1
我有一個用帕斯卡計算三次樣條的任務。我已經完成了大部分的程序，還有一個問題。我的任務是，對於一個函數y = f（x），找到樣條S1和S2，也可以找到
delta1 = max | f（x）-S（x）|，delta2 = max | f'（x）-S1 '（x）|，delta3 = max | f（x）-S2（x）|，delta4 = max | f'（x）-S2'（x）|三次樣條插值：如何計算二次樣條S2
對於S2，哪個公式適用？
這裏是我到目前爲止的代碼：
program spline_1_2;
uses crt;
const
k=1000;
var
input_file:text;
out1_file:text;
out2_file:text;
n,i:integer;
a_int,b_int,h:real;
delta_1, delta_2:real;
x, alpha, beta, y, a_coef, b_coef, c_coef, d_coef:array[0..k] of real;
S, S1, S2:array[0..k] of real;
A, B, C, F, z:array[0..k] of real;
Function f_given(x:real):real;
begin
f_given:=Cos((x*x)/3)-Exp(Sin(2*x));
end;
Function f_prime(x:real):real;
begin
f_prime:=-Cos((x*x)/3)*(1/3)*2*x-Exp(Sin(2*x))*Cos(2*x)*2;
end;
begin
clrscr;
Assign(input_file,'in.txt');
Reset(input_file);
Assign(out1_file,'out1.txt');
Rewrite(out1_file);
Assign(out2_file,'out2.txt');
Rewrite(out2_file);
while not EOF(input_file) do
begin
readln(input_file,a_int,b_int);
readln(input_file,h);
readln(input_file,n);
end;
for i:=0 to n do
begin
x[i]:=a_int+(i-0.5)*h;
f_given(x[i]);
y[i]:= f_given(x[i]);
f_prime(x[i]);
A[i]:= h;
C[i]:= 4*h;
B[i]:= h;
end;
for i:=1 to n-1 do
begin
a_coef[i]:=y[i];
F[i]:=((y[i+1]-2*y[i]+y[i-1])*3)/(h*h);
A[1]:=0;
C[n-1]:=0;
end;
alpha[1]:=-C[1]/B[1];
beta[1]:=F[1]/B[1];
for i:=1 to n-1 do
begin
z[i]:=(A[i]*alpha[i-1] + C[i]);
alpha[i]:= -B[i]/z[i];
beta[i]:= (F[i]-A[i]*beta[i-1])/z[i];
end;
c_coef[n-1]:=(F[i] - A[i] * beta[n-1])/(C[i] + A[i] * alpha[n-1]);
for i:=n-2 downto 1 do
begin
c_coef[i]:=alpha[i]*c_coef[i+1] + beta[i];
c_coef[0]:=0;
c_coef[n]:=0;
end;
for i:=0 to n-1 do
begin
d_coef[i]:=(c_coef[i+1]-c_coef[i])/(3*h);
end;
b_coef[0]:=f_prime(x[0]);
b_coef[n-1]:=f_prime(x[n-1]);
for i:=1 to n-2 do
begin
b_coef[i]:=(h/3)*(c_coef[i+1]+2*c_coef[i]) + (y[i+1] - y[i])/h;
end;
//our spline
for i:=1 to n do
begin
S[i]:= a_coef[i] + b_coef[i]*(x[i-1]-x[i]) + c_coef[i]*sqr(x[i-1]-x[i]) + d_coef[i]*sqr(x[i-1]-x[i])*(x[i-1]-x[i]);
S1[i]:=b_coef[i]+2*c_coef[i]*(x[i-1]-x[i])+3*d_coef[i]*(x[i-1]-x[i])*(x[i-1]-x[i]);
end;
for i:=1 to n-1 do
begin
Writeln(out1_file, x[i],' ',y[i]:3:6,' ',S[i]:3:6);
delta_1:=abs(y[i]-S[i]);
Writeln(out2_file,x[i],' ',f_prime(x[i]):3:6,' ', S1[i]:3:6);
delta_2:=abs(f_prime(x[i])-S1[i]);
end;
Writeln(out1_file,'"delta1" = ',delta_1);
Writeln(out2_file,'"delta2" = ',delta_2);
Readln;
Close(input_file);
Close(out1_file);
Close(out2_file);
end.