1. Numerical Analysis
Lecture 45

2. Summing up

3. Non-Linear
Equations

4. Bisection Method (Bolzano) Regula-Falsi Method Method of iteration Newton - Raphson Method Muller’s Method Graeffe’s Root Squaring Method

5. In the method of False Position, the first approximation to the root of f (x) = 0 is given by
(2.2)
Here f (xn-1) and f (xn+1) are of opposite sign. Successive approximations to the root of f (x) = 0 is given by Eq. (2.2).

6. METHOD OF ITERATION can be applied to find a real root of the equation f (x) = 0 by rewriting the same in the form,

7. N-R Formula In Newton – Raphson Method successive approximations
x2, x3, …, xn to the root are obtained from
N-R Formula

8. Secant Method
This sequence converges to the root ‘b’ of f (x) = 0 i.e. f( b ) = 0.

9. The Secant method converges faster than linear and slower than Newton’s quadratic.

10. In Muller’s Method we can get a better approximation to the root, by using

11. Where we defined

12. Systems of Linear
Equations

13. Gaussian Elimination Gauss-Jordon
Gaussian Elimination Gauss-Jordon Elimination Crout’s Reduction Jacobi’s Gauss- Seidal Iteration Relaxation Matrix Inversion

14. In Gaussian Elimination method, the solution to the system of equations is obtained in two stages.
the given system of equations is reduced to an equivalent upper triangular form using elementary transformations
the upper triangular system is solved using back substitution procedure

15. Gauss-Jordon method is a variation of Gaussian method
Gauss-Jordon method is a variation of Gaussian method. In this method, the elements above and below the diagonal are simultaneously made zero

16. In Crout’s Reduction Method the coefficient matrix [A] of the system of equations is decomposed into the product of two matrices [L] and [U], where [L] is a lower-triangular matrix and [U] is an upper-triangular matrix with 1’s on its main diagonal.

17. For the purpose of illustration, consider a general matrix in the form

18. Jacobi’s Method is an iterative method, where initial approximate solution to a given system of equations is assumed and is improved towards the exact solution in an iterative way.

19. In Jacobi’s method, the (r + 1)th approximation to the above system is given by Equations

21. Here we can observe that no element of. replaces
Here we can observe that no element of replaces entirely for the next cycle of computation.

22. In Gauss-Seidel method, the corresponding elements of
In Gauss-Seidel method, the corresponding elements of replaces those of as soon as they become available. It is also called method of Successive Displacement.

23. The Relaxation Method is also an iterative method and is due to Southwell.

24. Eigen Value Problems
Power Method Jcobi’s Method

25. In Power Method the result looks like
Here, is the desired largest eigen value and is the corresponding eigenvector.

26. Interpolation

27. Finite Difference Operators Newton’s Forward Difference
Finite Difference Operators Newton’s Forward Difference Interpolation Formula Newton’s Backward Difference Interpolation Formula Lagrange’s Interpolation Formula Divided Differences Interpolation in Two Dimensions Cubic Spline Interpolation

28. Finite Difference Operators. Forward Differences. Backward Differences
Finite Difference Operators Forward Differences Backward Differences Central Difference

30. Thus
Similarly

31. Shift operator, E

32. The inverse operator E-1 is defined as
Similarly,

33. Average Operator,

34. Differential Operator, D

35. Important Results

36. The Newton’s forward difference formula for interpolation, which gives the value of f (x0 + ph) in terms of f (x0) and its leading differences.

37. An alternate expression is

38. Newton’s Backward difference formula is,

39. Alternatively, this formula can also be written as
Here

40. The Lagrange’s formula for interpolation

41. Newton’s divided difference interpolation formula can be written as

42. Where the first order divided difference is defined as

43. Numerical Differentiation and Integration

44. We expressed D in terms of ∆ :

45. Using backward difference operator , we have
On expansion, we have

46. Using Central difference Operator
Differentiation Using Interpolation
Richardson’s Extrapolation

48. Thus, is approximated by
which is given by

51. Basic Issues in Integration
What does an integral
represent?
= AREA
= VOLUME

52. yn-1 y3 y2 y1 y0 yn
xn = b
xn-1 x3 x2 x1
x0 = a X O Y
(x2, y2)
(x1, y1)
(x0, y0)
y = f(x)

54. xn = b
xn-1 x3 x2 x1
x0 = a X O Y
(x2, y2)
(x0, y0) y2 y1 y0
y = f(x)

55. TRAPEZOIDAL RULE

56. DOUBLE INTEGRATION
We described procedure to evaluate numerically a double integral of the form

57. Differential Equations

58. Taylor Series Euler Method Runge-Kutta Method Predictor Corrector
Taylor Series Euler Method Runge-Kutta Method Predictor Corrector Method

59. In Taylor’s series we expanded y (t ) by Taylor’s series about the point t = t0 and obtain

60. In Euler Method we obtained the solution of the differential equation in the form of a recurrence relation

61. We derived the recurrence relation
Which is the modified Euler’s method.

62. The fourth-order R-K method was described as

63. where

64. In general, Milne’s predictor-corrector pair can be written as

65. This is known as Adam’s predictor formula.
Alternatively, it can be written as

66. Numerical Analysis
Lecture 45