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MATHEMATICAL METHODS
INTERPOLATION
I YEAR B
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Y
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Professor of Mathematics
Guru Nanak Engineering College
Ibrahimpatnam, Hyderabad
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Eigen values, Eigen vectors – properties – Condition number of Matrix, Cayley –
Hamilton Theorem (without proof) – Inverse and powers of a matrix by Cayley –
Hamilton theorem – Diagonalization of matrix – Calculation of powers of matrix –
Model and spectral matrices
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Complex Matrices, Hermition and skew Hermition
matrices, Unitary Matrices - Eigen values and Eigen vectors of complex matrices and
their properties
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Solution of Algebraic and Transcendental Equations- Introduction: The Bisection
Method – The Method of False Position – The Iteration Method - Newton –Raphson
Unit-IV
Solution of Nonlinear Systems
Method Interpolation:Introduction-Errors in Polynomial Interpolation - Finite
differences- Forward difference, Backward differences, Central differences, Symbolic
relations and separation of symbols-Difference equations – Differences of a
polynomial - Newton’s Formulae for interpolation - Central difference interpolation
formulae - Gauss Central Difference Formulae - Lagrange’s Interpolation formulae- B
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Unit-V
Curve fitting &
Curve Fitting: Fitting a straight line - Second degree curve - Exponential curve Power curve by method of least squares
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Unit-VI
Solution by Taylor’s series - Picard’s Method of successive approximation- Euler’s
Numerical
Method -Runge kutta Methods, Predictor Corrector Methods, Adams- Bashforth
solution of ODE
Unit-VII
Fourier Series
Unit-VIII
Partial
Differential
Equations
Method
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Introduction and formation of PDE by elimination of arbitrary constants and
arbitrary functions - Solutions of first order linear equation - Non linear equations Method of separation of variables for second order equations - Two dimensional
wave equation
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Interpolating polynomial passing through the given set of points is unique
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is not in the range of
and
, then the method to find
Equally Spaced
Arguments
is called as Extrapolation
...
If
be given set of observations and let
their corresponding values for the curve
be
, then
is called
as finite difference
...
e
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Forward differences
Backward differences
Central differences
Forward Difference
Let us consider
be given set of observations and let
corresponding values of the curve
by
, then the Forward difference operator is denoted
and is defined as
In this case
are
...
The difference of first forward differences will give us Second forward differences and it is
denoted by
and is defined as
Similarly, the difference of second forward differences will give us third forward difference and
it is denoted by
...
...
...
...
...
is common difference in the values of
...
In this case
are called as First Backward differences of
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Backward difference table
First Backward
differences
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...
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Note: If
...
...
is common difference in the values of
...
H
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H
...
H
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Proof:
Let us assume a polynomial equation by using the arrow marks shown in the above table
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Proof:
Let us assume a polynomial equation by using the arrow marks shown in the above table
...
in (1), we get
,
Stirling’s Formulae
Statement: If
and let
are given set of observations with common difference
are their corresponding values, where
be the given
function then
where
Proof: Stirling’s Formula will be obtained by taking the average of Gauss forward difference
formula and Gauss Backward difference formula
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