Interpolation and-its-application

Interpolation and It’s application

What is interpolation ?
Interpolation is the process of using points with known values or sample points to estimate
values at other unknown points. It can be used to predict unknown values for any geographic
point data, such as elevation, rainfall, chemical concentrations, noise levels, and so on.
Example:-Find value of x for y(x)=0.390 , from the following table

Application of interpolation in Computer Science
Iterative reconstruction in digital imaging
Approximation Using CAD Facilities
Interpolation in Grapics

1 • Approximation Using CAD Facilities

 Let us consider a problem where the formulation of the problem depends on some parameters and an iterative solution
is possible. It is well known that the final solution depends on the initial approximation in terms of accuracy and of
computer runtime.
If some solutions can be found using an initial solution which is not accurate, the next initial solutions can be evaluated by
the use of an interpolation method. In this way the computer time decreases and the accuracy is increased
Approximation of the Initial Solution for Iterative Methods

Moving Average in Experimental Data Processing
Important sources of the points considered in interpolation problems are the experiments. For instance, a thorough experimental and
numerical study was carried out regarding
The experimental data were used to calibrate the numerical FEM model, so the experimental data was checked, double checked and over
checked in order to offer a high degree of confidence. Several states of the running conditions were studied and several millions of
experimental values were acquired. In this way there was verified the repeatability of the experimental data for all the measurement points.
Figure 4 - Repeatability of the experimental values for the same measurement point

 The experimental data were used to calibrate the numerical FEM model, so the experimental data was checked, double
checked and over checked in order to offer a high degree of confidence. Several states of the running conditions were studied
and several millions of experimental values were acquired. In this way there was verified the repeatability of the experimental
data for all the measurement points.
Figure 5 : The “trend line” facility offered in Excel, in order to smooth the curves

The most facile method to smooth the curves was to use Excel in order to add a trend line which
might present the relevant variation of the phenomenon. Excel also offers a whole set of
facilities, such as:
 import of the data from several formats: text, Visual FoxPro tables, comma separated values,
etc;
graphics of the data presented in the sheet;
 record, store and run a macro (a set of consecutive commands),
in this way being possible to draw a graph by using a single click

Secure Multiparty Computation and Secret Sharing
schemes
“Secret sharing” or secret splitting describes the methodology of distributing secret information among a group of
participants in a way that a subset of this group is needed to reconstruct the secret. Whereby a single piece of
information cannot be used to reconstruct the secret.
Secret sharing has a wide operation range. It can be used in high security areas for validation reasons e.g. in the
military or in the banking area where it might be necessary to have a 4, 6 or 8 eye validation principle. It also can
be used to ensure the secure transport of information e.g. transport of a cryptographic key over several channels.
Last but not least it can be used in secure access regulation where only a combination of trustworthy people
might get access to a secure area

1) k is the least quantity of participants you would like
to be able to reconstruct the secret
2) Choose a polynomial of grade k-1
3) Define k random indices (a0…ak-1)
4) Define k points which can be calculated by the
polynomial
5) Distribute k points to the participants.
6) To calculate the secret out of the keys use e.g.
Lagrange interpolation [5] with x=0 .
Secret Sharing schemes step by step:

Secret sharing graphical explanation:
 Let’s start with a simple polynomial y = a 0+ a1x1.
 a0 is our secret and since the grade of the polynomial is 1 we need at least two points to
determine it (see figure 1).
figure 1: graph of a first grade polinomial. a0 is the secret e.g.
password. The points are the given keys
• If you have only one point given an infinite account of possible solutions can
be calculated (see figure 2).
figure 2: graph of a first grade polinomial. a0 is the secret e.g. password. Since
there is only one fix point the secret is not reconstructible, because an infinite
amount of lines are crossing the point

 If K is supposed to be bigger than 2 just take a polynomial with a higher grade e.g. y = a0-
a2x2+a0x4 in this case K is equal 5 (see figure 3).
figure 3: graph of a fourth grade polynomial. a0 is the
secret e.g. password. The points are the given keys.
 Now if you would like to add more participants to have a key you could just create a
new key by taking another point of the polynomial (see figure 4).
figure 4: graph of a fourth grade polynomial. a0 is the secret e.g.
password. The points are the given keys. The x are additional keys
that could be created without interfering with already existing keys.

Nonlinear programming
The Lagrange multiplier method has several generalizations. In nonlinear programming there are
several multiplier rules, e.g., the Caratheodory-John Multiplier Rule and the Convex Multiplier Rule,
for inequality constraints

ADVANTAGES AND DISADVANTAGES of LAGRANGE’S
INTERPOLATION
• ADVANTAGES
• The formula is simple and easy to remember.
• There is no need to construct the divided difference table.
• The application of the formula is not speedy.
• DISADVANTAGE
• There is always a chance to committing some error.
• The calculation provide no check whether the functional values
used the taken correctly or not.

Interpolation and-its-application

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