MATHEMATICAL METHODS
CONTENTS•   Matrices and Linear systems of equations•   Eigen values and eigen vectors•   Real and complex matrices and Qu...
TEXT BOOKS• 1.Mathematical Methods, T.K.V.Iyengar, B.Krishna  Gandhi and others, S.Chand and company• Mathematical Methods...
REFERENCES• 1. A text book of Engineering Mathematics,  B.V.Ramana, Tata Mc Graw Hill.• 2.Advanced Engineering Mathematics...
UNIT HEADER     Name of the Course:B.Tech        Code No:07A1BS02Year/Branch:I Year CSE,IT,ECE,EEE,            Unit No: I ...
UNIT INDEX                      UNIT-IVS.No.          Module        Lecture   PPT Slide                             No.   ...
UNIT-IV               CHAPTER-5Solutions of Algebraic and transcendental equations               CHAPTER-6                ...
LECTURE-1Method 1: Bisection method                  If a function f(x) is continuous b/w x0 and x1 and  f(x0) & f(x1) are...
LECTURE-2Method 2: Iteration method or successive approximation              Consider the equation f(x)=0 which can take i...
LECTURE-3  Method 3: Newton-Raphson method or Newton iteration                               method Let the given equation...
LECTURE-4Finite difference methods        Let (xi,yi),i=0,1,2…………..n be the equally spaced data of the unknown function y=...
Symbolic operatorsForward shift operator(E) :    It is defined as Ef(x)=f(x+h)    (or)     Eyx = yx+h  The second and high...
LECTURE-5Backward shift operator(E-1) :   It is defined as E-1f(x)=f(x-h)       (or) Eyx = yx-h  The second and higher ord...
Forward difference operator (∆) :          The first order forward difference operator of a function   f(x) with increment...
LECTURE-6                              ∇Backward difference operator (nabla ) :   The first order backward difference oper...
Central difference operator ( δ) :        The central difference operator isdefined as        δf(x)= f(x+h/2)-f(x-h/2)    ...
LECTURE-7INTERPOLATION : The process of finding a  missed value in the given table values of X, Y.FINITE DIFFERENCES : We ...
RELATIONS BETWEEN THE OPERATORS              IDENTITIES:    ∇1. ∆=E-1 or E=1+ ∆          ∇2. = 1- E -13. δ = E 1/2 – E -1/...
LECTURE-8,9Newtons Forward interpolation formula :y=f(x)=f(x0+ph)= y0 + p∆y0 + p(p-1)/2! ∆2y0 + p(p-1)(p-  2)/3! ∆3y0+…….....
LECTURE-10,11          GAUSS INTERPOLATIONThe Guass forward interpolation is given by yp = yo + p  ∆y0+p(p-1)/2! ∆2y-1 +(p...
LECTURE-12INTERPOLATIN WITH UNEQUAL INTERVALS:The various interpolation formulae Newton’s forwardformula, Newton’s backwar...
The Lagrange’s interpolation formula is given by Y = (X-X1)(X-X2)……..(X-Xn)           Y0 +     (X0-X1)(X0-X2)……..(X0-Xn)  ...
Upcoming SlideShare
Loading in...5
×

Unit iv

126

Published on

Published in: Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
126
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
5
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Unit iv

  1. 1. MATHEMATICAL METHODS
  2. 2. 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
  3. 3. 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.
  4. 4. 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, 3rd Edition, 2006.
  5. 5. UNIT HEADER Name of the Course:B.Tech Code No:07A1BS02Year/Branch:I Year CSE,IT,ECE,EEE, Unit No: I V No.of slides:23
  6. 6. UNIT INDEX UNIT-IVS.No. Module Lecture PPT Slide No. No. 1 Bisection Method, L1-4 8-11 Iteration method,Newton- Raphson,Regula Falsi method 2 Finite Differences L5-6 12-17 3 Interpolation. L7-12 18-24
  7. 7. UNIT-IV CHAPTER-5Solutions of Algebraic and transcendental equations CHAPTER-6 Interpolation
  8. 8. LECTURE-1Method 1: Bisection method If a function f(x) is continuous b/w x0 and x1 and f(x0) & f(x1) are of opposite signs, then there exsist at least one root b/w x0 and x1 Let f(x0) be –ve and f(x1) be +ve ,then the root lies b/w xo and x1 and its approximate value is given by x2=(x0+x1)/2 If f(x2)=0,we conclude that x2 is a root of the equ f(x)=0 Otherwise the root lies either b/w x2 and x1 (or) b/w x2 and x0 depending on wheather f(x2) is +ve or –ve Then as before, we bisect the interval and repeat the process untill the root is known to the desired accuracy
  9. 9. LECTURE-2Method 2: Iteration method or successive approximation Consider the equation f(x)=0 which can take in the form x = ø(x) -------------(1) where |ø1(x)|<1 for all values of x.Taking initial approximation is x0we put x1=ø(x0) and take x1 is the first approximation x2=ø(x1) , x2 is the second approximation x3=ø(x2) ,x3 is the third approximation . . xn=ø(xn-1) ,xn is the nth approximationSuch a process is called an iteration process
  10. 10. LECTURE-3 Method 3: Newton-Raphson method or Newton iteration method Let the given equation be f(x)=0 Find f1(x) and initial approximation x0The first approximation is x1= x0-f(x0)/ f1(x0)The second approximation is x2= x1-f(x1)/ f1(x1) . . The nth approximation is xn= xn-1-f(xn)/ f1(xn)
  11. 11. LECTURE-4Finite 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 analyzingthe differences of f(x). Let x1 = x0+h x2 = x0+2h . . . xn = x0+nh be equally spaced points where the function value of f(x)be y0, y1, y2…………….. yn
  12. 12. Symbolic operatorsForward shift operator(E) : It is defined as Ef(x)=f(x+h) (or) Eyx = yx+h The second and higher order forward shift operators are defined in similar manner as follows E2f(x)= E(Ef(x))= E(f(x+h)= f(x+2h)= yx+2h E3f(x)= f(x+3h) . . Ekf(x)= f(x+kh)
  13. 13. LECTURE-5Backward shift operator(E-1) : It is defined as E-1f(x)=f(x-h) (or) Eyx = yx-h The second and higher order backward shift operators are defined in similar manner as follows E-2f(x)= E-1(E-1f(x))= E-1(f(x-h)= f(x-2h)= yx-2h E-3f(x)= f(x-3h) . . E-kf(x)= f(x-kh)
  14. 14. 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) ∆fk =fk+1-fk ; k=0,1,2……… ∆2f(x)= ∆[∆f(x)]= ∆[f(x+h)-f(x)]= ∆fk+1- ∆fk ; 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+ ∆
  15. 15. 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) ∇ k =fk+1-fk ; k=0,1,2……… f ∇f(x)= ∇[ f(x)]= ∇ [f(x+h)-f(x)]=∇ fk+1- ∇k ; k=0,1,2………… f ……………………………. …………………………….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
  16. 16. Central difference operator ( δ) : The central difference operator isdefined as δf(x)= f(x+h/2)-f(x-h/2) δf(x)= E1/2f(x)-E-1/2f(x) = [E1/2-E-1/2]f(x) δ= E1/2-E-1/2
  17. 17. LECTURE-7INTERPOLATION : 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
  18. 18. RELATIONS BETWEEN THE OPERATORS IDENTITIES: ∇1. ∆=E-1 or E=1+ ∆ ∇2. = 1- E -13. δ = E 1/2 – E -1/24. µ= ½∇ 1/2-E-1/2) (E ∇5. ∆=E = ∇E= δE1/26. (1+ ∆)(1- )=1
  19. 19. LECTURE-8,9Newtons Forward interpolation formula :y=f(x)=f(x0+ph)= y0 + p∆y0 + p(p-1)/2! ∆2y0 + p(p-1)(p- 2)/3! ∆3y0+…….. ………………+p(p-1)(p2) …… [p- (n-1)]/n! ∆ny0 .Newtons Backward interpolatin formula :y=f(x)=f(xn+ph)= yn + p∆yn + p(p+1)/2! ∆2yn + p(p+1) (p+2)/3! ∆3yn+…….. ………………+p(p+1)(p+2) …… [p+(n-1)]/n! ∆nyn .
  20. 20. LECTURE-10,11 GAUSS INTERPOLATIONThe Guass forward interpolation is given by yp = yo + p ∆y0+p(p-1)/2! ∆2y-1 +(p+1)p(p-1)/3! ∆3y-1+(p+1)p(p- 1)(p-2)/4! ∆4y-2+……The Guass backward interpolation is given by yp = yo + p ∆y-1+p(p+1)/2! ∆2y-1 +(p+1)p(p-1)/3! ∆3y-2+ (p+2)(p+1)p(p-1)/4! ∆4y-2+……
  21. 21. LECTURE-12INTERPOLATIN WITH UNEQUAL INTERVALS:The various interpolation formulae Newton’s forwardformula, Newton’s backward formula possess can beapplied only to equal spaced values of argument. It istherefore, desirable to develop interpolation formula forunequally spaced values of x. We use Lagrange’sinterpolation formula.
  22. 22. The Lagrange’s interpolation formula is given by Y = (X-X1)(X-X2)……..(X-Xn) Y0 + (X0-X1)(X0-X2)……..(X0-Xn) (X-X0)(X-X2)……..(X-Xn) Y1 +………………….. (X1-X0)(X1-X2)……..(X1-Xn) (X-X0)(X-X1)……..(X-Xn-1) Yn + (Xn-X0)(Xn-X1)……..(Xn-Xn-1)
  1. A particular slide catching your eye?

    Clipping is a handy way to collect important slides you want to go back to later.

×