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MATHEMATICAL METHODS

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CONTENTS Matrices and Linear systems of equations Eigen values and eigen vectors Real and complex matrices and Quadratic forms Algebraic equations transcendental equations and Interpolation Curve Fitting, numerical differentiation & integration Numerical differentiation of O.D.E Fourier series and Fourier transforms Partial differential equation and Z-transforms

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TEXT BOOKS 1.Mathematical Methods, T.K.V.Iyengar, B.Krishna Gandhi and others, S.Chand and company Mathematical Methods, C.Sankaraiah, V.G.S.Book links. A text book of Mathametical Methods, V.Ravindranath, A.Vijayalakshmi, Himalaya Publishers. A text book of Mathametical Methods, Shahnaz Bathul, Right Publishers.

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REFERENCES 1. A text book of Engineering Mathematics, B.V.Ramana, Tata Mc Graw Hill. 2.Advanced Engineering Mathematics, Irvin Kreyszig Wiley India Pvt Ltd. 3. Numerical Methods for scientific and Engineering computation, M.K.Jain, S.R.K.Iyengar and R.K.Jain, New Age International Publishers Elementary Numerical Analysis, Aitkison and Han, Wiley India, 3 rd Edition, 2006.

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UNIT HEADER Name of the Course:B.Tech Code No:07A1BS02 Year/Branch:I Year CSE,IT,ECE,EEE, Unit No: I V No.of slides:23

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S.No.ModuleLecture No. PPT Slide No. 1Bisection Method, Iteration method,Newton- Raphson,Regula Falsi method L1-48-11 2Finite DifferencesL5-612-17 3Interpolation.L7-1218-24 UNIT INDEX UNIT-IV

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UNIT-IV CHAPTER-5 Solutions of Algebraic and transcendental equations CHAPTER-6Interpolation

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LECTURE-1 Method 1: Bisection method LECTURE-1 Method 1: Bisection method If a function f(x) is continuous b/w x 0 and x 1 and f(x 0 ) & f(x 1 ) are of opposite signs, then there exsist at least one root b/w x 0 and x 1 Let f(x 0 ) be –ve and f(x 1 ) be +ve,then the root lies b/w xo and x 1 and its approximate value is given by x 2 =(x 0 +x 1 )/2 If f(x 2 )=0,we conclude that x 2 is a root of the equ f(x)=0 Otherwise the root lies either b/w x 2 and x 1 (or) b/w x 2 and x 0 depending on wheather f(x 2 ) is +ve or –ve Then as before, we bisect the interval and repeat the process untill the root is known to the desired accuracy

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LECTURE-2 Method 2: Iteration method or successive approximation LECTURE-2 Method 2: Iteration method or successive approximation Consider the equation f(x)=0 which can take in the form x = ø(x) -------------(1) where |ø 1 (x)|<1 for all values of x. Taking initial approximation is x 0 we put x 1 =ø(x 0 ) and take x 1 is the first approximation x 2 =ø(x 1 ), x 2 is the second approximation x 3 =ø(x 2 ),x 3 is the third approximation. x n =ø(x n-1 ),x n is the n th approximation Such a process is called an iteration process

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LECTURE-3 Method 3: Newton-Raphson method or Newton iteration method Let the given equation be f(x)=0 Find f 1 (x) and initial approximation x 0 The first approximation is x 1 = x 0 -f(x 0 )/ f 1 (x 0 ) The second approximation is x 2 = x 1 -f(x 1 )/ f 1 (x 1 ). The n th approximation is x n = x n-1 -f(x n )/ f 1 (x n )

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LECTURE-4 Finite difference methods LECTURE-4 Finite difference methods Let (xi,yi),i=0,1,2…………..n be the equally spaced data of the unknown function y=f(x) then much of the f(x) can be extracted by analyzing the differences of f(x). Let x 1 = x 0 +h x 2 = x 0 +2h. x n = x 0 +nh be equally spaced points where the function value of f(x) be y 0, y 1, y 2 …………….. y n

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Symbolic operators Symbolic operators Forward shift operator(E) : It is defined as Ef(x)=f(x+h) (or) Ey x = y x+h The second and higher order forward shift operators are defined in similar manner as follows E 2 f(x)= E(Ef(x))= E(f(x+h)= f(x+2h)= y x+2h E 3 f(x)= f(x+3h). E k f(x)= f(x+kh)

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LECTURE-5 Backward shift operator(E -1 ) : It is defined as E -1 f(x)=f(x-h) (or) Ey x = y x-h The second and higher order backward shift operators are defined in similar manner as follows E -2 f(x)= E -1 (E -1 f(x))= E -1 (f(x-h)= f(x-2h)= y x-2h E -3 f(x)= f(x-3h). E -k f(x)= f(x-kh)

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Forward difference operator (∆) : Forward difference operator (∆) : The first order forward difference operator of a function f(x) with increment h in x is given by ∆f(x)=f(x+h)-f(x) (or) ∆f k =f k+1 -f k ; k=0,1,2……… ∆ 2 f(x)= ∆[∆f(x)]= ∆[f(x+h)-f(x)]= ∆f k+1 - ∆f k ; k=0,1,2………… ……………………………. Relation between E and ∆ : ∆f(x)=f(x+h)-f(x) =Ef(x)-f(x) [Ef(x)=f(x+h)] =(E-1)f(x) ∆=E-1 E=1+ ∆

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LECTURE-6 Backward difference operator (nabla ) : LECTURE-6 Backward difference operator (nabla ) : The first order backward difference operator of a function f(x) with increment h in x is given by f(x)=f(x)-f(x-h) (or) f k =f k+1 -f k ; k=0,1,2……… f(x)= [ f(x)]= [f(x+h)-f(x)]= f k+1 - f k ; k=0,1,2………… ……………………………. Relation between E and nabla : nabla f(x)=f(x+h)-f(x) =Ef(x)-f(x) [Ef(x)=f(x+h)] =(E-1)f(x) nabla=E-1 E=1+ nabla

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Central difference operator ( δ) : Central difference operator ( δ) : The central difference operator is defined as δf(x)= f(x+h/2)-f(x-h/2) δf(x)= E 1/2 f(x)-E -1/2 f(x) = [E 1/2 -E -1/2 ]f(x) δ= E 1/2 -E -1/2

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LECTURE-7 INTERPOLATION : The process of finding a missed value in the given table values of X, Y. FINITE DIFFERENCES : We have three finite differences 1. Forward Difference 2. Backward Difference 3. Central Difference

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RELATIONS BETWEEN THE OPERATORS IDENTITIES: 1. ∆=E-1 or E=1+ ∆ 2. = 1- E -1 3. δ = E 1/2 – E -1/2 4. µ= ½ (E 1/2 -E -1/2 ) 5. ∆=E = E= δE 1/2 6. (1+ ∆)(1- )=1

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LECTURE-8,9 Newtons Forward interpolation formula : LECTURE-8,9 Newtons Forward interpolation formula : y=f(x)=f(x 0 +ph)= y 0 + p∆y 0 + p(p-1)/2! ∆ 2 y 0 + p(p-1)(p- 2)/3! ∆ 3 y 0 +…….. ………………+p(p-1)(p2) …… [p-(n-1)]/n! ∆ n y 0. Newtons Backward interpolatin formula : y=f(x)=f(x n +ph)= y n + p∆y n + p(p+1)/2! ∆ 2 y n + p(p+1)(p+2)/3! ∆ 3 y n +…….. ………………+p(p+1)(p+2) …… [p+(n-1)]/n! ∆ n y n.

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LECTURE-10,11 GAUSS INTERPOLATION The Guass forward interpolation is given by y p = y o + p ∆y 0 +p(p-1)/2! ∆ 2 y -1 +(p+1)p(p-1)/3! ∆ 3 y - 1 +(p+1)p(p-1)(p-2)/4! ∆ 4 y -2 +…… The Guass backward interpolation is given by y p = y o + p ∆y -1 +p(p+1)/2! ∆ 2 y -1 +(p+1)p(p-1)/3! ∆ 3 y -2 + (p+2)(p+1)p(p-1)/4! ∆ 4 y -2 +……

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LECTURE-12 INTERPOLATIN WITH UNEQUAL INTERVALS: The various interpolation formulae Newton’s forward formula, Newton’s backward formula possess can be applied only to equal spaced values of argument. It is therefore, desirable to develop interpolation formula for unequally spaced values of x. We use Lagrange’s interpolation formula.

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The Lagrange’s interpolation formula is given by Y = (X-X 1 )(X-X 2 )……..(X-X n ) Y 0 + (X 0 -X 1 )(X 0 -X 2 )……..(X 0 -X n ) (X-X 0 )(X-X 2 )……..(X-X n ) Y 1 +………………….. (X 1 -X 0 )(X 1 -X 2 )……..(X 1 -X n ) (X-X 0 )(X-X 1 )……..(X-X n-1 ) Y n + (X n -X 0 )(X n -X 1 )……..(X n -X n-1 )

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