restart:

# TRAPEZOIDAL WITH NEWTON ITERATION ALGORITHM 5.8

# TO APPROXIMATE THE SOLUTION OF THE INITIAL VALUE PROBLEM:

# Y' = F(T,Y), A <= T <= B, Y(A) = ALPHA,

# AT (N+1) EQUALLY SPACED NUMBERS IN THE INTERVAL [A,B].

# INPUT: ENDPOINTS A,B: INITIAL CONDITION ALPHA: INTEGER N:

# TOLERANCE TOL: MAXIMUM NUMBER OF ITERATIONS M AT ANY ONE STEP.

# OUTPUT: APPROXIMATION W TO Y AT THE (N+1) VALUES OF T

# OR A MESSAGE OF FAILURE.

print(`This is the Implicit Trapezoidal Method. `):

print(`Input the function F(t,y) in terms of t and y `):

print(`For example: y-t^2+1 `):

F := scanf(`%a`)[1]:print(`F(x) = `):print(F):

FYP := diff(F,y):

F := unapply(F,t,y):

FYP := unapply(FYP,t,y):

OK := FALSE:

while OK = FALSE do

print(`Input left and right endpoints separated by blank. `):

A := scanf(`%f`)[1]:

B := scanf(`%f`)[1]:print(`a = `):print(A):print(`b = `):print(B):

if A >= B then

print(`Left endpoint must be less than right endpoint. `):

else

OK := TRUE:

fi:

od:

print(`Input the initial condition. `):

ALPHA := scanf(`%f`)[1]:print(`ALPHA = `):print(ALPHA):

OK := FALSE:

print(`Input a positive integer for the number of subintervals. `):

while OK = FALSE do

N := scanf(`%d`)[1]:print(`N = `):print(N):

if N <= 0 then

print(`Number must be a postiive integer. `):

else

OK := TRUE:

fi:

od:

OK := FALSE:

while OK = FALSE do

print(`Input tolerance. `):

TOL := scanf(`%f`)[1]:print(`Tolerance = `):print(TOL):

if TOL <= 0 then

print(`Tolerance must be positive. `):

else

OK := TRUE:

fi:

od:

OK := FALSE:

while OK = FALSE do

print(`Input maximum number of iterations. `):

M := scanf(`%f`)[1]:print(`Maximum number of iterations = `):print(M):

if M > 0 then

OK := TRUE:

else

print(`Number of iterations must be positive. `):

fi:

od:

if OK = TRUE then

print(`Choice of output method: `):

print(`1. Output to screen `):

print(`2. Output to text file `):

print(`Please enter 1 or 2 `):

FLAG := scanf(`%d`)[1]:print(`Entry = `):print(FLAG):

if FLAG = 2 then

print(`Input the file name in the form - drive:\134\134name.ext `):

print(`For example A:\134\134OUTPUT.DTA `):

NAME := scanf(`%s`)[1]:print(`Entry = `):print(NAME):

OUP := fopen(NAME,WRITE,TEXT):

else

OUP := default:

fi:

fprintf(OUP, `IMPLICIT TRAPEZOIDAL METHOD USING NEWTONS METHOD\134n\134n`):

fprintf(OUP, ` t w #iter\134n`):

# Step 1

W := ALPHA:

T := A:

H := (B-A)/N:

fprintf(OUP, `%5.3f %11.8f 0\134n`, T, W):

I1 := 1:

OK := TRUE:

# Step 2

while I1 <= N and OK = TRUE do

# Step 3

XK1 := W+0.5*H*F(T, W):

W0 := XK1:

J := 1:

IFLAG := 0:

# Step 4

while IFLAG = 0 and OK = TRUE do

# Step 5

W := W0-(W0-XK1-0.5*H*F(T+H, W0))/(1-0.5*H*FYP(T+H, W0)):

# Step 6

if abs(W-W0) < TOL then

# Step 7

IFLAG := 1:

T := A+I1*H:

fprintf(OUP,`%5.3f %11.8f %3d\134n`, T, W, J):

I1 := I1+1:

else

J := J+1:

W0 := W:

if J > M then

OK := FALSE:

fi:

od:

if OK = FALSE then

print(`Maximum Number of Iterations Exceeded`):

fi:

# Step 8

if OUP <> default then

fclose(OUP):

print(`Output file `,NAME,` created successfully`):

fi:

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IiIi

IMPLICIT TRAPEZOIDAL METHOD USING NEWTONS METHOD t w #iter 0.000 -1.00000000 0 0.200 -0.14149685 4 0.400 0.27486139 5 0.600 0.55398284 5 0.800 0.78307197 6 1.000 0.99377255 7

JSFH