restart:

# HERMITE INTERPOLATION ALGORITHM 3.3

# TO OBTAIN THE COEFFICIENTS OF THE HERMITE INTERPOLATING

# POLYNOMIAL H ON THE (N+1) DISTINCT NUMBERS X(0), ..., X(N)

# FOR THE FUNCTION F:

# INPUT: NUMBERS X(0), X(1), ..., X(N): VALUES F(X(0)),

# F(X(1)),..., F(X(N)) AND F'(X(0)), F'(X(1)), ...,

# F'(X(N)).

# OUTPUT: NUMBERS Q(0,0), Q(1,1), ..., Q(2N + 1,2N + 1) WHERE

# H(X) = Q(0,0) + Q(1,1) * ( X - X(0) ) + Q(2,2) *

# ( X - X(0) )**2 + Q(3,3) * ( X - X(0) )**2 *

# ( X - X(1) ) + Q(4,4) * ( X - X(0) )**2 *

# ( X - X(1) )**2 + ... + Q(2N + 1,2N + 1) *

# ( X - X(0) )**2 * ( X - X(1) )**2 * ... *

# ( X - X(N - 1) )**2 * (X - X(N) ).

print(`This is Hermite interpolation.\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 the number of data points minus 1 `):

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), F(X(i)), and F'(X(i)) separated by spaces for i = `):print(I1):

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

Q[2*I1,0] := scanf(`%f`)[1]:

Q[2*I1+1,1] := scanf(`%f`)[1]:print(`Entries are `):print(X[I1],Q[2*I1,0],Q[2*I1+1,1]):

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 three 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:\134\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 the number of data points minus 1 `):

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

Q[2*I1+1,1] := 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 three column `):

print(`form with the X values, F(X), 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: sin(x) `):

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

FP := diff(F,x):

F := unapply(F,x):

FP := unapply(FP,x):

OK := FALSE:

while OK = FALSE do

print(`Input the number of data points minus 1 `):

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[2*I1,0] := F(X[I1]):

Q[2*I1+1,1] := FP(X[I1]):

od:

OK := TRUE:

else

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

fi:

od:

fi:

if OK = TRUE then

# Step 1

for I1 from 0 to N do

# Step 2

Z[2*I1] := X[I1]:

Z[2*I1+1] := X[I1]:

Q[2*I1+1,0] := Q[2*I1,0]:

# Step 3

if I1 <> 0 then

Q[2*I1,1] := (Q[2*I1,0]-Q[2*I1-1,0])/(Z[2*I1]-Z[2*I1-1]):

fi:

od:

# Step 4

K := 2*N+1:

for I1 from 2 to K do

for J from 2 to I1 do

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

od:

# Step 5

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, `HERMITE INTERPOLATING POLYNOMIAL\134n\134n`):

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

fprintf(OUP, ` X, F(X), F'(X)\134n`):

for I1 from 0 to N do

fprintf(OUP, ` %12.10e %12.10e %12.10e\134n`, X[I1], Q[2*I1,0], Q[2*I1+1,1]):

od:

fprintf(OUP, `\134nThe Coefficients of the Hermite Interpolation `):

fprintf(OUP, `Polynomial\134n`):

fprintf(OUP, `in order of increasing exponent follow:\134n\134n`):

for I1 from 0 to K do

fprintf(OUP, ` %12.10e\134n`, Q[I1,I1]):

od:

print(`Do you wish to evaluate this polynomial? `):

print(`Enter Y or N `):

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

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

print(`Enter a point at which to evaluate `):

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

S := Q[K,K]*(XX-Z[K-1]):

for I1 from 2 to K do

J := K-I1+1:

S := (S+Q[J,J])*(XX-Z[J-1]):

od:

S := S + Q[0,0]:

fprintf(OUP, `x-value and interpolated-value\134n`):

fprintf(OUP, ` %12.10e %12.10e\134n`, XX, S):

fi:

if OUP <> default then

fclose(OUP):

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

fi:

SUBUaGlzfmlzfkhlcm1pdGV+aW50ZXJwb2xhdGlvbi58K0c2Ig==

STlDaG9pY2V+b2Z+aW5wdXR+bWV0aG9kOn5HNiI=

SUcxLn5JbnB1dH5lbnRyeX5ieX5lbnRyeX5mcm9tfmtleWJvYXJkfkc2Ig==

SUAyLn5JbnB1dH5kYXRhfmZyb21+YX50ZXh0fmZpbGV+RzYi

SUUzLn5HZW5lcmF0ZX5kYXRhfnVzaW5nfmF+ZnVuY3Rpb25+Rn5HNiI=

STpDaG9vc2V+MSx+Mix+b3J+M35wbGVhc2V+RzYi

SSlFbnRyeX49fkc2Ig==

IiIi

SUlJbnB1dH50aGV+bnVtYmVyfm9mfmRhdGF+cG9pbnRzfm1pbnVzfjF+RzYi

SSVufj1+RzYi

IiIj

SWluSW5wdXR+WChpKSx+RihYKGkpKSx+YW5kfkYnKFgoaSkpfnNlcGFyYXRlZH5ieX5zcGFjZXN+Zm9yfml+PX5HNiI=

IiIh

SS1FbnRyaWVzfmFyZX5HNiI=

NiUkIiM4ISIiJCInJzM/JyEiJyQhKEstQSYhIig=

SWluSW5wdXR+WChpKSx+RihYKGkpKSx+YW5kfkYnKFgoaSkpfnNlcGFyYXRlZH5ieX5zcGFjZXN+Zm9yfml+PX5HNiI=

IiIi

SS1FbnRyaWVzfmFyZX5HNiI=

NiUkIiM7ISIiJCIoQVNiJSEiKCQhKGYqKXAmRig=

SWluSW5wdXR+WChpKSx+RihYKGkpKSx+YW5kfkYnKFgoaSkpfnNlcGFyYXRlZH5ieX5zcGFjZXN+Zm9yfml+PX5HNiI=

IiIj

SS1FbnRyaWVzfmFyZX5HNiI=

NiUkIiM+ISIiJCIoJz09RyEiKCQhKHI6ImVGKA==

STpDaG9pY2V+b2Z+b3V0cHV0fm1ldGhvZDp+RzYi

STUxLn5PdXRwdXR+dG9+c2NyZWVufkc2Ig==

STgyLn5PdXRwdXR+dG9+dGV4dH5maWxlfkc2Ig==

STVQbGVhc2V+ZW50ZXJ+MX5vcn4yfkc2Ig==

SSlFbnRyeX49fkc2Ig==

IiIi

HERMITE INTERPOLATING POLYNOMIAL The input data follows: X, F(X), F'(X) 1.3000000000e+00 6.2008600000e-01 -5.2202320000e-01 1.6000000000e+00 4.5540220000e-01 -5.6989590000e-01 1.9000000000e+00 2.8181860000e-01 -5.8115710000e-01 The Coefficients of the Hermite Interpolation Polynomial in order of increasing exponent follow: 6.2008600000e-01 -5.2202320000e-01 -8.9742666670e-02 6.6365555570e-02 2.6666666330e-03 -2.7746912770e-03

SUpEb355b3V+d2lzaH50b35ldmFsdWF0ZX50aGlzfnBvbHlub21pYWw/fkc2Ig==

SS5FbnRlcn5Zfm9yfk5+RzYi

SSlFbnRyeX49fkc2Ig==

USJ5NiI=

SURFbnRlcn5hfnBvaW50fmF0fndoaWNofnRvfmV2YWx1YXRlfkc2Ig==

SSlFbnRyeX49fkc2Ig==

JCIjOiEiIg==

x-value and interpolated-value 1.5000000000e+00 5.1182770170e-01

JSFH