Getting Help - Tutorial (original) (raw)

Sage has extensive built-in documentation, accessible by typing the name of a function or a constant (for example), followed by a question mark:

Sage

sage: tan? Type: <class 'sage.calculus.calculus.Function_tan'> Definition: tan( [noargspec] ) Docstring:

The tangent function

EXAMPLES:
    sage: tan(pi)
    0
    sage: tan(3.1415)
    -0.0000926535900581913
    sage: tan(3.1415/4)
    0.999953674278156
    sage: tan(pi/4)
    1
    sage: tan(1/2)
    tan(1/2)
    sage: RR(tan(1/2))
    0.546302489843790

sage: log2? Type: <class 'sage.functions.constants.Log2'> Definition: log2( [noargspec] ) Docstring:

The natural logarithm of the real number 2.

EXAMPLES:
    sage: log2
    log2
    sage: float(log2)
    0.69314718055994529
    sage: RR(log2)
    0.693147180559945
    sage: R = RealField(200); R
    Real Field with 200 bits of precision
    sage: R(log2)
    0.69314718055994530941723212145817656807550013436025525412068
    sage: l = (1-log2)/(1+log2); l
    (1 - log(2))/(log(2) + 1)
    sage: R(l)
    0.18123221829928249948761381864650311423330609774776013488056
    sage: maxima(log2)
    log(2)
    sage: maxima(log2).float()
    .6931471805599453
    sage: gp(log2)
    0.6931471805599453094172321215             # 32-bit
    0.69314718055994530941723212145817656807   # 64-bit

sage: sudoku? File: sage/local/lib/python2.5/site-packages/sage/games/sudoku.py Type: <... 'function'> Definition: sudoku(A) Docstring:

Solve the 9x9 Sudoku puzzle defined by the matrix A.

EXAMPLE:
    sage: A = matrix(ZZ,9,[5,0,0, 0,8,0, 0,4,9, 0,0,0, 5,0,0,
0,3,0, 0,6,7, 3,0,0, 0,0,1, 1,5,0, 0,0,0, 0,0,0, 0,0,0, 2,0,8, 0,0,0,
0,0,0, 0,0,0, 0,1,8, 7,0,0, 0,0,4, 1,5,0,   0,3,0, 0,0,2,
0,0,0, 4,9,0, 0,5,0, 0,0,3])
    sage: A
    [5 0 0 0 8 0 0 4 9]
    [0 0 0 5 0 0 0 3 0]
    [0 6 7 3 0 0 0 0 1]
    [1 5 0 0 0 0 0 0 0]
    [0 0 0 2 0 8 0 0 0]
    [0 0 0 0 0 0 0 1 8]
    [7 0 0 0 0 4 1 5 0]
    [0 3 0 0 0 2 0 0 0]
    [4 9 0 0 5 0 0 0 3]
    sage: sudoku(A)
    [5 1 3 6 8 7 2 4 9]
    [8 4 9 5 2 1 6 3 7]
    [2 6 7 3 4 9 5 8 1]
    [1 5 8 4 6 3 9 7 2]
    [9 7 4 2 1 8 3 6 5]
    [3 2 6 7 9 5 4 1 8]
    [7 8 2 9 3 4 1 5 6]
    [6 3 5 1 7 2 8 9 4]
    [4 9 1 8 5 6 7 2 3]

Python

from sage.all import * tan? Type: <class 'sage.calculus.calculus.Function_tan'> Definition: tan( [noargspec] ) Docstring:

The tangent function

EXAMPLES:

tan(pi) 0 tan(RealNumber('3.1415')) -0.0000926535900581913 tan(RealNumber('3.1415')/Integer(4)) 0.999953674278156 tan(pi/Integer(4)) 1 tan(Integer(1)/Integer(2)) tan(1/2) RR(tan(Integer(1)/Integer(2))) 0.546302489843790 log2? Type: <class 'sage.functions.constants.Log2'> Definition: log2( [noargspec] ) Docstring:

The natural logarithm of the real number 2.

EXAMPLES:

log2 log2 float(log2) 0.69314718055994529 RR(log2) 0.693147180559945 R = RealField(Integer(200)); R Real Field with 200 bits of precision R(log2) 0.69314718055994530941723212145817656807550013436025525412068 l = (Integer(1)-log2)/(Integer(1)+log2); l (1 - log(2))/(log(2) + 1) R(l) 0.18123221829928249948761381864650311423330609774776013488056 maxima(log2) log(2) maxima(log2).float() .6931471805599453 gp(log2) 0.6931471805599453094172321215 # 32-bit 0.69314718055994530941723212145817656807 # 64-bit sudoku? File: sage/local/lib/python2.5/site-packages/sage/games/sudoku.py Type: <... 'function'> Definition: sudoku(A) Docstring:

Solve the 9x9 Sudoku puzzle defined by the matrix A.

EXAMPLE:

A = matrix(ZZ,Integer(9),[Integer(5),Integer(0),Integer(0), Integer(0),Integer(8),Integer(0), Integer(0),Integer(4),Integer(9), Integer(0),Integer(0),Integer(0), Integer(5),Integer(0),Integer(0), 0,3,0, 0,6,7, 3,0,0, 0,0,1, 1,5,0, 0,0,0, 0,0,0, 0,0,0, 2,0,8, 0,0,0, 0,0,0, 0,0,0, 0,1,8, 7,0,0, 0,0,4, 1,5,0, 0,3,0, 0,0,2, 0,0,0, 4,9,0, 0,5,0, 0,0,3]) A [5 0 0 0 8 0 0 4 9] [0 0 0 5 0 0 0 3 0] [0 6 7 3 0 0 0 0 1] [1 5 0 0 0 0 0 0 0] [0 0 0 2 0 8 0 0 0] [0 0 0 0 0 0 0 1 8] [7 0 0 0 0 4 1 5 0] [0 3 0 0 0 2 0 0 0] [4 9 0 0 5 0 0 0 3] sudoku(A) [5 1 3 6 8 7 2 4 9] [8 4 9 5 2 1 6 3 7] [2 6 7 3 4 9 5 8 1] [1 5 8 4 6 3 9 7 2] [9 7 4 2 1 8 3 6 5] [3 2 6 7 9 5 4 1 8] [7 8 2 9 3 4 1 5 6] [6 3 5 1 7 2 8 9 4] [4 9 1 8 5 6 7 2 3]

Sage also provides ‘Tab completion’: type the first few letters of a function and then hit the Tab key. For example, if you typeta followed by Tab, Sage will print tachyon, tan, tanh, taylor. This provides a good way to find the names of functions and other structures in Sage.

Functions, Indentation, and Counting¶

To define a new function in Sage, use the def command and a colon after the list of variable names. For example:

Sage

sage: def is_even(n): ....: return n%2 == 0 sage: is_even(2) True sage: is_even(3) False

Python

from sage.all import * def is_even(n): ... return n%Integer(2) == Integer(0) is_even(Integer(2)) True is_even(Integer(3)) False

Note: Depending on which version of the tutorial you are viewing, you may see three dots ....: on the second line of this example. Do not type them; they are just to emphasize that the code is indented. Whenever this is the case, press [Return/Enter] once at the end of the block to insert a blank line and conclude the function definition.

You do not specify the types of any of the input arguments. You can specify multiple inputs, each of which may have an optional default value. For example, the function below defaults to divisor=2 ifdivisor is not specified.

Sage

sage: def is_divisible_by(number, divisor=2): ....: return number%divisor == 0 sage: is_divisible_by(6,2) True sage: is_divisible_by(6) True sage: is_divisible_by(6, 5) False

Python

from sage.all import * def is_divisible_by(number, divisor=Integer(2)): ... return number%divisor == Integer(0) is_divisible_by(Integer(6),Integer(2)) True is_divisible_by(Integer(6)) True is_divisible_by(Integer(6), Integer(5)) False

You can also explicitly specify one or either of the inputs when calling the function; if you specify the inputs explicitly, you can give them in any order:

Sage

sage: is_divisible_by(6, divisor=5) False sage: is_divisible_by(divisor=2, number=6) True

Python

from sage.all import * is_divisible_by(Integer(6), divisor=Integer(5)) False is_divisible_by(divisor=Integer(2), number=Integer(6)) True

In Python, blocks of code are not indicated by curly braces or begin and end blocks as in many other languages. Instead, blocks of code are indicated by indentation, which must match up exactly. For example, the following is a syntax error because the returnstatement is not indented the same amount as the other lines above it.

Sage

sage: def even(n): ....: v = [] ....: for i in range(3,n): ....: if i % 2 == 0: ....: v.append(i) ....: return v Syntax Error: return v

Python

from sage.all import * def even(n): ... v = [] ... for i in range(Integer(3),n): ... if i % Integer(2) == Integer(0): ... v.append(i) ... return v Syntax Error: return v

If you fix the indentation, the function works:

Sage

sage: def even(n): ....: v = [] ....: for i in range(3,n): ....: if i % 2 == 0: ....: v.append(i) ....: return v sage: even(10) [4, 6, 8]

Python

from sage.all import * def even(n): ... v = [] ... for i in range(Integer(3),n): ... if i % Integer(2) == Integer(0): ... v.append(i) ... return v even(Integer(10)) [4, 6, 8]

Semicolons are not needed at the ends of lines; a line is in most cases ended by a newline. However, you can put multiple statements on one line, separated by semicolons:

Sage

sage: a = 5; b = a + 3; c = b^2; c 64

Python

from sage.all import * a = Integer(5); b = a + Integer(3); c = b**Integer(2); c 64

If you would like a single line of code to span multiple lines, use a terminating backslash:

Sage

Python

from sage.all import * Integer(2) + Integer(3) 5

In Sage, you count by iterating over a range of integers. For example, the first line below is exactly like for(i=0; i<3; i++) in C++ or Java:

Sage

sage: for i in range(3): ....: print(i) 0 1 2

Python

from sage.all import * for i in range(Integer(3)): ... print(i) 0 1 2

The first line below is like for(i=2;i<5;i++).

Sage

sage: for i in range(2,5): ....: print(i) 2 3 4

Python

from sage.all import * for i in range(Integer(2),Integer(5)): ... print(i) 2 3 4

The third argument controls the step, so the following is likefor(i=1;i<6;i+=2).

Sage

sage: for i in range(1,6,2): ....: print(i) 1 3 5

Python

from sage.all import * for i in range(Integer(1),Integer(6),Integer(2)): ... print(i) 1 3 5

Often you will want to create a nice table to display numbers you have computed using Sage. One easy way to do this is to use string formatting. Below, we create three columns each of width exactly 6 and make a table of squares and cubes.

Sage

sage: for i in range(5): ....: print('%6s %6s %6s' % (i, i^2, i^3)) 0 0 0 1 1 1 2 4 8 3 9 27 4 16 64

Python

from sage.all import * for i in range(Integer(5)): ... print('%6s %6s %6s' % (i, iInteger(2), iInteger(3))) 0 0 0 1 1 1 2 4 8 3 9 27 4 16 64

The most basic data structure in Sage is the list, which is – as the name suggests – just a list of arbitrary objects. For example, using range, the following command creates a list:

Sage

sage: list(range(2,10)) [2, 3, 4, 5, 6, 7, 8, 9]

Python

from sage.all import * list(range(Integer(2),Integer(10))) [2, 3, 4, 5, 6, 7, 8, 9]

Here is a more complicated list:

Sage

sage: v = [1, "hello", 2/3, sin(x^3)] sage: v [1, 'hello', 2/3, sin(x^3)]

Python

from sage.all import * v = [Integer(1), "hello", Integer(2)/Integer(3), sin(x**Integer(3))] v [1, 'hello', 2/3, sin(x^3)]

List indexing is 0-based, as in many programming languages.

Sage

sage: v[0] 1 sage: v[3] sin(x^3)

Python

from sage.all import * v[Integer(0)] 1 v[Integer(3)] sin(x^3)

Use len(v) to get the length of v, use v.append(obj) to append a new object to the end of v, and use del v[i] to delete the \(i^{th}\) entry of v:

Sage

sage: len(v) 4 sage: v.append(1.5) sage: v [1, 'hello', 2/3, sin(x^3), 1.50000000000000] sage: del v[1] sage: v [1, 2/3, sin(x^3), 1.50000000000000]

Python

from sage.all import * len(v) 4 v.append(RealNumber('1.5')) v [1, 'hello', 2/3, sin(x^3), 1.50000000000000] del v[Integer(1)] v [1, 2/3, sin(x^3), 1.50000000000000]

Another important data structure is the dictionary (or associative array). This works like a list, except that it can be indexed with almost any object (the indices must be immutable):

Sage

sage: d = {'hi':-2, 3/8:pi, e:pi} sage: d['hi'] -2 sage: d[e] pi

Python

from sage.all import * d = {'hi':-Integer(2), Integer(3)/Integer(8):pi, e:pi} d['hi'] -2 d[e] pi

You can also define new data types using classes. Encapsulating mathematical objects with classes is a powerful technique that can help to simplify and organize your Sage programs. Below, we define a class that represents the list of even positive integers up to n; it derives from the builtin type list.

Sage

sage: class Evens(list): ....: def init(self, n): ....: self.n = n ....: list.init(self, range(2, n+1, 2)) ....: def repr(self): ....: return "Even positive numbers up to n."

Python

from sage.all import * class Evens(list): ... def init(self, n): ... self.n = n ... list.init(self, range(Integer(2), n+Integer(1), Integer(2))) ... def repr(self): ... return "Even positive numbers up to n."

The __init__ method is called to initialize the object when it is created; the __repr__ method prints the object out. We call the list constructor method in the second line of the__init__ method. We create an object of class Evens as follows:

Sage

sage: e = Evens(10) sage: e Even positive numbers up to n.

Python

from sage.all import * e = Evens(Integer(10)) e Even positive numbers up to n.

Note that e prints using the __repr__ method that we defined. To see the underlying list of numbers, use the listfunction:

Sage

sage: list(e) [2, 4, 6, 8, 10]

Python

from sage.all import * list(e) [2, 4, 6, 8, 10]

We can also access the n attribute or treat e like a list.

Sage

sage: e.n 10 sage: e[2] 6

Python

from sage.all import * e.n 10 e[Integer(2)] 6