(b) Compare the answers you obtain in (a) with the exact by computing the relati

Question

(b) Compare the answers you obtain in (a) with the exact by computing the relative error in each case.

(c) Comment on the results you obtain.

(* SIMPSON'S COMPOSITE ALGORITHM 4.1
*
* To approximate I=integral ( ( f(x) dx ) ) from a to b:
*
* INPUT: endpoints a, b; even positive integer n.
*
* OUTPUT: approximation XI to I
*)
TEMP = Input["This is Simpson's Method.
Input the function F(X) in terms of x.

For example: Cos[x] "];
F[x_] := Evaluate[TEMP];
OK = 0;
While[OK == 0,
A = Input["Input the lower limit of integration "];
B = Input["Input the upper limit of integration "];
If[A > B,
Input["Lower limit must be less than upper limit

Press 1 [enter] to continue "],
OK = 1;
];
];
OK = 0;
While[OK == 0,
n = Input["Input an even positive integer N. "];
If[ n > 0 && Mod[n,2]==0,
OK = 1,
Input["Number must be even and positive

Press 1 [enter] to continue "];
];
];
If[OK == 1,
(* Step 1 *)
H = (B-A)/n;
(* Step 2 *)
Xi0 = F[A]+F[B];
(* Summation of f(x(2*i-1)) *)
Xi1 = 0.0;
(* Summation of f(x(2*i)) *)
Xi2 = 0.0;
(* Step 3 *)
NN = n-1;
For[i = 1,
i <= NN,
i++,
(* Step 4 *)
X = A+i*H;
(* Step 5 *)
If[Mod[i,2] == 0,
Xi2 = Xi2+F[X],
Xi1 = Xi1+F[X];
];
];
(* Step 6 *)
Xi = (Xi0+2.0*Xi2+4.0*Xi1)*H/3.0;
(* Step 7 *)
Print[" "];
Print["The integral of F from ",A," to ",B,
" is ",N[Xi,9]];
];
.1a

Explanation / Answer

TEMP = Input["This is Simpson's Method.
Input the function F(X) in terms of x.

For example: Cos[x] "];
F[x_] := Evaluate[TEMP];
OK = 0;
While[OK == 0,
A = Input["Input the lower limit of integration "];
B = Input["Input the upper limit of integration "];
If[A > B,
Input["Lower limit must be less than upper limit

Press 1 [enter] to continue "],
OK = 1;
];
];
OK = 0;
While[OK == 0,
n = Input["Input an even positive integer N. "];
If[ n > 0 && Mod[n,2]==0,
OK = 1,
Input["Number must be even and positive

Press 1 [enter] to continue "];
];
];
If[OK == 1,
(* Step 1 *)
H = (B-A)/n;
(* Step 2 *)
Xi0 = F[A]+F[B];
(* Summation of f(x(2*i-1)) *)
Xi1 = 0.0;
(* Summation of f(x(2*i)) *)
Xi2 = 0.0;
(* Step 3 *)
NN = n-1;
For[i = 1,
i <= NN,
i++,
(* Step 4 *)
X = A+i*H;
(* Step 5 *)
If[Mod[i,2] == 0,
Xi2 = Xi2+F[X],
Xi1 = Xi1+F[X];
];
];
(* Step 6 *)
Xi = (Xi0+2.0*Xi2+4.0*Xi1)*H/3.0;
(* Step 7 *)
Print[" "];
Print["The integral of F from ",A," to ",B,
" is ",N[Xi,9]];
];

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