Multivariate Taylor approximation in sympy

Question

I aim to write a multidimensional Taylor approximation using sympy, which

• uses as many builtin code as possible,
• computes the truncated T

Answer 1

multivariate taylor expansion

def mtaylor(funexpr,x,mu,order=1):

    nvars = len(x)
    hlist = ['__h' + str(i+1) for i in range(nvars)]
    command=''
    command="symbols('"+'  '.join(hlist) +"')"
    hvar = eval(command)
    #mtaylor is utaylor for specificly defined function
    t = symbols('t')
    #substitution
    loc_funexpr = funexpr
    for i in range(nvars):
        locvar = x[i]
        locsubs = mu[i]+t*hvar[i]
        loc_funexpr = loc_funexpr.subs(locvar,locsubs)
    #calculate taylorseries
    g = 0
    for i in range(order+1):
        g+=loc_funexpr.diff(t,i).subs(t,0)*t**i/math.factorial(i)

    #resubstitute
    for i in range(nvars):
        g = g.subs(hlist[i],x[i]-mu[i])

    g = g.subs(t,1)    
    return g

test for some function

x1,x2,x3,x4,x5 = symbols('x1 x2 x3 x4 x5')
funexpr=1+x1+x2+x1*x2+x1**3
funexpr=cos(funexpr)
x=[x1,x2,x3,x4,x5]
mu=[1,1,1,1,1]
mygee = mtaylor(funexpr,x,mu,order=4)
print(mygee)