restart:

# NONLINEAR SHOOTING ALGORITHM 11.2

# To approximate the solution of the nonlinear boundary-value problem

# Y'' = F(X,Y,Y'), A<=X<=B, Y(A) = ALPHA, Y(B) = BETA:

# INPUT: Endpoints A,B: boundary conditions ALPHA, BETA: number of

# subintervals N: tolerance TOL: maximum number of iterations M.

# OUTPUT: Approximations W(1,I) TO Y(X(I)): W(2,I) TO Y'(X(I))

# for each I=0,1,...,N or a message that the maximum

# number of iterations was exceeded.

print(`This is the Nonlinear Shooting Method.`):

print(`Input the function F(X,Y,Z) in terms of x, y, z.`):

print(`For example: (32+2*x^3-y*z)/8`):

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

FY := diff(F,y):

FYP := diff(F,z):

F := unapply(F,x,y,z):

FY := unapply(FY,x,y,z):

FYP := unapply(FYP,x,y,z):

OK := FALSE:

while OK = FALSE do

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

A := scanf(`%e`)[1]: print(`Left endpoint = `):print(A):

B := scanf(`%e`)[1]: print(`Right endpoint = `):print(B):

if A >= B then

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

else OK := TRUE:

fi:

od:

print(`Input Y(A).`):

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

print(`Input Y(B).`):

BETA := scanf(`%e`)[1]: print(`beta = `):print(BETA):

TK := (BETA-ALPHA)/(B-A):

print(`TK := `): print(TK):

print(`Input new TK? Enter 1 for yes or 2 for no.`):

AA := scanf(`%d`)[1]: print(`Input is `):print(AA):

if AA = 1 then

print(`Input new TK`):

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

fi:

OK := FALSE:

while OK = FALSE do

print(`Input an integer > 1 for the number of subintervals.`):

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

if N <= 1 then

print(`Number must exceed 1.`):

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.`):

NN := scanf(`%d`)[1]:print(`Maximum number of iterations = `):print(NN):

if NN <= 0 then

print(`Must be positive integer.`):

else

OK := TRUE:

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(`Input is `):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(`Output file is `):print(NAME):

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

else

OUP := default:

fi:

fprintf(OUP, `NONLINEAR SHOOTING METHOD\134n\134n`):

fprintf(OUP, ` I X(I) W1(I) W2(I)\134n`):

# Step 1

H := (B-A)/N:

K := 1:

# TK is already computed.

OK := FALSE:

# Step 2

while K <= NN and OK = FALSE do

# Step 3

W1[0] := ALPHA:

W2[0] := TK:

U1 := 0 :

U2 := 1:

# Step 4

# Runge-Kutta method for systems is used in Steps 5 and 6

for I2 from 1 to N do

# Step 5

X := A+(I2-1)*H:

T := X+0.5*H:

# Step 6

K11 := H*W2[I2-1]:

K12 := H*F(X,W1[I2-1],W2[I2-1]):

K21 := H*(W2[I2-1]+0.5*K12):

K22 := H*F(T,W1[I2-1]+0.5*K11,W2[I2-1]+0.5*K12):

K31 := H*(W2[I2-1]+0.5*K22):

K32 := H*F(T,W1[I2-1]+0.5*K21,W2[I2-1]+0.5*K22):

K41 := H*(W2[I2-1]+K32):

K42 := H*F(X+H,W1[I2-1]+K31,W2[I2-1]+K32):

W1[I2] := W1[I2-1]+(K11+2*(K21+K31)+K41)/6:

W2[I2] := W2[I2-1]+(K12+2*(K22+K32)+K42)/6:

K11 := H*U2:

K12 := H*(FY(X,W1[I2-1],W2[I2-1])*U1+FYP(X,W1[I2-1],W2[I2-1])*U2):

K21 := H*(U2+0.5*K12):

K22 := H*(FY(T,W1[I2-1],W2[I2-1])*(U1+0.5*K11)+FYP(T,W1[I2-1],W2[I2-1])*(U2+0.5*K21)):

K31 := H*(U2+0.5*K22):

K32 := H*(FY(T,W1[I2-1],W2[I2-1])*(U1+0.5*K21)+FYP(T,W1[I2-1],W2[I2-1])*(U2+0.5*K22)):

K41 := H*(U2+K32):

K42 := H*(FY(X+H,W1[I2-1],W2[I2-1])*(U1+K31)+FYP(X+H,W1[I2-1],W2[I2-1])*(U2+K32)):

U1 := U1+(K11+2*(K21+K31)+K41)/6:

U2 := U2+(K12+2*(K22+K32)+K42)/6:

od:

# Step 7

# Test for accuracy

if abs(W1[N]-BETA) < TOL then

# Step 8

I2 := 0:

fprintf(OUP, `%3d %13.8f %13.8f %13.8f\134n`, I2, A, ALPHA, TK):

for I2 from 1 to N do

J := I2+1:

X := A+I2*H:

fprintf(OUP, `%3d %13.8f %13.8f %13.8f\134n`, I2, X, W1[J-1], W2[J-1]):

od:

fprintf(OUP, `Convergence in %d iterations\134n`, K):

fprintf(OUP, ` t = %14.7e\134n`, TK):

# Step 9

OK := TRUE:

else

# Step 10

# Newton's method applied to improve TK

TK := TK-(W1[N]-BETA)/U1:

K := K+1:

fi:

od:

# Step 11

# Procedure completed unsuccessfully.

if OK = FALSE then

fprintf(OUP, `Method failed after %d iterations\134n`, NN):

fi:

if OUP <> default then

fclose(OUP):

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

fi:

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IiIi

NONLINEAR SHOOTING METHOD I X(I) W1(I) W2(I)

0 1.00000000 17.00000000 -14.00018772 1 1.10000000 15.75549654 -11.02333504 2 1.20000000 14.77339188 -8.71129092 3 1.30000000 13.99775528 -6.86761495 4 1.40000000 13.38863301 -5.36340417 5 1.50000000 12.91672415 -4.11123148 6 1.60000000 12.56005225 -3.05010428 7 1.70000000 12.30181137 -2.13642267 8 1.80000000 12.12893005 -1.33835032 9 1.90000000 12.03108856 -0.63220162 10 2.00000000 12.00003106 -0.00006000 11 2.10000000 12.02907423 0.57182956 12 2.20000000 12.11274983 1.09416887 13 2.30000000 12.24654065 1.57538505 14 2.40000000 12.42668231 2.02218700 15 2.50000000 12.65001273 2.43996928 16 2.60000000 12.91385628 2.83310940 17 2.70000000 13.21593375 3.20518954 18 2.80000000 13.55429153 3.55916386 19 2.90000000 13.92724542 3.89748611 20 3.00000000 14.33333583 4.22220804 Convergence in 6 iterations t = -1.4000188e+01

JSFH