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

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

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

```
>>> 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: 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'>
```

```
>>> 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: 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'>
```

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

```
>>> 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: 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
```

```
>>> 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.

## Lists, Tuples, and Sequences#

The list data type stores elements of arbitrary type. Like in C, C++, etc. (but unlike most standard computer algebra systems), the elements of the list are indexed starting from \(0\):

```
sage: v = [2, 3, 5, 'x', SymmetricGroup(3)]; v
[2, 3, 5, 'x', Symmetric group of order 3! as a permutation group]
sage: type(v)
<... 'list'>
sage: v[0]
2
sage: v[2]
5
```

```
>>> from sage.all import *
>>> v = [Integer(2), Integer(3), Integer(5), 'x', SymmetricGroup(Integer(3))]; v
[2, 3, 5, 'x', Symmetric group of order 3! as a permutation group]
>>> type(v)
<... 'list'>
>>> v[Integer(0)]
2
>>> v[Integer(2)]
5
```

(When indexing into a list, it is OK if the index is
not a Python int!)
A Sage Integer (or Rational, or anything with an `__index__`

method)
will work just fine.

```
sage: v = [1,2,3]
sage: v[2]
3
sage: n = 2 # Sage Integer
sage: v[n] # Perfectly OK!
3
sage: v[int(n)] # Also OK.
3
```

```
>>> from sage.all import *
>>> v = [Integer(1),Integer(2),Integer(3)]
>>> v[Integer(2)]
3
>>> n = Integer(2) # Sage Integer
>>> v[n] # Perfectly OK!
3
>>> v[int(n)] # Also OK.
3
```

The `range`

function creates a list of Python int’s (not Sage
Integers):

```
sage: list(range(1, 15))
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]
```

```
>>> from sage.all import *
>>> list(range(Integer(1), Integer(15)))
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]
```

This is useful when using list comprehensions to construct lists:

```
sage: L = [factor(n) for n in range(1, 15)]
sage: L
[1, 2, 3, 2^2, 5, 2 * 3, 7, 2^3, 3^2, 2 * 5, 11, 2^2 * 3, 13, 2 * 7]
sage: L[12]
13
sage: type(L[12])
<class 'sage.structure.factorization_integer.IntegerFactorization'>
sage: [factor(n) for n in range(1, 15) if is_odd(n)]
[1, 3, 5, 7, 3^2, 11, 13]
```

```
>>> from sage.all import *
>>> L = [factor(n) for n in range(Integer(1), Integer(15))]
>>> L
[1, 2, 3, 2^2, 5, 2 * 3, 7, 2^3, 3^2, 2 * 5, 11, 2^2 * 3, 13, 2 * 7]
>>> L[Integer(12)]
13
>>> type(L[Integer(12)])
<class 'sage.structure.factorization_integer.IntegerFactorization'>
>>> [factor(n) for n in range(Integer(1), Integer(15)) if is_odd(n)]
[1, 3, 5, 7, 3^2, 11, 13]
```

For more about how to create lists using list comprehensions, see [PyT].

List slicing is a wonderful feature. If `L`

is a list, then `L[m:n]`

returns the sublist of `L`

obtained by
starting at the \(m^{th}\) element and stopping at the
\((n-1)^{st}\) element, as illustrated below.

```
sage: L = [factor(n) for n in range(1, 20)]
sage: L[4:9]
[5, 2 * 3, 7, 2^3, 3^2]
sage: L[:4]
[1, 2, 3, 2^2]
sage: L[14:4]
[]
sage: L[14:]
[3 * 5, 2^4, 17, 2 * 3^2, 19]
```

```
>>> from sage.all import *
>>> L = [factor(n) for n in range(Integer(1), Integer(20))]
>>> L[Integer(4):Integer(9)]
[5, 2 * 3, 7, 2^3, 3^2]
>>> L[:Integer(4)]
[1, 2, 3, 2^2]
>>> L[Integer(14):Integer(4)]
[]
>>> L[Integer(14):]
[3 * 5, 2^4, 17, 2 * 3^2, 19]
```

Tuples are similar to lists, except they are immutable, meaning once they are created they can’t be changed.

```
sage: v = (1,2,3,4); v
(1, 2, 3, 4)
sage: type(v)
<... 'tuple'>
sage: v[1] = 5
Traceback (most recent call last):
...
TypeError: 'tuple' object does not support item assignment
```

```
>>> from sage.all import *
>>> v = (Integer(1),Integer(2),Integer(3),Integer(4)); v
(1, 2, 3, 4)
>>> type(v)
<... 'tuple'>
>>> v[Integer(1)] = Integer(5)
Traceback (most recent call last):
...
TypeError: 'tuple' object does not support item assignment
```

Sequences are a third list-oriented Sage
type. Unlike lists and tuples, Sequence is not a built-in Python
type. By default, a sequence is mutable, but using the `Sequence`

class method `set_immutable`

, it can be set to be immutable, as
the following example illustrates. All elements of a sequence have
a common parent, called the sequences universe.

```
sage: v = Sequence([1,2,3,4/5])
sage: v
[1, 2, 3, 4/5]
sage: type(v)
<class 'sage.structure.sequence.Sequence_generic'>
sage: type(v[1])
<class 'sage.rings.rational.Rational'>
sage: v.universe()
Rational Field
sage: v.is_immutable()
False
sage: v.set_immutable()
sage: v[0] = 3
Traceback (most recent call last):
...
ValueError: object is immutable; please change a copy instead.
```

```
>>> from sage.all import *
>>> v = Sequence([Integer(1),Integer(2),Integer(3),Integer(4)/Integer(5)])
>>> v
[1, 2, 3, 4/5]
>>> type(v)
<class 'sage.structure.sequence.Sequence_generic'>
>>> type(v[Integer(1)])
<class 'sage.rings.rational.Rational'>
>>> v.universe()
Rational Field
>>> v.is_immutable()
False
>>> v.set_immutable()
>>> v[Integer(0)] = Integer(3)
Traceback (most recent call last):
...
ValueError: object is immutable; please change a copy instead.
```

Sequences derive from lists and can be used anywhere a list can be used:

```
sage: v = Sequence([1,2,3,4/5])
sage: isinstance(v, list)
True
sage: list(v)
[1, 2, 3, 4/5]
sage: type(list(v))
<... 'list'>
```

```
>>> from sage.all import *
>>> v = Sequence([Integer(1),Integer(2),Integer(3),Integer(4)/Integer(5)])
>>> isinstance(v, list)
True
>>> list(v)
[1, 2, 3, 4/5]
>>> type(list(v))
<... 'list'>
```

As another example, basis for vector spaces are immutable sequences, since it’s important that you don’t change them.

```
sage: V = QQ^3; B = V.basis(); B
[
(1, 0, 0),
(0, 1, 0),
(0, 0, 1)
]
sage: type(B)
<class 'sage.structure.sequence.Sequence_generic'>
sage: B[0] = B[1]
Traceback (most recent call last):
...
ValueError: object is immutable; please change a copy instead.
sage: B.universe()
Vector space of dimension 3 over Rational Field
```

```
>>> from sage.all import *
>>> V = QQ**Integer(3); B = V.basis(); B
[
(1, 0, 0),
(0, 1, 0),
(0, 0, 1)
]
>>> type(B)
<class 'sage.structure.sequence.Sequence_generic'>
>>> B[Integer(0)] = B[Integer(1)]
Traceback (most recent call last):
...
ValueError: object is immutable; please change a copy instead.
>>> B.universe()
Vector space of dimension 3 over Rational Field
```

## 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: 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
```

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

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

```
>>> 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: 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}
```

```
>>> 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: 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
```

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

```
>>> 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: 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
```

```
>>> 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: [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)
```

```
>>> 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)
```

## Loops, Functions, Control Statements, and Comparisons#

We have seen a few examples already of some common uses of `for`

loops. In Python, a `for`

loop has an indented structure, such as

```
>>> for i in range(5):
... print(i)
...
0
1
2
3
4
```

Note the colon at the end of the for statement (there is no “do” or
“od” as in GAP or Maple), and the indentation before the “body” of
the loop, namely `print(i)`

. This indentation is important. In
Sage, the indentation is automatically put in for you when you hit
`enter`

after a “:”, as illustrated below.

```
sage: for i in range(5):
....: print(i) # now hit enter twice
....:
0
1
2
3
4
```

```
>>> from sage.all import *
>>> for i in range(Integer(5)):
... print(i) # now hit enter twice
....:
0
1
2
3
4
```

The symbol `=`

is used for assignment.
The symbol `==`

is used to check for equality:

```
sage: for i in range(15):
....: if gcd(i,15) == 1:
....: print(i)
....:
1
2
4
7
8
11
13
14
```

```
>>> from sage.all import *
>>> for i in range(Integer(15)):
... if gcd(i,Integer(15)) == Integer(1):
... print(i)
....:
1
2
4
7
8
11
13
14
```

Keep in mind how indentation determines the block structure for
`if`

, `for`

, and `while`

statements:

```
sage: def legendre(a,p):
....: is_sqr_modp=-1
....: for i in range(p):
....: if a % p == i^2 % p:
....: is_sqr_modp=1
....: return is_sqr_modp
sage: legendre(2,7)
1
sage: legendre(3,7)
-1
```

```
>>> from sage.all import *
>>> def legendre(a,p):
... is_sqr_modp=-Integer(1)
... for i in range(p):
... if a % p == i**Integer(2) % p:
... is_sqr_modp=Integer(1)
... return is_sqr_modp
>>> legendre(Integer(2),Integer(7))
1
>>> legendre(Integer(3),Integer(7))
-1
```

Of course this is not an efficient implementation of the Legendre symbol! It is meant to illustrate various aspects of Python/Sage programming. The function {kronecker}, which comes with Sage, computes the Legendre symbol efficiently via a C-library call to PARI.

Finally, we note that comparisons, such as `==`

,
`!=`

, `<=`

, `>=`

, `>`

, `<`

, between numbers will automatically
convert both numbers into the same type if possible:

```
sage: 2 < 3.1; 3.1 <= 1
True
False
sage: 2/3 < 3/2; 3/2 < 3/1
True
True
```

```
>>> from sage.all import *
>>> Integer(2) < RealNumber('3.1'); RealNumber('3.1') <= Integer(1)
True
False
>>> Integer(2)/Integer(3) < Integer(3)/Integer(2); Integer(3)/Integer(2) < Integer(3)/Integer(1)
True
True
```

Use bool for symbolic inequalities:

```
sage: x < x + 1
x < x + 1
sage: bool(x < x + 1)
True
```

```
>>> from sage.all import *
>>> x < x + Integer(1)
x < x + 1
>>> bool(x < x + Integer(1))
True
```

When comparing objects of different types in Sage, in most cases
Sage tries to find a canonical coercion of both objects to a common
parent (see Parents, Conversion and Coercion for more details). If successful,
the comparison is performed between the coerced objects; if not successful,
the objects are considered not equal. For testing whether two variables
reference the same object use `is`

. As we see in this example,
the Python int `1`

is unique, but the Sage Integer `1`

is not:

```
sage: 1 is 2/2
False
sage: 1 is 1
False
sage: 1 == 2/2
True
```

```
>>> from sage.all import *
>>> Integer(1) is Integer(2)/Integer(2)
False
>>> Integer(1) is Integer(1)
False
>>> Integer(1) == Integer(2)/Integer(2)
True
```

In the following two lines, the first equality is `False`

because
there is no canonical morphism \(\QQ\to \GF{5}\), hence no
canonical way to compare the \(1\) in \(\GF{5}\) to the
\(1 \in \QQ\). In contrast, there is a canonical map
\(\ZZ \to \GF{5}\), hence the second comparison is `True`

. Note
also that the order doesn’t matter.

```
sage: GF(5)(1) == QQ(1); QQ(1) == GF(5)(1)
False
False
sage: GF(5)(1) == ZZ(1); ZZ(1) == GF(5)(1)
True
True
sage: ZZ(1) == QQ(1)
True
```

```
>>> from sage.all import *
>>> GF(Integer(5))(Integer(1)) == QQ(Integer(1)); QQ(Integer(1)) == GF(Integer(5))(Integer(1))
False
False
>>> GF(Integer(5))(Integer(1)) == ZZ(Integer(1)); ZZ(Integer(1)) == GF(Integer(5))(Integer(1))
True
True
>>> ZZ(Integer(1)) == QQ(Integer(1))
True
```

WARNING: Comparison in Sage is more restrictive than in Magma, which declares the \(1 \in \GF{5}\) equal to \(1 \in \QQ\).

```
sage: magma('GF(5)!1 eq Rationals()!1') # optional - magma
true
```

```
>>> from sage.all import *
>>> magma('GF(5)!1 eq Rationals()!1') # optional - magma
true
```

## Profiling#

“Premature optimization is the root of all evil.” - Donald Knuth

*Section author: Martin Albrecht <malb@informatik.uni-bremen.de>*

Sometimes it is useful to check for bottlenecks in code to understand which parts take the most computational time; this can give a good idea of which parts to optimize. Python and therefore Sage offers several profiling–as this process is called–options.

The simplest to use is the `prun`

command in the interactive
shell. It returns a summary describing which functions took how
much computational time. To profile (the currently slow! - as of
version 1.0) matrix multiplication over finite fields, for example,
do:

```
sage: k,a = GF(2**8, 'a').objgen()
sage: A = Matrix(k,10,10,[k.random_element() for _ in range(10*10)])
```

```
>>> from sage.all import *
>>> k,a = GF(Integer(2)**Integer(8), 'a').objgen()
>>> A = Matrix(k,Integer(10),Integer(10),[k.random_element() for _ in range(Integer(10)*Integer(10))])
```

```
sage: %prun B = A*A
32893 function calls in 1.100 CPU seconds
Ordered by: internal time
ncalls tottime percall cumtime percall filename:lineno(function)
12127 0.160 0.000 0.160 0.000 :0(isinstance)
2000 0.150 0.000 0.280 0.000 matrix.py:2235(__getitem__)
1000 0.120 0.000 0.370 0.000 finite_field_element.py:392(__mul__)
1903 0.120 0.000 0.200 0.000 finite_field_element.py:47(__init__)
1900 0.090 0.000 0.220 0.000 finite_field_element.py:376(__compat)
900 0.080 0.000 0.260 0.000 finite_field_element.py:380(__add__)
1 0.070 0.070 1.100 1.100 matrix.py:864(__mul__)
2105 0.070 0.000 0.070 0.000 matrix.py:282(ncols)
...
```

```
>>> from sage.all import *
>>> %prun B = A*A
32893 function calls in 1.100 CPU seconds
Ordered by: internal time
ncalls tottime percall cumtime percall filename:lineno(function)
12127 0.160 0.000 0.160 0.000 :0(isinstance)
2000 0.150 0.000 0.280 0.000 matrix.py:2235(__getitem__)
1000 0.120 0.000 0.370 0.000 finite_field_element.py:392(__mul__)
1903 0.120 0.000 0.200 0.000 finite_field_element.py:47(__init__)
1900 0.090 0.000 0.220 0.000 finite_field_element.py:376(__compat)
900 0.080 0.000 0.260 0.000 finite_field_element.py:380(__add__)
1 0.070 0.070 1.100 1.100 matrix.py:864(__mul__)
2105 0.070 0.000 0.070 0.000 matrix.py:282(ncols)
...
```

Here `ncalls`

is the number of calls, `tottime`

is the total time
spent in the given function (and excluding time made in calls to
sub-functions), `percall`

is the quotient of `tottime`

divided by
`ncalls`

. `cumtime`

is the total time spent in this and all
sub-functions (i.e., from invocation until exit), `percall`

is the
quotient of `cumtime`

divided by primitive calls, and
`filename:lineno(function)`

provides the respective data of each
function. The rule of thumb here is: The higher the function in
that listing, the more expensive it is. Thus it is more interesting
for optimization.

As usual, `prun?`

provides details on how to use the profiler and
understand the output.

The profiling data may be written to an object as well to allow closer examination:

```
sage: %prun -r A*A
sage: stats = _
sage: stats?
```

```
>>> from sage.all import *
>>> %prun -r A*A
>>> stats = _
>>> stats?
```

Note: entering `stats = prun -r A\*A`

displays a syntax error
message because prun is an IPython shell command, not a regular
function.

For a nice graphical representation of profiling data, you can use
the hotshot profiler, a small script called `hotshot2cachetree`

and
the program `kcachegrind`

(Unix only). The same example with the
hotshot profiler:

```
sage: k,a = GF(2**8, 'a').objgen()
sage: A = Matrix(k,10,10,[k.random_element() for _ in range(10*10)])
sage: import hotshot
sage: filename = "pythongrind.prof"
sage: prof = hotshot.Profile(filename, lineevents=1)
```

```
>>> from sage.all import *
>>> k,a = GF(Integer(2)**Integer(8), 'a').objgen()
>>> A = Matrix(k,Integer(10),Integer(10),[k.random_element() for _ in range(Integer(10)*Integer(10))])
>>> import hotshot
>>> filename = "pythongrind.prof"
>>> prof = hotshot.Profile(filename, lineevents=Integer(1))
```

```
sage: prof.run("A*A")
<hotshot.Profile instance at 0x414c11ec>
sage: prof.close()
```

```
>>> from sage.all import *
>>> prof.run("A*A")
<hotshot.Profile instance at 0x414c11ec>
>>> prof.close()
```

This results in a file `pythongrind.prof`

in the current working
directory. It can now be converted to the cachegrind format for
visualization.

On a system shell, type

```
$ hotshot2calltree -o cachegrind.out.42 pythongrind.prof
```

The output file `cachegrind.out.42`

can now be examined with
`kcachegrind`

. Please note that the naming convention
`cachegrind.out.XX`

needs to be obeyed.