使用 sagemaths 的 Gram-Schmidt 过程

Question

我正在尝试实现一个函数 myGramSchmidt(L)，它采用存在于某个内积空间中的向量列表 L，并返回一个已实现上述 Gram-Schmidt 过程的新列表。

我的代码：

def myGramSchmidt(L):

    n = len(L)
    V = L.copy()
    for j in range(n):
        V[j]= V[j].norm() #normalised vector
        for i in range(j):
            V[j] -= ip(V[i],V[j]) * V[i] #find value of V[j] which is orthogonal to every previous vectors

    return V

我有错误

---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-69-b1221e18fe68> in <module>()
     14 
     15 if M.det() != Integer(0):
---> 16     GSL = myGramSchmidt(L)
     17     GSM = Matrix(Integer(6),Integer(6),GSL)
     18     assert len(GSL)==Integer(6)

<ipython-input-68-a8904262801b> in myGramSchmidt(L)
      8         V[j]= V[j].norm() #normalised vector
      9         for i in range(j):
---> 10             V[j] -= ip(V[i],V[j]) * V[i] #find value of V[j] which is orthogonal to every previous vectors
     11     print(ip(V[Integer(1)],V[Integer(0)]))
     12     print(ip(V[Integer(1)],V[Integer(2)]))

<ipython-input-4-f47230187e38> in ip(v, w)
      6 #to save us some typing...
      7 def ip(v,w):
----> 8     return v.inner_product(w)
      9 def length(v):
     10     return sqrt(ip(v,v))

/home/sage/sage/local/lib/python3.7/site-packages/sage/structure/element.pyx in sage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4614)()
    485             AttributeError: 'LeftZeroSemigroup_with_category.element_class' object has no attribute 'blah_blah'
    486         """
--> 487         return self.getattr_from_category(name)
    488 
    489     cdef getattr_from_category(self, name):

/home/sage/sage/local/lib/python3.7/site-packages/sage/structure/element.pyx in sage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4723)()
    498         else:
    499             cls = P._abstract_element_class
--> 500         return getattr_from_other_class(self, cls, name)
    501 
    502     def __dir__(self):

/home/sage/sage/local/lib/python3.7/site-packages/sage/cpython/getattr.pyx in sage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2614)()
    392         dummy_error_message.cls = type(self)
    393         dummy_error_message.name = name
--> 394         raise AttributeError(dummy_error_message)
    395     attribute = <object>attr
    396     # Check for a descriptor (__get__ in Python)

AttributeError: 'sage.rings.rational.Rational' object has no attribute 'inner_product'

有人可以指出哪里出了问题吗？