复合梯形规则和标准正态分布所需的节点数

如何解决复合梯形规则和标准正态分布所需的节点数

我使用不同的数值方法来近似正态分布。我的第一种方法是梯形规则。为什么需要我的节点数量表现如此奇怪？我用一个表格来找出 n 的最小值，这样误差就会小于 0.0001。我会期待一个模式。有没有我想念它？ 标准差为 1 的 n 为 6。 标准差为 2 的 n 为 54。 标准差为 3 时的 n 为 29。

我的代码如下：

s=1---------------------------------------------------------
print("For Standard Deviation,s,of 1,we have a=-1 and b=1")


def CompositeTrapezoidal(f,a,b,n):
    #compute the size of each subinterval
    h = (b-a)/n
    #here we add all the terms
    sum = 1/2*f(a)
    #add the values for all intermediate nodes
    for i in [1,2,..,n-1] :
        sum += f(a+i*h)
    #add the last node
    sum +=1/2*f(b)
    #multiply by h to complete the computation
    sum = sum*h
    #return the sum
    return sum
#define the function we will be using
f(x) =(e^-((x^2)/2))/(sqrt(2*pi))
#print the top part of the table
print ("n","\t","trapezoidal","\t\t","error")
print ("--------------------------------------------------")
#call the function several times
for power in [11,12,45]:
    n = power
    trapezoidal = CompositeTrapezoidal(f,-1,1,n).n()
    error = (trapezoidal - integral(f(x),x,1)).n()
    if abs(error) >= 0.0001:
        print (n,trapezoidal,error)
    if abs(error) <= 0.0001:
        print(n,"nodes")

print("by the table above we need n=41 for s=1 to get error less than 1/100.")

trapezoidal = CompositeTrapezoidal(f,41).n()
error = (trapezoidal - integral(f(x),1)).n()
print("integral(f,1) ≈","by Composite Trapezoid Rule")
print()

#s=2---------------------------------------------------------------------------

print("For Standard Deviation,of 2,we have a=-2 and b=2")


def CompositeTrapezoidal(f,"error")
print ("--------------------------------------------------")
#call the function several times
for power in [21,22,55]:
    n = power
    trapezoidal = CompositeTrapezoidal(f,-2,2)).n()
    if abs(error) >= 0.0001:
        print (n,"nodes")
print("By the table above we need n=54 when s=2 to get error less than 1/100.")

trapezoidal = CompositeTrapezoidal(f,54).n()
error = (trapezoidal - integral(f(x),2)).n()
print("integral(f,2) ≈","by Composite Trapezoid Rule")
print()




#s=3 ---------------------------------------------------------------

print("For Standard Deviation,of 3,we have a=-3 and b=3")

def CompositeTrapezoidal(f,-3,3,3)).n()
    if abs(error) >= 0.0001:
        print (n,"nodes")
print("By the table above we need n=54 when s=3 to get error less than 1/100.")

trapezoidal = CompositeTrapezoidal(f,3)).n()
print("integral(f,3) ≈","by Composite Trapezoid Rule")