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MATHEMATICAL METHODS




             NUMERICAL SOLUTIONS OF
      ALGEBRAIC AND TRANSDENTIAL EQUATIONS
                            I YEAR B.Tech




By
Mr. Y. Prabhaker Reddy
Asst. Professor of Mathematics
Guru Nanak Engineering College
Ibrahimpatnam, Hyderabad.
SYLLABUS OF MATHEMATICAL METHODS (as per JNTU Hyderabad)
Name of the Unit                                      Name of the Topic
                     Matrices and Linear system of equations: Elementary row transformations – Rank
      Unit-I
                     – Echelon form, Normal form       – Solution of Linear Systems    – Direct Methods        – LU
Solution of Linear
                     Decomposition from Gauss Elimination – Solution of Tridiagonal systems – Solution
    systems
                     of Linear Systems.
                     Eigen values, Eigen vectors     – properties      – Condition number of Matrix, Cayley            –
     Unit-II
                     Hamilton Theorem (without proof)        – Inverse and powers of a matrix by Cayley            –
Eigen values and
                     Hamilton theorem       – Diagonalization of matrix Calculation of powers of matrix
                                                                         –                                     –
  Eigen vectors
                     Model and spectral matrices.
                     Real Matrices, Symmetric, skew symmetric, Orthogonal, Linear Transformation -
                     Orthogonal Transformation. Complex Matrices, Hermition and skew Hermition
     Unit-III
                     matrices, Unitary Matrices - Eigen values and Eigen vectors of complex matrices and
     Linear
                     their properties. Quadratic forms - Reduction of quadratic form to canonical form,
Transformations
                     Rank, Positive, negative and semi definite, Index, signature, Sylvester law, Singular
                     value decomposition.
                     Solution of Algebraic and Transcendental Equations- Introduction: The Bisection
                     Method – The Method of False Position       – The Iteration Method - Newton
                                                                                               –Raphson
                     Method Interpolation: Introduction-Errors in Polynomial Interpolation - Finite
     Unit-IV
                     differences- Forward difference, Backward differences, Central differences, Symbolic
Solution of Non-
                     relations and separation of symbols-Difference equations           – Differences of a
 linear Systems
                     polynomial - Newton’s Formulae for interpolation - Central difference interpolation
                     formulae - Gauss Central Difference Formulae - Lagrange’s Interpolation formulae- B.
                     Spline interpolation, Cubic spline.
     Unit-V          Curve Fitting: Fitting a straight line - Second degree curve - Exponential curve -
 Curve fitting &     Power curve by method of least squares.
   Numerical         Numerical Integration: Numerical Differentiation-Simpson’s         3/8    Rule,      Gaussian
   Integration       Integration, Evaluation of Principal value integrals, Generalized Quadrature.
     Unit-VI         Solution   by   Taylor’s    series -   Picard’s   Method of approximation-
                                                                         successive                    Euler’s
   Numerical         Method -Runge kutta Methods, Predictor Corrector Methods, Adams- Bashforth
 solution of ODE     Method.
                     Determination of Fourier coefficients - Fourier series-even and odd functions -
    Unit-VII
                     Fourier series in an arbitrary interval - Even and odd periodic continuation - Half-
 Fourier Series
                     range Fourier sine and cosine expansions.
    Unit-VIII        Introduction and formation of PDE by elimination of arbitrary constants and
     Partial         arbitrary functions - Solutions of first order linear equation - Non linear equations -
   Differential      Method of separation of variables for second order equations - Two dimensional
   Equations         wave equation.
CONTENTS
UNIT-IV (a)
SOLUTIONS OF NON-LINEAR SYSTEMS
a) Numerical Solutions of Algebraic and Transcendental Equation

          Introduction

          Bisection Method

          Regular Folsi Method

          Newton Raphson Method

          Iteration Method
NUMERICAL SOLUTIONS OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS

Aim: To find a root of



                                          Algebraic                          Highest power
                                          Equation                            of x is finite
      f(x)=0
                                       Transcendental                        Highest power
                                          Equation                           of x is Infinite



Algebraic Equation: An Equation which contains algebraic terms is called as an algebraic
Equation.

Example:                  , Here Highest power of x is finite. So it is an algebraic Equation.

Transcendental Equation: An equation which contains trigonometric ratios, exponential
function and logarithmic functions is called as a Transcendental Equation.

Example:                                              etc.

In order to solve above type of equations following methods exist

     Directive Methods: The methods which are used to find solutions of given equations in the
     direct process is called as directive methods.
     Example: Synthetic division, remainder theorem, Factorization method etc
    Note: By using Directive Methods, it is possible to find exact solutions of the given equation.
     Iterative Methods (Indirect Methods): The methods which are used to find solutions of
     the given equation in some indirect process is called as Iterative Methods
     Note: By using Iterative methods, it is possible to find approximate solution of the given
     equation and also it is possible to find single solution of the given equation at the same time.

To find a root of the given equation, we have following methods

       Bisection Method
       The Method of false position (Or) Regular folsi Method
       Iteration Method (Successive approximation Method)
       Newton Raphson Method.
Bisection Method

 Consider                 be the given equation.

 Let us choose the constants             and       in such a way that

 (I.e.        and         are of opposite signs) i.e.



 Then the curve                   crosses -            in between     and , so that there exists a root of the
 given equation in between              and .

 Let us define initial approximation

 Now, If                  then     is root of the given equation.

 If              then either

 Case (i): Let us consider that                       and       Case (ii): Let us consider that            and



 Since                                                          Since

         Root lies between        and                               Root lies between      and




                                           Here

Let us consider that             Let us consider that           Let us consider that        Let us consider that
             &                                 &                            &                          &

Since                            Since                          Since                       Since

      Root lies between             Root lies between               Root lies between            Root lies between
and                              and                            and                         and



 Here, the logic is that, we have to select first negative or first positive from bottom
 approximations only, but not from top. i.e. the approximation which we have recently found
 should be selected.

 If              , then     is root of the given equation. Otherwise repeat above process until we obtain
 solution of the given equation.
Regular Folsi Method (Or) Method of False position

Let us consider that                be the given equation.

Let us choose two points      and in such a way that                     and     are of opposite signs.

i.e.

Consider                           , so that the graph                   crosses -axis in between            . then,
there exists a root of the given equation in between            and .

Let us define a straight line joining the points                                    , then the equation of

straight line is given by                            ----- > ( I )


This method consists of replacing the part of the curve by means of a straight line and then takes
the point of intersection of the straight line with -axis, which give an approximation to the
required root.

In the present case, it can be obtained by substituting                    in equation I, and it is denoted by     .

From ( I )

Now, any point on -axis                    and let initial approximation be          i.e.

     (I)




                                  ---- > ( II )


If           , then    is root of the given equation.

If           , then either                 (or)

Case (i): Let us consider that

We know that                and

Hence root of the given equation lies between            and         .
In order to obtain next approximation        , replace      with    in equation ( II )

Hence


Case (ii): Let us consider that

We know that                and
Hence root of the given equation lies between         and     .
In order to obtain next approximation        , replace      with    in equation ( II )

Hence

If          , then is root of the given equation. Otherwise repeat above process until we
obtain a root of the given equation to the desired accuracy.

Newton Raphson Method

Let us consider that              be the given equation.

Let us choose initial approximation to be       .

Let us assume that                 be exact root of                where        , so that

Expanding above relation by means of Taylor’s expansion method, we get




Since   is very small,     and higher powers of      are neglected. Then the above relation becomes




Hence




If          , then       is root of the given equation. Otherwise repeat above process until we
obtain a root of the given equation to the desired accuracy.

Successive approximations are given by

                                        and so on.
Iteration Method

Let us consider that         be the given equation.
Let us choose initial approximation to be   .
Rewrite           as          such that               .
Then successive approximations are given by




Repeat the above process until we get successive approximation equal, which will gives the
required root of the given equation.

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algebraic&transdential equations

  • 1. MATHEMATICAL METHODS NUMERICAL SOLUTIONS OF ALGEBRAIC AND TRANSDENTIAL EQUATIONS I YEAR B.Tech By Mr. Y. Prabhaker Reddy Asst. Professor of Mathematics Guru Nanak Engineering College Ibrahimpatnam, Hyderabad.
  • 2. SYLLABUS OF MATHEMATICAL METHODS (as per JNTU Hyderabad) Name of the Unit Name of the Topic Matrices and Linear system of equations: Elementary row transformations – Rank Unit-I – Echelon form, Normal form – Solution of Linear Systems – Direct Methods – LU Solution of Linear Decomposition from Gauss Elimination – Solution of Tridiagonal systems – Solution systems of Linear Systems. Eigen values, Eigen vectors – properties – Condition number of Matrix, Cayley – Unit-II Hamilton Theorem (without proof) – Inverse and powers of a matrix by Cayley – Eigen values and Hamilton theorem – Diagonalization of matrix Calculation of powers of matrix – – Eigen vectors Model and spectral matrices. Real Matrices, Symmetric, skew symmetric, Orthogonal, Linear Transformation - Orthogonal Transformation. Complex Matrices, Hermition and skew Hermition Unit-III matrices, Unitary Matrices - Eigen values and Eigen vectors of complex matrices and Linear their properties. Quadratic forms - Reduction of quadratic form to canonical form, Transformations Rank, Positive, negative and semi definite, Index, signature, Sylvester law, Singular value decomposition. Solution of Algebraic and Transcendental Equations- Introduction: The Bisection Method – The Method of False Position – The Iteration Method - Newton –Raphson Method Interpolation: Introduction-Errors in Polynomial Interpolation - Finite Unit-IV differences- Forward difference, Backward differences, Central differences, Symbolic Solution of Non- relations and separation of symbols-Difference equations – Differences of a linear Systems polynomial - Newton’s Formulae for interpolation - Central difference interpolation formulae - Gauss Central Difference Formulae - Lagrange’s Interpolation formulae- B. Spline interpolation, Cubic spline. Unit-V Curve Fitting: Fitting a straight line - Second degree curve - Exponential curve - Curve fitting & Power curve by method of least squares. Numerical Numerical Integration: Numerical Differentiation-Simpson’s 3/8 Rule, Gaussian Integration Integration, Evaluation of Principal value integrals, Generalized Quadrature. Unit-VI Solution by Taylor’s series - Picard’s Method of approximation- successive Euler’s Numerical Method -Runge kutta Methods, Predictor Corrector Methods, Adams- Bashforth solution of ODE Method. Determination of Fourier coefficients - Fourier series-even and odd functions - Unit-VII Fourier series in an arbitrary interval - Even and odd periodic continuation - Half- Fourier Series range Fourier sine and cosine expansions. Unit-VIII Introduction and formation of PDE by elimination of arbitrary constants and Partial arbitrary functions - Solutions of first order linear equation - Non linear equations - Differential Method of separation of variables for second order equations - Two dimensional Equations wave equation.
  • 3. CONTENTS UNIT-IV (a) SOLUTIONS OF NON-LINEAR SYSTEMS a) Numerical Solutions of Algebraic and Transcendental Equation  Introduction  Bisection Method  Regular Folsi Method  Newton Raphson Method  Iteration Method
  • 4. NUMERICAL SOLUTIONS OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS Aim: To find a root of Algebraic Highest power Equation of x is finite f(x)=0 Transcendental Highest power Equation of x is Infinite Algebraic Equation: An Equation which contains algebraic terms is called as an algebraic Equation. Example: , Here Highest power of x is finite. So it is an algebraic Equation. Transcendental Equation: An equation which contains trigonometric ratios, exponential function and logarithmic functions is called as a Transcendental Equation. Example: etc. In order to solve above type of equations following methods exist Directive Methods: The methods which are used to find solutions of given equations in the direct process is called as directive methods. Example: Synthetic division, remainder theorem, Factorization method etc Note: By using Directive Methods, it is possible to find exact solutions of the given equation. Iterative Methods (Indirect Methods): The methods which are used to find solutions of the given equation in some indirect process is called as Iterative Methods Note: By using Iterative methods, it is possible to find approximate solution of the given equation and also it is possible to find single solution of the given equation at the same time. To find a root of the given equation, we have following methods Bisection Method The Method of false position (Or) Regular folsi Method Iteration Method (Successive approximation Method) Newton Raphson Method.
  • 5. Bisection Method Consider be the given equation. Let us choose the constants and in such a way that (I.e. and are of opposite signs) i.e. Then the curve crosses - in between and , so that there exists a root of the given equation in between and . Let us define initial approximation Now, If then is root of the given equation. If then either Case (i): Let us consider that and Case (ii): Let us consider that and Since Since Root lies between and Root lies between and Here Let us consider that Let us consider that Let us consider that Let us consider that & & & & Since Since Since Since Root lies between Root lies between Root lies between Root lies between and and and and Here, the logic is that, we have to select first negative or first positive from bottom approximations only, but not from top. i.e. the approximation which we have recently found should be selected. If , then is root of the given equation. Otherwise repeat above process until we obtain solution of the given equation.
  • 6. Regular Folsi Method (Or) Method of False position Let us consider that be the given equation. Let us choose two points and in such a way that and are of opposite signs. i.e. Consider , so that the graph crosses -axis in between . then, there exists a root of the given equation in between and . Let us define a straight line joining the points , then the equation of straight line is given by ----- > ( I ) This method consists of replacing the part of the curve by means of a straight line and then takes the point of intersection of the straight line with -axis, which give an approximation to the required root. In the present case, it can be obtained by substituting in equation I, and it is denoted by . From ( I ) Now, any point on -axis and let initial approximation be i.e. (I) ---- > ( II ) If , then is root of the given equation. If , then either (or) Case (i): Let us consider that We know that and Hence root of the given equation lies between and .
  • 7. In order to obtain next approximation , replace with in equation ( II ) Hence Case (ii): Let us consider that We know that and Hence root of the given equation lies between and . In order to obtain next approximation , replace with in equation ( II ) Hence If , then is root of the given equation. Otherwise repeat above process until we obtain a root of the given equation to the desired accuracy. Newton Raphson Method Let us consider that be the given equation. Let us choose initial approximation to be . Let us assume that be exact root of where , so that Expanding above relation by means of Taylor’s expansion method, we get Since is very small, and higher powers of are neglected. Then the above relation becomes Hence If , then is root of the given equation. Otherwise repeat above process until we obtain a root of the given equation to the desired accuracy. Successive approximations are given by and so on.
  • 8. Iteration Method Let us consider that be the given equation. Let us choose initial approximation to be . Rewrite as such that . Then successive approximations are given by Repeat the above process until we get successive approximation equal, which will gives the required root of the given equation.