Python maxima.eval函数代码示例

    def assume(self):
        """
        TEST::

            sage: from sage.symbolic.assumptions import GenericDeclaration
            sage: decl = GenericDeclaration(x, 'even')
            sage: decl.assume()
            sage: cos(x*pi).simplify()
            1
            sage: decl2 = GenericDeclaration(x, 'odd')
            sage: decl2.assume()
            Traceback (most recent call last):
            ...
            ValueError: Assumption is inconsistent
            sage: decl.forget()
        """
        from sage.calculus.calculus import maxima
        if self._context is None:
            # We get the list here because features may be added with time.
            valid_features = list(maxima("features"))
            if self._assumption not in [repr(x).strip() for x in list(valid_features)]:
                raise ValueError, "%s not a valid assumption, must be one of %s" % (self._assumption, valid_features)
            cur = maxima.get("context")
            self._context = maxima.newcontext('context' + maxima._next_var_name())
            try:
                maxima.eval("declare(%s, %s)" % (repr(self._var), self._assumption))
#            except TypeError, mess:
#                if 'inconsistent' in str(mess): # note Maxima doesn't tell you if declarations are redundant
#                    raise ValueError, "Assumption is inconsistent"
            except RuntimeError, mess:
                if 'inconsistent' in str(mess): # note Maxima doesn't tell you if declarations are redundant
                    raise ValueError, "Assumption is inconsistent"
                else:
                    raise
            maxima.set("context", cur)

def _init():
    """
    Internal function which checks if Maxima has loaded the
    "orthopoly" package.  All functions using this in this
    file should call this function first.

    TEST:

    The global starts ``False``::

        sage: sage.functions.orthogonal_polys._done
        False

    Then after using one of these functions, it changes::

        sage: from sage.functions.orthogonal_polys import chebyshev_T
        sage: chebyshev_T(2,x)
        2*(x - 1)^2 + 4*x - 3
        sage: sage.functions.orthogonal_polys._done
        True

    Note that because here we use a Pynac variable ``x``,
    the representation of the function is different from
    its actual doctest, where a polynomial indeterminate
    ``x`` is used.
    """
    global _done
    if _done:
        return
    maxima.eval('load("orthopoly");')
    # TODO -- make it possible to use the intervals returned
    # instead of just discarding this info!
    maxima.eval("orthopoly_returns_intervals:false;")
    _done = True

def _init():
    """
    Internal function which checks if Maxima has loaded the
    "orthopoly" package.  All functions using this in this
    file should call this function first.
    
    TEST:

    The global starts ``False``:: 

        sage: sage.functions.special._done
        False

    Then after using one of these functions, it changes::

        sage: from sage.functions.special import airy_ai
        sage: airy_ai(1.0)
        0.135292416313
        sage: sage.functions.special._done
        True
    """
    global _done
    if _done:
        return
    maxima.eval('load("orthopoly");')
    maxima.eval('orthopoly_returns_intervals:false;')
    _done = True

def _init():
    """
    Internal function which checks if Maxima has loaded the
    "orthopoly" package.  All functions using this in this
    file should call this function first.

    TEST:

    The global starts ``False``::

        sage: sage.functions.special._done
        False

    Then after using one of the MaximaFunctions, it changes::

        sage: from sage.functions.special import elliptic_ec
        sage: elliptic_ec(0.1)
        1.53075763689776

        sage: sage.functions.special._done
        True
    """
    global _done
    if _done:
        return
    maxima.eval('load("orthopoly");')
    maxima.eval('orthopoly_returns_intervals:false;')
    _done = True

def gen_laguerre(n, a, x):
    """
    Returns the generalized Laguerre polynomial for integers `n > -1`.
    Typically, a = 1/2 or a = -1/2.

    REFERENCE:

    - table on page 789 in AS.

    EXAMPLES::

        sage: x = PolynomialRing(QQ, 'x').gen()
        sage: gen_laguerre(2,1,x)
        1/2*x^2 - 3*x + 3
        sage: gen_laguerre(2,1/2,x)
        1/2*x^2 - 5/2*x + 15/8
        sage: gen_laguerre(2,-1/2,x)
        1/2*x^2 - 3/2*x + 3/8
        sage: gen_laguerre(2,0,x)
        1/2*x^2 - 2*x + 1
        sage: gen_laguerre(3,0,x)
        -1/6*x^3 + 3/2*x^2 - 3*x + 1
    """
    from sage.functions.all import sqrt

    _init()
    return sage_eval(maxima.eval("gen_laguerre(%s,%s,x)" % (ZZ(n), a)), locals={"x": x})

def legendre_P(n, x):
    """
    Returns the Legendre polynomial of the first kind for integers
    `n > -1`.

    REFERENCE:

    - AS 22.5.35 page 779.

    EXAMPLES::

        sage: P.<t> = QQ[]
        sage: legendre_P(2,t)
        3/2*t^2 - 1/2
        sage: legendre_P(3, 1.1)
        1.67750000000000
        sage: legendre_P(3, 1 + I)
        7/2*I - 13/2
        sage: legendre_P(3, MatrixSpace(ZZ, 2)([1, 2, -4, 7]))
        [-179  242]
        [-484  547]
        sage: legendre_P(3, GF(11)(5))
        8
    """
    _init()
    return sage_eval(maxima.eval("legendre_p(%s,x)" % ZZ(n)), locals={"x": x})

def gen_laguerre(n, a, x):
    """
    Returns the generalized Laguerre polynomial for integers `n > -1`.
    Typically, `a = 1/2` or `a = -1/2`.

    REFERENCES:

    - Table on page 789 in [ASHandbook]_.

    EXAMPLES::

        sage: x = PolynomialRing(QQ, 'x').gen()
        sage: gen_laguerre(2,1,x)
        1/2*x^2 - 3*x + 3
        sage: gen_laguerre(2,1/2,x)
        1/2*x^2 - 5/2*x + 15/8
        sage: gen_laguerre(2,-1/2,x)
        1/2*x^2 - 3/2*x + 3/8
        sage: gen_laguerre(2,0,x)
        1/2*x^2 - 2*x + 1
        sage: gen_laguerre(3,0,x)
        -1/6*x^3 + 3/2*x^2 - 3*x + 1
    """
    _init()
    return sage_eval(maxima.eval("gen_laguerre(%s,%s,x)" % (ZZ(n), a)), locals={"x": x})

def ultraspherical(n, a, x):
    """
    Returns the ultraspherical (or Gegenbauer) polynomial for integers
    `n > -1`.

    Computed using Maxima.

    REFERENCE:

    - AS 22.5.27

    EXAMPLES::

        sage: x = PolynomialRing(QQ, 'x').gen()
        sage: ultraspherical(2,3/2,x)
        15/2*x^2 - 3/2
        sage: ultraspherical(2,1/2,x)
        3/2*x^2 - 1/2
        sage: ultraspherical(1,1,x)
        2*x
        sage: t = PolynomialRing(RationalField(),"t").gen()
        sage: gegenbauer(3,2,t)
        32*t^3 - 12*t
    """
    _init()
    return sage_eval(maxima.eval("ultraspherical(%s,%s,x)" % (ZZ(n), a)), locals={"x": x})

def hermite(n, x):
    """
    Returns the Hermite polynomial for integers `n > -1`.

    REFERENCE:

    - AS 22.5.40 and 22.5.41, page 779.

    EXAMPLES::

        sage: x = PolynomialRing(QQ, 'x').gen()
        sage: hermite(2,x)
        4*x^2 - 2
        sage: hermite(3,x)
        8*x^3 - 12*x
        sage: hermite(3,2)
        40
        sage: S.<y> = PolynomialRing(RR)
        sage: hermite(3,y)
        8.00000000000000*y^3 - 12.0000000000000*y
        sage: R.<x,y> = QQ[]
        sage: hermite(3,y^2)
        8*y^6 - 12*y^2
        sage: w = var('w')
        sage: hermite(3,2*w)
        8*(8*w^2 - 3)*w
    """
    _init()
    return sage_eval(maxima.eval("hermite(%s,x)" % ZZ(n)), locals={"x": x})

def bessel_Y(nu,z,algorithm="maxima", prec=53):
    r"""
    Implements the "Y-Bessel function", or "Bessel function of the 2nd
    kind", with index (or "order") nu and argument z.
    
    .. note::

       Currently only prec=53 is supported.
    
    Defn::
    
                    cos(pi n)*bessel_J(nu, z) - bessel_J(-nu, z)
                   -------------------------------------------------
                                     sin(nu*pi)
    
    if nu is not an integer and by taking a limit otherwise.
    
    Sometimes bessel_Y(n,z) is denoted Y_n(z) in the literature.
    
    This is computed using Maxima by default.
    
    EXAMPLES::
    
        sage: bessel_Y(2,1.1,"scipy")
        -1.4314714939...
        sage: bessel_Y(2,1.1)   
        -1.4314714939590...
        sage: bessel_Y(3.001,2.1) 
        -1.0299574976424...

    TESTS::

        sage: bessel_Y(2,1.1, algorithm="pari")
        Traceback (most recent call last):
        ...
        NotImplementedError: The Y-Bessel function is only implemented for the maxima and scipy algorithms
    """
    if algorithm=="scipy":
        if prec != 53:
            raise ValueError, "for the scipy algorithm the precision must be 53"
        import scipy.special
        ans = str(scipy.special.yv(float(nu),complex(real(z),imag(z))))
        ans = ans.replace("(","")
        ans = ans.replace(")","")
        ans = ans.replace("j","*I")
        ans = sage_eval(ans)
        return real(ans) if z in RR else ans
    elif algorithm == "maxima":
        if prec != 53:
            raise ValueError, "for the maxima algorithm the precision must be 53"
        return RR(maxima.eval("bessel_y(%s,%s)"%(float(nu),float(z))))
    elif algorithm == "pari":
        raise NotImplementedError, "The Y-Bessel function is only implemented for the maxima and scipy algorithms"
    else:
        raise ValueError, "unknown algorithm '%s'"%algorithm

    def _maxima_init_evaled_(self, n, x):
        """
        Evaluate the Chebyshev polynomial ``self`` with maxima.

        EXAMPLES::

            sage: var('n, x')
            (n, x)
            sage: chebyshev_T._maxima_init_evaled_(1,x)
            '_SAGE_VAR_x'
            sage: maxima(chebyshev_T(n, chebyshev_T(n, x)))
            chebyshev_t(_SAGE_VAR_n,chebyshev_t(_SAGE_VAR_n,_SAGE_VAR_x))
        """
        return maxima.eval("chebyshev_t({0},{1})".format(n._maxima_init_(), x._maxima_init_()))

def chebyshev_U(n, x):
    """
    Returns the Chebyshev function of the second kind for integers `n>-1`.

    REFERENCE:

    - AS, 22.8.3 page 783 and AS 6.1.22 page 256.

    EXAMPLES::

        sage: x = PolynomialRing(QQ, 'x').gen()
        sage: chebyshev_U(2,x)
        4*x^2 - 1
    """
    _init()
    return sage_eval(maxima.eval("chebyshev_u(%s,x)" % ZZ(n)), locals={"x": x})

def hermite(n,x):
    """
    Returns the Hermite polynomial for integers `n > -1`.

    REFERENCE:

    - [ASHandbook]_ 22.5.40 and 22.5.41, page 779.

    EXAMPLES::

        sage: x = PolynomialRing(QQ, 'x').gen()
        sage: hermite(2,x)
        4*x^2 - 2
        sage: hermite(3,x)
        8*x^3 - 12*x
        sage: hermite(3,2)
        40
        sage: S.<y> = PolynomialRing(RR)
        sage: hermite(3,y)
        8.00000000000000*y^3 - 12.0000000000000*y
        sage: R.<x,y> = QQ[]
        sage: hermite(3,y^2)
        8*y^6 - 12*y^2
        sage: w = var('w')
        sage: hermite(3,2*w)
        8*(8*w^2 - 3)*w

    Check that :trac:`17192` is fixed::
        sage: x = PolynomialRing(QQ, 'x').gen()
        sage: hermite(0,x)
        1

        sage: hermite(-1,x)
        Traceback (most recent call last):
        ...
        ValueError: n must be greater than -1, got n = -1

        sage: hermite(-7,x)
        Traceback (most recent call last):
        ...
        ValueError: n must be greater than -1, got n = -7
    """
    if not (n > -1):
        raise ValueError("n must be greater than -1, got n = {0}".format(n))

    _init()
    return sage_eval(maxima.eval('hermite(%s,x)'%ZZ(n)), locals={'x':x})

def chebyshev_T(n, x):
    """
    Returns the Chebyshev function of the first kind for integers
    `n>-1`.

    REFERENCE:

    - AS 22.5.31 page 778 and AS 6.1.22 page 256.

    EXAMPLES::

        sage: x = PolynomialRing(QQ, 'x').gen()
        sage: chebyshev_T(2,x)
        2*x^2 - 1
    """
    _init()
    return sage_eval(maxima.eval("chebyshev_t(%s,x)" % ZZ(n)), locals={"x": x})

def ultraspherical(n,a,x):
    """
    Returns the ultraspherical (or Gegenbauer) polynomial for integers
    `n > -1`.

    Computed using Maxima.

    REFERENCE:

    - [ASHandbook]_ 22.5.27

    EXAMPLES::

        sage: x = PolynomialRing(QQ, 'x').gen()
        sage: ultraspherical(2,3/2,x)
        15/2*x^2 - 3/2
        sage: ultraspherical(2,1/2,x)
        3/2*x^2 - 1/2
        sage: ultraspherical(1,1,x)
        2*x
        sage: t = PolynomialRing(RationalField(),"t").gen()
        sage: gegenbauer(3,2,t)
        32*t^3 - 12*t

    Check that :trac:`17192` is fixed::
        sage: x = PolynomialRing(QQ, 'x').gen()
        sage: ultraspherical(0,1,x)
        1

        sage: ultraspherical(-1,1,x)
        Traceback (most recent call last):
        ...
        ValueError: n must be greater than -1, got n = -1

        sage: ultraspherical(-7,1,x)
        Traceback (most recent call last):
        ...
        ValueError: n must be greater than -1, got n = -7
    """
    if not (n > -1):
        raise ValueError("n must be greater than -1, got n = {0}".format(n))

    _init()
    return sage_eval(maxima.eval('ultraspherical(%s,%s,x)'%(ZZ(n),a)), locals={'x':x})