编辑:我不是在找你调试这段代码。如果您熟悉这个众所周知的算法,那么您可能会有所帮助。请注意,该算法会正确生成系数。
三次样条插值的这段代码正在产生线性样条,我似乎无法弄清楚为什么(还)。该算法来自 Burden 的数值分析,与此处的伪代码几乎相同,或者您可以从评论中的链接中找到该书(参见第 3 章,无论如何都值得拥有)。该代码正在产生正确的系数;我相信我误解了实施。非常感谢任何反馈。另外,我是编程新手,所以也欢迎任何关于我的编码有多糟糕的反馈。我尝试用 h、a 和 c 上传线性系统的图片,但作为新用户,我不能。如果您想要算法求解的三对角线性系统的视觉效果,并且由 var alpha 设置,请参阅本书评论中的链接,请参阅第 3 章。该系统是严格对角占优的,所以我们知道那里存在唯一解 c0,...,cn。一旦我们知道了 ci 值,其他系数就会随之而来。
import matplotlib.pyplot as plt
# need some zero vectors...
def zeroV(m):
z = [0]*m
return(z)
#INPUT: n; x0, x1, ... ,xn; a0 = f(x0), a1 =f(x1), ... , an = f(xn).
def cubic_spline(n, xn, a, xd):
"""function cubic_spline(n,xn, a, xd) interpolates between the knots
specified by lists xn and a. The function computes the coefficients
and outputs the ranges of the piecewise cubic splines."""
h = zeroV(n-1)
# alpha will be values in a system of eq's that will allow us to solve for c
# and then from there we can find b, d through substitution.
alpha = zeroV(n-1)
# l, u, z are used in the method for solving the linear system
l = zeroV(n+1)
u = zeroV(n)
z = zeroV(n+1)
# b, c, d will be the coefficients along with a.
b = zeroV(n)
c = zeroV(n+1)
d = zeroV(n)
for i in range(n-1):
# h[i] is used to satisfy the condition that
# Si+1(xi+l) = Si(xi+l) for each i = 0,..,n-1
# i.e., the values at the knots are "doubled up"
h[i] = xn[i+1]-xn[i]
for i in range(1, n-1):
# Sets up the linear system and allows us to find c. Once we have
# c then b and d follow in terms of it.
alpha[i] = (3./h[i])*(a[i+1]-a[i])-(3./h[i-1])*(a[i] - a[i-1])
# I, II, (part of) III Sets up and solves tridiagonal linear system...
# I
l[0] = 1
u[0] = 0
z[0] = 0
# II
for i in range(1, n-1):
l[i] = 2*(xn[i+1] - xn[i-1]) - h[i-1]*u[i-1]
u[i] = h[i]/l[i]
z[i] = (alpha[i] - h[i-1]*z[i-1])/l[i]
l[n] = 1
z[n] = 0
c[n] = 0
# III... also find b, d in terms of c.
for j in range(n-2, -1, -1):
c[j] = z[j] - u[j]*c[j+1]
b[j] = (a[j+1] - a[j])/h[j] - h[j]*(c[j+1] + 2*c[j])/3.
d[j] = (c[j+1] - c[j])/(3*h[j])
# This is my only addition, which is returning values for Sj(x). The issue I'm having
# is related to this implemention, i suspect.
for j in range(n-1):
#OUTPUT:S(x)=Sj(x)= aj + bj(x - xj) + cj(x - xj)^2 + dj(x - xj)^3; xj <= x <= xj+1)
return(a[j] + b[j]*(xd - xn[j]) + c[j]*((xd - xn[j])**2) + d[j]*((xd - xn[j])**3))
对于无聊的人,或成绩优异的人...
下面是测试代码,区间为 x: [1, 9], y:[0, 19.7750212]。测试函数是 xln(x),所以我们从 1 开始并增加 0.1 直到 9。
ln = []
ln_dom = []
cub = []
step = 1.
X=[1., 9.]
FX=[0, 19.7750212]
while step <= 9.:
ln.append(step*log(step))
ln_dom.append(step)
cub.append(cubic_spline(2, x, fx, step))
step += 0.1
...并用于绘图:
plt.plot(ln_dom, cub, color='blue')
plt.plot(ln_dom, ln, color='red')
plt.axis([1., 9., 0, 20], 'equal')
plt.axhline(y=0, color='black')
plt.axvline(x=0, color='black')
plt.show()