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###     TOPIC           Data Analysis based on the Newton Interpolation Polynomial (NIP)
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###     FILE NAME       NIP_input_StudyCase_1_Polynomial.txt 
###     DESCRIPTION     INPUT FILE of NIP.pl (perl script)
###     CREATED         So 24 Mai 2015 09:07:18 CEST
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###     AUTHOR          NIP.pl written by Amar Khelil, 
###                     www.khelil.de, 2014-2015, All rights reserved
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###     FURTHER INFOS   See project description http://www.khelil.de/Ref_Projekte_Intern/Projekte_perl_NIP_EN.html
###                     Chapter 'Study Cases' 
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###    GENERAL CONDITIONS
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###     The treatment steps:
###     -1- READ the Input Data stored in SECTION INPUT_DATA 
###         (limited by literals __BEGIN_INPUT_DATA__, __END_INPUT_DATA__)
###     -2- CALCULATE the b-coefficients of the NIP (Newton Interpolation Polynomial) N(X)
###         N(X) = b0 + b1 (X-X0) + b2 (X-X0) (X-X1) + ... + bn (X-X0)...(X-X(n-1))
###     -3- ANALYSE the results
###         The calculated coefficients will be used to calculate y-values as defined in SECTION ANALYSIS
###         (limited by literals __BEGIN_ANALYSIS__, __END_ANALYSIS__)
###
###      General reading principles:
###      -1- Lines starting with '#' are comments. Comments are ignored.
###      -2- Each data category is read within a dedicated SECTION {__BEGIN_SECTION__, __END_SECTION__}
###      -3- Each single data is identified by means of a pattern (data_identifier data_value)
###          Advantage  : The items must not be ordered - except for the (xi,yi) values (data indices are ignored) 
###          Shortcoming: Data cannot be read, if the pattern has been damaged i.e. the data_identifier cannot be recognized! 
###
__BEGIN_INPUT_DATA__
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###    DESCRIPTION OF SECTION 'INPUT DATA'
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###   To define the measurements points out of which the NIP is calculated
###   -1- Number of (measurement) points to read
###   -2- x-value of the first measurement point P0 = (x0, y0)
###   -3- Delta_x = (xi+1 - xi) is the equi-distance between 2 consecutive x-values
###   -4- The y-values starting from y0 to yN (Number of measures - 1): It is assumed Pi=(xi,yi)
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Number of measures : 10      # (X0 to X9) (b0 to b9: NIP is a polynomial of degree 9) 
x-value_00         : 0.0 
Delta-x            : 0.1 
#
# Beware:
# -1- y-values MUST be entered in increasing order y0, y1,... yi, yi+1,...
#     the value-indices _i are not parsed in the current program version.
# -2- The yi-values have been estimated using LibreOffice calc Y= 1 + X + 0.5 X**2 + 0.25 X**3 + 2 X**5
# -3- ALL legal values (21) stored will be READ, BUT NOT MORE than the given number in 'Number of measures' (10) will be CONSIDERED, when solving the problem.
#
y-value_00         :1,000000
y-value_01         :1,105270
y-value_02         :1,222640
y-value_03         :1,356610
y-value_04         :1,516480
y-value_05         :1,718750
y-value_06         :1,989520
y-value_07         :2,366890
y-value_08         :2,903360
y-value_09         :3,668230
y-value_10         :4,750000
y-value_11         :6,258770
y-value_12         :8,328640
y-value_13         :11,120110
y-value_14         :14,822480
y-value_15         :19,656250
y-value_16         :25,875520
y-value_17         :33,770390
y-value_18         :43,669360
y-value_19         :55,941730
y-value_20         :71,000000
__END_INPUT_DATA__
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###     DESCRIPTION OF SECTION ANALYSIS
###     The following data determines which y-values will be recalculated using the (complete or incomplete) NIPs
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__BEGIN_ANALYSIS__
# Characteristics of the Data Analysis
# -1- Name of the function that has been interpolated is known
#
# -2- The x-values to recalculate have been chosen in such a manner that
#       -   All the original values are recalculated
#           At least for the complete NIP there should be no differences between original and interpolated data.
#
#       -   New values INSIDE the original data range are recalculated
#           Check whether the NIP starts waving between the interpolation points  
#           - which is a negative sign
#
#       -   New values OUTSIDE the original data range are recalculated 
#           In most cases 'extrapolation' proves to work very poorly.
#           In this particular case it should work fine, because
#           the NIP must quite equalize the original function.
#
Name of interpolated function : 1 + X + 0.5 X**2 + 0.25 X**3 + 2 X**5 
Number of calculation points  : 41
x-value_00                    : 0.0
Delta-x                       : 0.05
b-max_00                      : 5       # NIP=N5(x)=b0 +..+ b5(X-X0)...(X-X4)
b-max_01                      : 9       # NIP=N9(x)=b0 +..+ b5(X-X0)...(X-X4)+...+ b9 (X-X0)...(X-X8)
# 
__END_ANALYSIS__

__END_OF_FILE__