def zeta(self, n=2):
"""
Return a root of unity in ``self``.
INPUT:
- ``n`` -- integer (default: 2) order of the root of
unity
EXAMPLES::
sage: QQ.zeta()
-1
sage: QQ.zeta(2)
-1
sage: QQ.zeta(1)
1
sage: QQ.zeta(3)
Traceback (most recent call last):
...
ValueError: no n-th root of unity in rational field
"""
if n == 1:
return rational.Rational(1)
elif n == 2:
return rational.Rational(-1)
else:
raise ValueError, "no n-th root of unity in rational field"
def frac(n, d):
"""
Return the fraction ``n/d``.
EXAMPLES::
sage: from sage.rings.rational_field import frac
sage: frac(1,2)
1/2
"""
return rational.Rational(n) / rational.Rational(d)
def some_elements(self):
r"""
Return some elements of `\QQ`.
See :func:`TestSuite` for a typical use case.
OUTPUT:
An iterator over 100 elements of `\QQ`.
EXAMPLES::
sage: tuple(QQ.some_elements())
(1/2, -1/2, 2, -2,
0, 1, -1, 42,
2/3, -2/3, 3/2, -3/2,
4/5, -4/5, 5/4, -5/4,
6/7, -6/7, 7/6, -7/6,
8/9, -8/9, 9/8, -9/8,
10/11, -10/11, 11/10, -11/10,
12/13, -12/13, 13/12, -13/12,
14/15, -14/15, 15/14, -15/14,
16/17, -16/17, 17/16, -17/16,
18/19, -18/19, 19/18, -19/18,
20/441, -20/441, 441/20, -441/20,
22/529, -22/529, 529/22, -529/22,
24/625, -24/625, 625/24, -625/24,
...)
"""
yield self.an_element()
yield -self.an_element()
yield 1 / self.an_element()
yield -1 / self.an_element()
yield self(0)
yield self(1)
yield self(-1)
yield self(42)
for n in range(1, 24):
a = 2 * n
b = (2 * n + 1)**(n // 10 + 1)
yield rational.Rational((a, b))
yield rational.Rational((-a, b))
yield rational.Rational((b, a))
yield rational.Rational((-b, a))
def _an_element_(self):
r"""
Return an element of `\QQ`.
EXAMPLES::
sage: QQ.an_element() # indirect doctest
1/2
"""
return rational.Rational((1, 2))
def getEquilibrium(t, p1SCount):
"""Returns the equilibrium from the given tableaux. The returned result
might contain mixed strategies like (1/3, 0/1), so normalization is need to
be performed on the result.
t - tableaux (Matrix)
p1SCount - number of strategies of player 1 (number)
Preconditions:
- 0 < p1SCount < t.getNumRows()
- first column of the matrix must contain each number from 1 to
t.getNumRows (inclusive, in absolute value)
Raises ValueError if some of the preconditions are not met.
"""
# 1st precondition
if p1SCount < 0 or t.getNumRows() <= p1SCount:
raise ValueError('Invalid number of strategies of player 1.')
# 2nd precondition
firstColNums = []
for i in range(1, t.getNumRows() + 1):
firstColNums.append(abs(t.getItem(i, 1)))
for i in range(1, t.getNumRows() + 1):
if not i in firstColNums:
raise ValueError(
'Invalid indices in the first column of the tableaux.')
# I decided to use a list instead of a tuple, because I need
# to modify it (tuples are immutable)
eqs = t.getNumRows() * [0]
# Equilibrium is in the second column of the tableaux
for i in range(1, t.getNumRows() + 1):
# Strategy
strat = t.getItem(i, 1)
# Strategy probability
prob = t.getItem(i, 2)
# If the strategy index or the probability is lower than zero,
# set it to zero instead
eqs[abs(strat) -
1] = rational.Rational(0) if (strat < 0 or prob < 0) else prob
# Convert the found equilibrium into a tuple
return (tuple(eqs[0:p1SCount]), tuple(eqs[p1SCount:]))
for elem in diff(a):
print('%6.2f' % elem, end=' '.ljust(4))
print(' ')
print('Dif of Dif :', end=' '.ljust(10))
for elem in diff(diff(a)):
print('%6.2f' % elem, end=' '.ljust(4))
print('\n' * 2, "--------" * 9)
###########################
## Operation on Mixed
###########################
an = nom(o)
ad = denom(o)
frac = vstack((an, ad))
a = [rational.Rational(frac[0][i], frac[1][i]).mixedp() for i in arange(n)]
al = [rational.Rational(frac[0][i], frac[1][i]).mixed() for i in arange(n)]
print('Num Fract :', end="")
for elem in a:
print(' %s' % elem, end=' '.ljust(7))
a = [al[i][0] for i in arange(n)]
print(' ')
print('Whole-Whole:', end=' '.ljust(7))
for elem in diff(a):
print('%3.f' % elem, end=' '.ljust(8))
a = [al[i][1] for i in arange(n)]
print(' ')
print('Num-Num rel:', end=' '.ljust(7))
def makePivotingStep(t, p1SCount, ebVar):
"""Makes a single pivoting step in the selected tableaux by
bringing the selected variable into the basis. All changes are done
in the original tableaux. Returns the variable that left the basis.
t - tableaux (Matrix)
p1SCount - number of strategies of player 1 (number)
ebVar - variable that will enter the basis (number)
Preconditions:
- 0 < abs(ebVar) <= t.getNumRows()
- 0 < p1SCount < t.getNumRows()
Raises ValueError if some of the preconditions are not met.
"""
# 1st precondition
if abs(ebVar) <= 0 or abs(ebVar) > t.getNumRows():
raise ValueError, 'Selected variable index is invalid.'
# 2nd precondition
if p1SCount < 0 or t.getNumRows() <= p1SCount:
raise ValueError, 'Invalid number of strategies of player 1.'
# Returns the column corresponding to the selected variable
def varToCol(var):
# Apart from players matrices values, there are 2 additional
# columns in the tableaux
return 2 + abs(var)
# Returns the list of row numbers which corresponds
# to the selected variable
def getRowNums(var):
# Example (for a game 3x3):
# -1,-2,-3,4,5,6 corresponds to the first part of the tableaux
# 1,2,3,-4,-5,-6 corresponds to the second part of the tableaux
if -p1SCount <= var < 0 or var > p1SCount:
return xrange(1, p1SCount + 1)
else:
return xrange(p1SCount + 1, t.getNumRows() + 1)
# Check which variable should leave the basis using the min-ratio rule
# (it will have the lowest ratio)
lbVar = None
minRatio = None
# Check only rows in the appropriate part of the tableaux
for i in getRowNums(ebVar):
if t.getItem(i, varToCol(ebVar)) < 0:
ratio = -rational.Rational(t.getItem(i, 2)) /\
t.getItem(i, varToCol(ebVar))
if minRatio == None or ratio < minRatio:
minRatio = ratio
lbVar = t.getItem(i, 1)
lbVarRow = i
lbVarCoeff = t.getItem(i, varToCol(ebVar))
# Update the row in which the variable that will leave the basis was
# found in the previous step
t.setItem(lbVarRow, 1, ebVar)
t.setItem(lbVarRow, varToCol(ebVar), 0)
t.setItem(lbVarRow, varToCol(lbVar), -1)
for j in xrange(2, t.getNumCols() + 1):
newVal = rational.Rational(t.getItem(lbVarRow, j)) / abs(lbVarCoeff)
t.setItem(lbVarRow, j, newVal)
# Update other rows in the appropriate part of the tableaux
for i in getRowNums(ebVar):
if t.getItem(i, varToCol(ebVar)) != 0:
for j in xrange(2, t.getNumCols() + 1):
newVal = t.getItem(i, j) + t.getItem(i, varToCol(ebVar)) *\
t.getItem(lbVarRow, j)
t.setItem(i, j, newVal)
t.setItem(i, varToCol(ebVar), 0)
return lbVar
def main():
"""Test the rational.Rational class methods."""
print('This program tests the Rational class.\n')
print('Read the output of this program CAREFULLY!\n')
number0 = rational.Rational()
print('Testing the existence of certain methods...')
if hasattr(number0, 'get_numerator'):
print('Correct: Has a get_numerator method')
else:
print('ERROR: Does not have a get_numerator method')
if hasattr(number0, 'get_denominator'):
print('Correct: Has a get_denominator method')
else:
print('ERROR: Does not have a get_denominator method')
if hasattr(number0, 'set_value'):
print('Correct: Has a set_value method')
else:
print('ERROR: Does not have a set_value method')
if hasattr(number0, '__float__'):
print('Correct: Has a __float__ method')
else:
print('ERROR: Does not have a __float__ method')
if hasattr(number0, '__add__'):
print('Correct: Has a __add__ method')
else:
print('ERROR: Does not have a __add__ method')
if hasattr(number0, '__iadd__'):
print('Correct: Has an __iadd__ method')
else:
print('ERROR: Does not have an __iadd__ method')
if hasattr(number0, '__mul__'):
print('Correct: Has a __mul__ method')
else:
print('ERROR: Does not have a __mul__ method')
if hasattr(number0, '__imul__'):
print('Correct: Has an __imul__ method')
else:
print('ERROR: Does not have an __imul__ method')
if hasattr(number0, '__eq__'):
print('Correct: Has an __eq__ method')
else:
print('ERROR: Does not have an __eq__ method')
if hasattr(number0, '__str__'):
print('Correct: Has a __str__ method')
else:
print('ERROR: Does not have a __str__ method')
print('\nTesting the constructor and __str__ method:\n')
# Created number0 = rational.Rational() above
if str(number0) == '0/1':
print('Correct: default object = 0/1')
else:
print('ERROR: default object should be 0/1 but is %s' % number0)
number1 = rational.Rational(3)
if str(number1) == '3/1':
print('Correct: 3 is represented as 3/1')
else:
print('ERROR: 3 should be represented as 3/1 but is %s' % number1)
number2 = rational.Rational(2, 3)
if str(number2) == '2/3':
print('Correct: 2/3 is represented as 2/3')
else:
print('ERROR: 2/3 should be 2/3 but is %s' % number2)
number3 = rational.Rational(2, 6)
if str(number3) == '1/3':
print('Correct: 2/6 reduces to 1/3')
else:
print('ERROR: 2/6 should reduce to 1/3 but is %s' % number3)
number4 = rational.Rational(12, 9)
if str(number4) == '4/3':
print('Correct: 12/9 reduces to 4/3')
else:
print('ERROR: 12/9 should reduce to 4/3 but is %s' % number4)
number5 = rational.Rational(1, -6)
if str(number5) == '-1/6':
print('Correct: 1/-6 reduces to -1/6')
else:
print('ERROR: 1/-6 should reduce to -1/6 but is %s' % number5)
number6 = rational.Rational(-5, -9)
if str(number6) == '5/9':
print('Correct: -5/-9 reduces to 5/9')
else:
print('ERROR: -5/-9 should reduce to 5/9 but is %s' % number6)
print('\nAttempting to create a rational number with 0 denominator...')
print('Expecting a ValueError...')
try:
dummy = rational.Rational(5, 0)
print('ERROR -- No exception was thrown from the constructor!')
except ValueError:
print('Correct: Caught a ValueError from the constructor')
except Exception:
print('ERROR: Wrong type of exception thrown from the constructor')
print('\nTesting the accessor and mutator methods:\n')
numerator = number1.get_numerator()
denominator = number1.get_denominator()
if numerator == 3 and denominator == 1:
print('Correct: 3/1 has numerator 3 and denominator 1')
else:
print('ERROR: %s has numerator %d and denominator %d' \
% (number1, numerator, denominator))
numerator = number2.get_numerator()
denominator = number2.get_denominator()
if numerator == 2 and denominator == 3:
print('Correct: 2/3 has numerator 2 and denominator 3')
else:
print('ERROR: %s has numerator %d and denominator %d' \
% (number2, numerator, denominator))
print('\nChanging 2/3 to 5/4...')
number2.set_value(5, 4)
if str(number2) == '5/4':
print('Correct: 2/3 has been changed to 5/4')
else:
print('ERROR: 2/3 should have been changed to 5/4 but is %s' \
% number2)
print('\nChanging 5/4 to 4/12 = 1/3...')
number2.set_value(4, 12)
if str(number2) == '1/3':
print('Correct: 5/4 has been changed to 1/3')
else:
print('ERROR: 5/4 should have been changed to 1/3 but is %s' \
% number2)
print('\nChanging 1/3 to 4 = 4/1...')
number2.set_value(4)
if str(number2) == '4/1':
print('Correct: 1/3 has been changed to 4/1')
else:
print('ERROR: 1/3 should have been changed to 4/1 but is %s' \
% number2)
print('\nAttempting to set a rational number with 0 denominator...')
print('Expecting a ValueError...')
dummy = rational.Rational(10)
try:
dummy.set_value(8, 0)
print('ERROR -- No exception was thrown from set_value!')
except ValueError:
print('Correct: Caught a ValueError from set_value')
except Exception:
print('ERROR: Wrong type of exception thrown from set_value')
print('\nTesting the __float__ method:\n')
if float(number1) == 3.0:
print('Correct: 3/1 == 3.0')
else:
print('ERROR: 3/1 = %.6f' % float(number1))
number2 = rational.Rational(2, 3)
if float(number2) == 2.0 / 3:
print('Correct: 2/3 == 0.6666667')
else:
print('ERROR: 2/3 = %f' % float(number2))
print('\nTesting the __add__ and __mul__ methods:\n')
number1.set_value(1, 3)
number2.set_value(1, 6)
number3 = number1 + number2
if str(number3) == '1/2':
print('Correct: 1/3 + 1/6 = 1/2')
else:
print('ERROR: 1/3 + 1/6 should be 1/2 but is %s' % number3)
number1.set_value(3, 4)
number3 = number1 + number2
if str(number3) == '11/12':
print('Correct: 3/4 + 1/6 = 11/12')
else:
print('ERROR: 3/4 + 1/6 should be 11/12 but is %s' % number3)
number1.set_value(-5, 4)
number2.set_value(6, -8)
number3 = number1 + number2
if str(number3) == '-2/1':
print('Correct: -5/4 + 6/-8 = -2/1\n')
else:
print('ERROR: -5/4 + 6/-8 should be -2/1 but is %s\n' % number3)
number1.set_value(2, 3)
number2.set_value(3, 2)
number3 = number1 * number2
if str(number3) == '1/1':
print('Correct: 2/3 * 3/2 = 1/1')
else:
print('ERROR: 2/3 * 3/2 should be 1/1 but is %s' % number3)
number2.set_value(1, 2)
number3 = number1 * number2
if str(number3) == '1/3':
print('Correct: 2/3 * 1/2 = 1/3')
else:
print('ERROR: 2/3 * 1/2 should be 1/3 but is %s' % number3)
number1.set_value(-1, 6)
number2.set_value(2, 1)
number3 = number1 * number2
if str(number3) == '-1/3':
print('Correct: -1/6 * 2/1 = -1/3')
else:
print('ERROR: -1/5 * 2/1 should be -1/3 but is %s' % number3)
print('\nTesting the __iadd__ and __imul__ methods:\n')
number1.set_value(1, 3)
number2.set_value(1, 6)
number1 += number2
if str(number1) == '1/2':
print('Correct: 1/3 += 1/6 is 1/2')
else:
print('ERROR: 1/3 += 1/6 should be 1/2 but is %s' % number1)
number1.set_value(3, 4)
number1 += number2
if str(number1) == '11/12':
print('Correct: 3/4 += 1/6 is 11/12')
else:
print('ERROR: 3/4 += 1/6 should be 11/12 but is %s' % number1)
number1.set_value(-5, 4)
number2.set_value(6, -8)
number1 += number2
if str(number1) == '-2/1':
print('Correct: -5/4 += 6/-8 is -2/1\n')
else:
print('ERROR: -5/4 += 6/-8 should be -2/1 but is %s\n' % number1)
number1.set_value(2, 3)
number2.set_value(3, 2)
number1 *= number2
if str(number1) == '1/1':
print('Correct: 2/3 *= 3/2 is 1/1')
else:
print('ERROR: 2/3 *= 3/2 should be 1/1 but is %s' % number1)
number1.set_value(2, 3)
number2.set_value(1, 2)
number1 *= number2
if str(number1) == '1/3':
print('Correct: 2/3 *= 1/2 is 1/3')
else:
print('ERROR: 2/3 *= 1/2 should be 1/3 but is %s' % number1)
number1.set_value(-1, 6)
number2.set_value(2, 1)
number1 *= number2
if str(number1) == '-1/3':
print('Correct: -1/6 *= 2/1 is -1/3')
else:
print('ERROR: -1/5 *= 2/1 should be -1/3 but is %s' % number1)
print('\nTesting the __eq__ method:\n')
number1.set_value(4, 9)
number2.set_value(4, 9)
if number1 == number2:
print('Correct: 4/9 == 4/9')
else:
print('ERROR: 4/9 != 4/9')
number1.set_value(4, 9)
number2.set_value(4, 8)
if number1 != number2:
print('Correct: 4/9 != 4/8')
else:
print('ERROR: 4/9 == 4/8')
number1.set_value(3, 4)
number2.set_value(3, 5)
if number1 != number2:
print('Correct: 3/4 != 3/5')
else:
print('ERROR: 3/4 == 3/5')
number1.set_value(2, 5)
number2.set_value(3, 5)
if number1 != number2:
print('Correct: 2/5 != 3/5')
else:
print('ERROR: 2/5 == 3/5')
print('\n*** End of testing for the Rational class ***\n')
print('Note that passing all of these tests provides some confidence')
print('but does not necessarily guarantee that all the code is correct.')
def frac(n, d):
return rational.Rational(n) / rational.Rational(d)
def _an_element_(self):
return rational.Rational((1, 2))
"""
350112
a2 3.py
MICHAEL MAGAISA
[email protected]
"""
import rational
r1 = rational.Rational(1, 2)
print("Initializing 1/2")
r2 = rational.Rational(1, 8)
print("Initializing 1/8")
res = r1 + r2
print(r1, "+", r2, "=", res)
