an example usage:

$ ./factor 2006
2 * 17 * 59

Data Types¶

Every object in Sage has a well-defined type. Python has a wide range of basic built-in types, and the Sage library adds many more. Some built-in Python types include strings, lists, tuples, ints and floats, as illustrated:

Sage

sage: s = "sage"; type(s)
<... 'str'>
sage: s = 'sage'; type(s)      # you can use either single or double quotes
<... 'str'>
sage: s = [1,2,3,4]; type(s)
<... 'list'>
sage: s = (1,2,3,4); type(s)
<... 'tuple'>
sage: s = int(2006); type(s)
<... 'int'>
sage: s = float(2006); type(s)
<... 'float'>

Python

>>> from sage.all import *
>>> s = "sage"; type(s)
<... 'str'>
>>> s = 'sage'; type(s)      # you can use either single or double quotes
<... 'str'>
>>> s = [Integer(1),Integer(2),Integer(3),Integer(4)]; type(s)
<... 'list'>
>>> s = (Integer(1),Integer(2),Integer(3),Integer(4)); type(s)
<... 'tuple'>
>>> s = int(Integer(2006)); type(s)
<... 'int'>
>>> s = float(Integer(2006)); type(s)
<... 'float'>

To this, Sage adds many other types. E.g., vector spaces:

Sage

sage: V = VectorSpace(QQ, 1000000); V
Vector space of dimension 1000000 over Rational Field
sage: type(V)
<class 'sage.modules.free_module.FreeModule_ambient_field_with_category'>

Python

>>> from sage.all import *
>>> V = VectorSpace(QQ, Integer(1000000)); V
Vector space of dimension 1000000 over Rational Field
>>> type(V)
<class 'sage.modules.free_module.FreeModule_ambient_field_with_category'>

Only certain functions can be called on V. In other math software systems, these would be called using the “functional” notation foo(V,...). In Sage, certain functions are attached to the type (or class) of V, and are called using an object-oriented syntax like in Java or C++, e.g., V.foo(...). This helps keep the global namespace from being polluted with tens of thousands of functions, and means that many different functions with different behavior can be named foo, without having to use type-checking of arguments (or case statements) to decide which to call. Also, if you reuse the name of a function, that function is still available (e.g., if you call something zeta, then want to compute the value of the Riemann-Zeta function at 0.5, you can still type s=.5; s.zeta()).

Sage

sage: zeta = -1
sage: s=.5; s.zeta()
-1.46035450880959

Python

>>> from sage.all import *
>>> zeta = -Integer(1)
>>> s=RealNumber('.5'); s.zeta()
-1.46035450880959

In some very common cases, the usual functional notation is also supported for convenience and because mathematical expressions might look confusing using object-oriented notation. Here are some examples.

Sage

sage: n = 2; n.sqrt()
sqrt(2)
sage: sqrt(2)
sqrt(2)
sage: V = VectorSpace(QQ,2)
sage: V.basis()
    [
    (1, 0),
    (0, 1)
    ]
sage: basis(V)
    [
    (1, 0),
    (0, 1)
    ]
sage: M = MatrixSpace(GF(7), 2); M
Full MatrixSpace of 2 by 2 dense matrices over Finite Field of size 7
sage: A = M([1,2,3,4]); A
[1 2]
[3 4]
sage: A.charpoly('x')
x^2 + 2*x + 5
sage: charpoly(A, 'x')
x^2 + 2*x + 5

Python

>>> from sage.all import *
>>> n = Integer(2); n.sqrt()
sqrt(2)
>>> sqrt(Integer(2))
sqrt(2)
>>> V = VectorSpace(QQ,Integer(2))
>>> V.basis()
    [
    (1, 0),
    (0, 1)
    ]
>>> basis(V)
    [
    (1, 0),
    (0, 1)
    ]
>>> M = MatrixSpace(GF(Integer(7)), Integer(2)); M
Full MatrixSpace of 2 by 2 dense matrices over Finite Field of size 7
>>> A = M([Integer(1),Integer(2),Integer(3),Integer(4)]); A
[1 2]
[3 4]
>>> A.charpoly('x')
x^2 + 2*x + 5
>>> charpoly(A, 'x')
x^2 + 2*x + 5

To list all member functions for $A$, use tab completion. Just type A., then type the [tab] key on your keyboard, as explained in Reverse Search and Tab Completion.

Dictionaries¶

A dictionary (also sometimes called an associative array) is a mapping from ‘hashable’ objects (e.g., strings, numbers, and tuples of such; see the Python documentation http://docs.python.org/tut/node7.html and http://docs.python.org/lib/typesmapping.html for details) to arbitrary objects.

Sage

sage: d = {1:5, 'sage':17, ZZ:GF(7)}
sage: type(d)
<... 'dict'>
sage: list(d.keys())
[1, 'sage', Integer Ring]
sage: d['sage']
17
sage: d[ZZ]
Finite Field of size 7
sage: d[1]
5

Python

>>> from sage.all import *
>>> d = {Integer(1):Integer(5), 'sage':Integer(17), ZZ:GF(Integer(7))}
>>> type(d)
<... 'dict'>
>>> list(d.keys())
[1, 'sage', Integer Ring]
>>> d['sage']
17
>>> d[ZZ]
Finite Field of size 7
>>> d[Integer(1)]
5

The third key illustrates that the indexes of a dictionary can be complicated, e.g., the ring of integers.

You can turn the above dictionary into a list with the same data:

Sage

sage: list(d.items())
[(1, 5), ('sage', 17), (Integer Ring, Finite Field of size 7)]

Python

>>> from sage.all import *
>>> list(d.items())
[(1, 5), ('sage', 17), (Integer Ring, Finite Field of size 7)]

A common idiom is to iterate through the pairs in a dictionary:

Sage

sage: d = {2:4, 3:9, 4:16}
sage: [a*b for a, b in d.items()]
[8, 27, 64]

Python

>>> from sage.all import *
>>> d = {Integer(2):Integer(4), Integer(3):Integer(9), Integer(4):Integer(16)}
>>> [a*b for a, b in d.items()]
[8, 27, 64]

A dictionary is unordered, as the last output illustrates.

Sets¶

Python has a built-in set type. The main feature it offers is very fast lookup of whether an element is in the set or not, along with standard set-theoretic operations.

Sage

sage: X = set([1,19,'a']);   Y = set([1,1,1, 2/3])
sage: X   # random sort order
{1, 19, 'a'}
sage: X == set(['a', 1, 1, 19])
True
sage: Y
{2/3, 1}
sage: 'a' in X
True
sage: 'a' in Y
False
sage: X.intersection(Y)
{1}

Python

>>> from sage.all import *
>>> X = set([Integer(1),Integer(19),'a']);   Y = set([Integer(1),Integer(1),Integer(1), Integer(2)/Integer(3)])
>>> X   # random sort order
{1, 19, 'a'}
>>> X == set(['a', Integer(1), Integer(1), Integer(19)])
True
>>> Y
{2/3, 1}
>>> 'a' in X
True
>>> 'a' in Y
False
>>> X.intersection(Y)
{1}

Sage also has its own set type that is (in some cases) implemented using the built-in Python set type, but has a little bit of extra Sage-related functionality. Create a Sage set using Set(...). For example,

Sage

sage: X = Set([1,19,'a']);   Y = Set([1,1,1, 2/3])
sage: X   # random sort order
{'a', 1, 19}
sage: X == Set(['a', 1, 1, 19])
True
sage: Y
{1, 2/3}
sage: X.intersection(Y)
{1}
sage: print(latex(Y))
\left\{1, \frac{2}{3}\right\}
sage: Set(ZZ)
Set of elements of Integer Ring

Python

>>> from sage.all import *
>>> X = Set([Integer(1),Integer(19),'a']);   Y = Set([Integer(1),Integer(1),Integer(1), Integer(2)/Integer(3)])
>>> X   # random sort order
{'a', 1, 19}
>>> X == Set(['a', Integer(1), Integer(1), Integer(19)])
True
>>> Y
{1, 2/3}
>>> X.intersection(Y)
{1}
>>> print(latex(Y))
\left\{1, \frac{2}{3}\right\}
>>> Set(ZZ)
Set of elements of Integer Ring

Iterators¶

Iterators are a recent addition to Python that are particularly useful in mathematics applications. Here are several examples; see [PyT] for more details. We make an iterator over the squares of the nonnegative integers up to $10000000$.

Sage

sage: v = (n^2 for n in range(10000000))
sage: next(v)
0
sage: next(v)
1
sage: next(v)
4

Python

>>> from sage.all import *
>>> v = (n**Integer(2) for n in range(Integer(10000000)))
>>> next(v)
0
>>> next(v)
1
>>> next(v)
4

We create an iterate over the primes of the form $4p+1$ with $p$ also prime, and look at the first few values.

Sage

sage: w = (4*p + 1 for p in Primes() if is_prime(4*p+1))
sage: w         # in the next line, 0xb0853d6c is a random 0x number
<generator object at 0xb0853d6c>
sage: next(w)
13
sage: next(w)
29
sage: next(w)
53

Python

>>> from sage.all import *
>>> w = (Integer(4)*p + Integer(1) for p in Primes() if is_prime(Integer(4)*p+Integer(1)))
>>> w         # in the next line, 0xb0853d6c is a random 0x number
<generator object at 0xb0853d6c>
>>> next(w)
13
>>> next(w)
29
>>> next(w)
53

Certain rings, e.g., finite fields and the integers have iterators associated to them:

Sage

sage: [x for x in GF(7)]
[0, 1, 2, 3, 4, 5, 6]
sage: W = ((x,y) for x in ZZ for y in ZZ)
sage: next(W)
(0, 0)
sage: next(W)
(0, 1)
sage: next(W)
(0, -1)

Python

>>> from sage.all import *
>>> [x for x in GF(Integer(7))]
[0, 1, 2, 3, 4, 5, 6]
>>> W = ((x,y) for x in ZZ for y in ZZ)
>>> next(W)
(0, 0)
>>> next(W)
(0, 1)
>>> next(W)
(0, -1)

Programming¶

Loading and Attaching Sage files¶

Creating Compiled Code¶

Accessing C Functions in Separate Files¶

Standalone Python/Sage Scripts¶

Data Types¶

Lists, Tuples, and Sequences¶

Dictionaries¶

Sets¶

Iterators¶

Loops, Functions, Control Statements, and Comparisons¶

Profiling¶