用三次樣條插值求斜率
三次樣條插值的matlabt程序9 c" N- I7 b; c# z* yfunction x=followup(a,b,c,d)n=length(d);
a(1)=0;% M" u/ r: t! p6 v) f
%“追”的過程# p, F# G$ ^* | n( w
L(1)=b(1);
y(1)=d(1)/L(1);0 j7 J4 p+ [5 g; \% [' \1 u+ Y4 a
u(1)=c(1)/L(1); m, ]" e. a( a' }$ c
for i=2
n-1)L(i)=b(i)-a(i)*u(i-1);
y(i)=(d(i)-y(i-1)*a(i))/L(i);' e; X" \ I+ e& A2 C! a
u(i)=c(i)/L(i);5 \; `% p b2 U2 \
end& ^+ L# d$ {5 m7 V+ A# u
L(n)=b(n)-a(n)*u(n-1);
y(n)=(d(n)-y(n-1)*a(n))/L(n);% `# w" @+ ^& R6 p3 B2 J+ G; b+ \* m2 Q# e
%“趕”的過程0 ]/ k; v: j3 j0 O l
x(n)=y(n);
for i=(n-1):-1:1* Q& G/ S2 B: z' v
x(i)=y(i)-u(i)*x(i+1);
end" I9 V0 y, ]2 Y: J
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function[s,y0]=spline3 (x,y,x0)
%x,y為數(shù)表x0為插值點(diǎn)s表示插值函數(shù)y0為x0對(duì)應(yīng)的插值函數(shù)值" Y& X" w, V r$ f" W$ l+ {- V
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n=length(x);
%得出n
for i=1:n-1;1 k2 ^- _" d$ N5 y
h(i)=x(i+1)-x(i);+ W6 B; Q* r \% a# `
end) `& Y) |6 q+ d' n1 H
for i=2:n-1;
lamda(i)=h(i)/(h(i-1)+h(i));1 i9 V! H5 x% F5 s* W% L
miu(i)=1-lamda(i);
g(i)=3*(lamda(i)*((y(i)-y(i-1))/h(i-1))+miu(i)*((y(i+1)-y(i))/h(i)));$ e! R, B* U2 ?1 k& M* c& z
end
g(1)=3*((y(2)-y(1))/h(1));% i& w$ m1 h: X+ s/ K
g(n)=3*((y(n)-y(n-1))/h(n-1));5 S! {1 a( x( O3 e6 w# h& j
%前邊求出lamda miu和g從而可以確定系數(shù)矩陣3 j, X% D1 d, ~6 |2 K: p
miu(1)=1;
miu(4)=0;+ P: \5 U. p, L a4 B8 u
lamda(n)=1;
lamda(1)=0;
%根據(jù)第二邊界條件補(bǔ)充兩個(gè)lamda和miu的值
for i=1:n
beta(i)=2;( ^! g, o( ?. M/ N' @
end
m=followup(lamda,beta,miu,g)3 j5 K$ v$ x. x! W0 B! ^/ b: T5 F
%解出m的值從而可確定st st為各段的插值多項(xiàng)式
for i=1:n-1+ _2 y, N/ K# W- C5 B* g
st(i)=(t-x(i+1))^2*(h(i)+2*(t-x(i)))*y(i)/(h(i)^3)...; D1 u e8 @# K1 o3 f5 p" Q1 m
+(t-x(i))^2*(h(i)+2*(x(i+1)-t))*y(i+1)/(h(i)^3)...
+(t-x(i))^2*(t-x(i+1))*m(i+1)/(h(i)^2)...8 u h, N% D# S$ \
+(t-x(i+1))^2*(t-x(i))*m(i)/(h(i)^2);, j: a) _$ _8 _* S- R5 C( R
end
%得到插值的結(jié)果各段的t的表達(dá)式& y2 ~9 x+ f$ J, t
%接下來要將插值點(diǎn)x0代入首先確定x0所在的插值區(qū)間0 N, P3 |3 e% R* }# y6 C
for i=1:n-18 _( ^( L) M7 ^
if (x(i)>x0)
in=i;
end
end% M( ]6 k4 T/ k1 ~
s=st(in);
s=expand(s);
s=collect(s,'t');
y0=subs(s,'t',x0): x# F5 j$ }, ~- {$ k
%s是插值多項(xiàng)式y(tǒng)0是插值點(diǎn)的函數(shù)值
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在matlab中輸入+ k6 n/ Z7 h; y) a+ g
x=[1 2 4 5]；
y=[1 3 4 2]；
spline3(x,y,2)
會(huì)得到各點(diǎn)的斜率
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