restart:

# NEWTON'S INTERPOLATORY DIVIDED-DIFFERENCE FORMULA ALGORITHM 3.2

# To obtain the divided-difference coefficients of the

# interpolatory polynomial P on the (n+1) distinct numbers x(0),

# x(1), ..., x(n) for the function f:

# INPUT: numbers x(0), x(1), ..., x(n); values f(x(0)), f(x(1)),

# ..., f(x(n)) as the first column Q(0,0), Q(1,0), ...,

# Q(N,0) of Q, or may be computed if function f is supplied.

# OUTPUT: the numbers Q(0,0), Q(1,1), ..., Q(N,N) where

# P(x) = Q(0,0) + Q(1,1)*(x-x(0)) + Q(2,2)*(x-x(0))*

# (x-x(1)) +... + Q(N,N)*(x-x(0))*(x-x(1))*...*(x-x(N-1)).

print(`Newtons form of the interpolation polynomial\134n`):

OK := FALSE:

while OK = FALSE do

print(`Choice of input method: `):

print(`1. Input entry by entry from keyboard `):

print(`2. Input data from a text file `):

print(`3. Generate data using a function F `):

print(`Choose 1, 2, or 3 please `):

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

if FLAG = 1 or FLAG = 2 or FLAG = 3 then

OK := TRUE:

fi:

od:

if FLAG = 1 then

OK := FALSE:

while OK = FALSE do

print(`Input n `):

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

if N > 0 then

OK := TRUE:

for I1 from 0 to N do

print(`Input X(i) and F(X(i)) for i = `):print(I1):

print(`separated by space `):

X[I1] := scanf(`%f`)[1]:

Q[I1,0] := scanf(`%f`)[1]: print(`Entries are `): print(X[I1],Q[I1,0]):

od:

else

print(`Number must be a positive integer `):

fi:

od:

fi:

if FLAG = 2 then

print(`Has a text file been created with the data in two columns ? `):

print(`Enter Y or N `):

A := scanf(` %c`)[1]:print(`Entry = `):print(A):

if A = "Y" or A = "y" then

print(`Input the file name in the form - `):

print(`drive:\134name.ext `):

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

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

INP := fopen(NAME,READ,TEXT):

OK := FALSE:

while OK = FALSE do

print(`Input n `):

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

if N > 0 then

for I1 from 0 to N do

X[I1] := fscanf(INP, `%f`)[1]:

Q[I1,0] := fscanf(INP, `%f`)[1]:

od:

fclose(INP):

OK := TRUE:

else

print(`Number must be a positive integer `)

fi:

od:

else

print(`Please create the input file in two column `):

print(`form with the X values and `):

print(`F(X) values in the corresponding columns. `):

print(`The program will end so the input file can `):

print(`be created. `):

OK := FALSE:

fi:

if FLAG = 3 then

print(`Input the function F(x) in terms of x `):

print(`For example: cos(x) `):

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

F := unapply(F,x):

OK := FALSE:

while OK = FALSE do

print(`Input n `):

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

if N > 0 then

for I1 from 0 to N do

print(`Input X(i) for i = `):print(I1):

X[I1] := scanf(`%f`)[1]:

Q[I1,0] := F(X[I1]):

od:

OK := TRUE:

else

print(`Number must be a positive integer `):

fi:

od:

fi:

if OK = TRUE then

print(`Select output destination `):

print(`1. Screen `):

print(`2. Text file `):

print(`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, `NEWTONS INTERPOLATION POLYNOMIAL\134n\134n`):

# STEP 1

for I1 from 1 to N do

for J from 1 to I1 do

Q[I1,J] := (Q[I1,J-1] - Q[I1-1,J-1]) / (X[I1] - X[I1-J]):

od:

# STEP 2

fprintf(OUP, `Input data follows:\134n`):

for I1 from 0 to N do

fprintf(OUP, `X(%d) = %12.8f F(X(%d)) = %12.8f\134n`, I1, X[I1], I1, Q[I1,0]):

od:

fprintf(OUP, `\134nThe coefficients Q(0,0), ..., Q(N,N) are:\134n`):

for I1 from 0 to N do

fprintf(OUP, `%12.8f\134n`, Q[I1,I1]):

od:

if OUP <> default then

fclose(OUP):

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

fi:

SU5OZXd0b25zfmZvcm1+b2Z+dGhlfmludGVycG9sYXRpb25+cG9seW5vbWlhbHwrRzYi

STlDaG9pY2V+b2Z+aW5wdXR+bWV0aG9kOn5HNiI=

SUcxLn5JbnB1dH5lbnRyeX5ieX5lbnRyeX5mcm9tfmtleWJvYXJkfkc2Ig==

SUAyLn5JbnB1dH5kYXRhfmZyb21+YX50ZXh0fmZpbGV+RzYi

SUUzLn5HZW5lcmF0ZX5kYXRhfnVzaW5nfmF+ZnVuY3Rpb25+Rn5HNiI=

STpDaG9vc2V+MSx+Mix+b3J+M35wbGVhc2V+RzYi

SSlFbnRyeX49fkc2Ig==

IiIi

SSlJbnB1dH5ufkc2Ig==

SSVufj1+RzYi

IiIk

SUBJbnB1dH5YKGkpfmFuZH5GKFgoaSkpfmZvcn5pfj1+RzYi

IiIh

STRzZXBhcmF0ZWR+Ynl+c3BhY2V+RzYi

SS1FbnRyaWVzfmFyZX5HNiI=

NiQkIiM1ISIiJCIoeD5sKCEiKA==

SUBJbnB1dH5YKGkpfmFuZH5GKFgoaSkpfmZvcn5pfj1+RzYi

IiIi

STRzZXBhcmF0ZWR+Ynl+c3BhY2V+RzYi

SS1FbnRyaWVzfmFyZX5HNiI=

NiQkIiM4ISIiJCInJzM/JyEiJw==

SUBJbnB1dH5YKGkpfmFuZH5GKFgoaSkpfmZvcn5pfj1+RzYi

IiIj

STRzZXBhcmF0ZWR+Ynl+c3BhY2V+RzYi

SS1FbnRyaWVzfmFyZX5HNiI=

NiQkIiM7ISIiJCIoQVNiJSEiKA==

SUBJbnB1dH5YKGkpfmFuZH5GKFgoaSkpfmZvcn5pfj1+RzYi

IiIk

STRzZXBhcmF0ZWR+Ynl+c3BhY2V+RzYi

SS1FbnRyaWVzfmFyZX5HNiI=

NiQkIiM+ISIiJCIoJz09RyEiKA==

STtTZWxlY3R+b3V0cHV0fmRlc3RpbmF0aW9ufkc2Ig==

SSsxLn5TY3JlZW5+RzYi

SS4yLn5UZXh0fmZpbGV+RzYi

SS5FbnRlcn4xfm9yfjJ+RzYi

SSlFbnRyeX49fkc2Ig==

IiIi

NEWTONS INTERPOLATION POLYNOMIAL Input data follows: X(0) = 1.00000000 F(X(0)) = 0.76519770 X(1) = 1.30000000 F(X(1)) = 0.62008600 X(2) = 1.60000000 F(X(2)) = 0.45540220 X(3) = 1.90000000 F(X(3)) = 0.28181860 The coefficients Q(0,0), ..., Q(N,N) are: 0.76519770 -0.48370567 -0.10873389 0.06587839

JSFH