三次样条插值自然边界条件

例：已知组数点编写程序求解三次样条插值函数满足

针面组具体实验数

025
03
039
045
053

05000
05477
06245
06708
07280
求解中边界条件
1）三次样条插值然边界条件源程序：
function sspline3(xydy1dyn)
x节点y节点函数值dy1dyn分x025053处二阶导
mlength(x)nlength(y)
if m~n
error(＇x or y输入误＇)
return
end
hzeros(1n1)
h(n1)x(n)x(n1)
for k1n2
h(k)x(k+1)x(k)
v(k)h(k+1)(h(k+1)+h(k))
u(k)1v(k)
end
g(1)3*(y(2)y(1))h(1)h(1)2*dy1
g(n)3*(y(n)y(n1))h(n1)+h(n1)2*dyn
for i2n1
g(i)3*(u(i1)*(y(i+1)y(i))h(i)+v(i1)*(y(i)y(i1))h(i1))
end
for i2n1
A(ii1)v(i1)
A(ii+1)u(i1)
end
A(nn1)1
A(12)1
AA+2*eye(n)
Mzhuigf(Ag) 调函数追赶法求M
fprintf(＇三次样条（三角）插值函数表达式＼n＇)
syms X
for k1n1
fprintf(＇Sdd＼n＇kk+1)
s(k)(h(k)+2*(Xx(k)))h(k)^3*(Xx(k+1))^2*y(k)
+(h(k)2*(Xx(k+1)))h(k)^3*(Xx(k))^2*y(k+1)
+(Xx(k))*(Xx(k+1))^2h(k)^2*M(k)+(Xx(k+1))*
(Xx(k))^2h(k)^2*M(k+1)
end
ss＇
svpa(s4)
画三次样条插值函数图
for i1n1
Xx(i)001x(i+1)
st(h(i)+2*(Xx(i)))(h(i)^3)*(Xx(i+1))^2*y(i)
+(h(i)2*(Xx(i+1)))(h(i)^3)*(Xx(i))^2*y(i+1)
+(Xx(i))*(Xx(i+1))^2h(i)^2*M(i)+(Xx(i+1))*
(Xx(i))^2h(i)^2*M(i+1)
plot(xy＇o＇Xst)
hold on
End
plot(xy)
grid on


调函数：
追赶法
function Mzhuigf(Ag)
nlength(A)
Leye(n)
Uzeros(n)
for i1n1
U(ii+1)A(ii+1)
end
U(11)A(11)
for i2n
L(ii1)A(ii1)U(i1i1)
U(ii)A(ii)L(ii1)*A(i1i)
end
Y(1)g(1)
for i2n
Y(i)g(i)L(ii1)*Y(i1)
end
M(n)Y(n)U(nn)
for in111
M(i)(Y(i)A(ii+1)*M(i+1))U(ii)
end

2)命令窗口输入xydy1dyn三次样条函数：

>>x[02503039045053]
>>y[0505477062450670807280]
>>dy10
>>dyn0
>>sspline3(xydy1dyn)

运行结果：

三次样条（三角）插值函数表达式
S12
S23
S34
S45
5000*(3600+1600e5*X)*(X3000)^2+5477*(52001600e5*X)*(X2500)^2+3948*(X2500)*(X3000)^2+3552*(X3000)*(X2500)^2
5477*(6996+2743*X)*(X3900)^2+6245*(11932743*X)*(X3000)^2+1096*(X3000)*(X3900)^2+9677*(X3900)*(X3000)^2
6245*(3333+9259*X)*(X4500)^2+6708*(44449259*X)*(X3900)^2+2177*(X3900)*(X4500)^2+2076*(X4500)*(X3900)^2
6708*(1602+3906*X)*(X5300)^2+7280*(22273906*X)*(X4500)^2+1168*(X4500)*(X5300)^2+1092*(X5300)*(X4500)^2

三次样条函数加整理程序：
scollect(s)
输出结果：
s
459500000000013200*X^3+99000000*X^2148800000000*X
3745930680000042924*X^3+366332200*X^2137976090000*X
518068170000033917*X^3+390576600*X^2733130370000*X
408614400000e1+25768*X^3403188800*X^2+2868770320000*X

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