Tutorial Release 10.4 The Sage Development Team https://doc.sagemath.org/pdf/en/tutorial/sage_tutorial.pdf

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Blaec (CSO) Tasks and Strategy Outline

tutorials/CoCalc's Official Sage Tutorial / Chapter Two / Chapter 2-2: Getting Help.ipynb

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2.2 Getting Help

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

In [31]:

tan?

Out[31]:

Type:           Function_tan
String form:    tan
File:           /ext/sage/10.4/src/sage/functions/trig.py
Docstring:     
   The tangent function.

   EXAMPLES:

      sage: tan(3.1415)
      -0.0000926535900581913
      sage: tan(3.1415/4)
      0.999953674278156

      sage: tan(pi)
      0
      sage: tan(pi/4)
      1
      sage: tan(1/2)
      tan(1/2)
      sage: RR(tan(1/2))
      0.546302489843790

   We can prevent evaluation using the "hold" parameter:

      sage: tan(pi/4, hold=True)
      tan(1/4*pi)

   To then evaluate again, we currently must use Maxima via
   "sage.symbolic.expression.Expression.simplify()":

      sage: a = tan(pi/4, hold=True); a.simplify()
      1

   If possible, the argument is also reduced modulo the period length
   \pi, and well-known identities are directly evaluated:

      sage: k = var('k', domain='integer')
      sage: tan(1 + 2*k*pi)
      tan(1)
      sage: tan(k*pi)
      0
Init docstring:
   The tangent function.

   EXAMPLES:

      sage: tan(3.1415)
      -0.0000926535900581913
      sage: tan(3.1415/4)
      0.999953674278156

      sage: tan(pi)
      0
      sage: tan(pi/4)
      1
      sage: tan(1/2)
      tan(1/2)
      sage: RR(tan(1/2))
      0.546302489843790

   We can prevent evaluation using the "hold" parameter:

      sage: tan(pi/4, hold=True)
      tan(1/4*pi)

   To then evaluate again, we currently must use Maxima via
   "sage.symbolic.expression.Expression.simplify()":

      sage: a = tan(pi/4, hold=True); a.simplify()
      1

   If possible, the argument is also reduced modulo the period length
   \pi, and well-known identities are directly evaluated:

      sage: k = var('k', domain='integer')
      sage: tan(1 + 2*k*pi)
      tan(1)
      sage: tan(k*pi)
      0
Call docstring:
   Evaluate this function on the given arguments and return the
   result.

   EXAMPLES:

      sage: exp(5)
      e^5
      sage: gamma(15)
      87178291200

   Python float, Python complex, mpmath mpf and mpc as well as numpy
   inputs are sent to the relevant "math", "cmath", "mpmath" or
   "numpy" function:

      sage: cos(1.r)
      0.5403023058681398
      sage: assert type(_) is float
      sage: gamma(4.r)
      6.0
      sage: assert type(_) is float

      sage: cos(1jr)  # abstol 1e-15
      (1.5430806348152437-0j)
      sage: assert type(_) is complex

      sage: import mpmath
      sage: cos(mpmath.mpf('1.321412'))
      mpf('0.24680737898640387')
      sage: cos(mpmath.mpc(1,1))
      mpc(real='0.83373002513114902', imag='-0.98889770576286506')

      sage: import numpy
      sage: sin(numpy.int32(0))
      0.0
      sage: type(_)
      <class 'numpy.float64'>

EXAMPLES:

Calculate $\tan(\pi)$ :

In [32]:

tan(pi)

Out[32]:

0

Calculate $\tan(3.1415)$ :

In [33]:

tan(3.1415)

Out[33]:

-0.0000926535900581913

Calculate $\tan(3.1415/4)$ :

In [34]:

tan(3.1415/4)

Out[34]:

0.999953674278156

Calculate $\tan(\pi/4)$ :

In [35]:

tan(pi/4)

Out[35]:

1

Evaluate $\tan(\frac{1}{2})$ as an element in the Real Field with infinite precision:

In [36]:

tan(1/2)

Out[36]:

tan(1/2)

Convert $\tan(\frac{1}{2})$ to a real number with fixed floating-point precision:

In [37]:

RR(tan(1/2))

Out[37]:

0.546302489843790

Query documentation for $\log_2$ :

In [38]:

log2?

Out[38]:

Type:           Expression
String form:    log2
File:           /ext/sage/10.4/src/sage/symbolic/expression.pyx
Docstring:     
   Nearly all expressions are created by calling
   new_Expression_from_*, but we need to make sure this at least does
   not leave self._gobj uninitialized and segfault.
Init docstring:
   Nearly all expressions are created by calling
   new_Expression_from_*, but we need to make sure this at least does
   not leave self._gobj uninitialized and segfault.
Call docstring:
   Call the "subs()" on this expression.

   EXAMPLES:

      sage: var('x,y,z')
      (x, y, z)
      sage: (x+y)(x=z^2, y=x^y)
      z^2 + x^y

EXAMPLES:

Retrieve the value of $\log_2$ :

In [39]:

log2

Out[39]:

log2

Convert $\log_2$ to a floating-point number:

In [40]:

float(log2)

Out[40]:

0.6931471805599453

Convert $\log_2$ to a real number with fixed floating-point precision:

In [41]:

RR(log2)

Out[41]:

0.693147180559945

Create a real field with 200 bits of precision and convert $\log_2$ to it:

In [42]:

R = RealField(200); R
R(log2)

Out[42]:

0.69314718055994530941723212145817656807550013436025525412068

Define and evaluate an expression involving $\log_2$ :

In [43]:

l = (1 - log2) / (1 + log2); l
R(l)

Out[43]:

0.18123221829928249948761381864650311423330609774776013488056

Use Maxima to convert $\log_2$ :

In [44]:

maxima(log2)
maxima(log2).float()

Out[44]:

0.6931471805599453

Use GP (PARI/GP) to convert $\log_2$ :

In [45]:

gp(log2)

Out[45]:

0.69314718055994530941723212145817656807

Query documentation for the sudoku function:

In [46]:

sudoku?

Out[46]:

Signature:      sudoku(m)
Docstring:     
   Solves Sudoku puzzles described by matrices.

   INPUT:

   * "m" -- a square Sage matrix over \ZZ, where zeros are blank
     entries

   OUTPUT:

   A Sage matrix over \ZZ containing the first solution found,
   otherwise "None".

   This function matches the behavior of the prior Sudoku solver and
   is included only to replicate that behavior.  It could be safely
   deprecated, since all of its functionality is included in the
   "Sudoku" class.

   EXAMPLES:

   An example that was used in previous doctests.

      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]

   Using inputs that are possible with the "Sudoku" class, other than
   a matrix, will cause an error.

      sage: sudoku('.4..32....14..3.')
      Traceback (most recent call last):
      ...
      ValueError: sudoku function expects puzzle to be a matrix, perhaps use the Sudoku class
Init docstring: Initialize self.  See help(type(self)) for accurate signature.
File:           /ext/sage/10.4/src/sage/games/sudoku.py
Type:           function

EXAMPLE:

Create a 9x9 Sudoku puzzle matrix and display it:

In [47]:

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
])
A

Out[47]:

[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]

Solve the Sudoku puzzle:

In [48]:

sudoku(A)

Out[48]:

[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]

2.2 Getting Help

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

EXAMPLES:

EXAMPLES:

EXAMPLE:

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