1. Advanced Numerical Methods (S. A. Sahu) Code: AMC 51151

2. We can divide our syllabus in following four major sections:
Method of solution of system of equations
Solution of Non-linear Simultaneous Equations
(Newton-Raphson Method)
Tridiagonal system of equations
(Thomas algorithm)

3. 2. Numerical Integration
Evaluation of Double & Triple Integrals
(a) Constant Limit (b) Variable Limit
Simpson’s Rule & Trapezoidal Rule

4. 3. Methods of Interpolation
Newton’s F/W and B/W formula
Gauss’s F/W and B/W formula
Bessel’s Formula
Stirling Formula
Lagrange’s Interpolation Formula
Newton’s Divided Difference Formula

5. 4. Numerical Solution of Differential Equations O.D.E. P.D.E.
First Order & Higher Order Ordinary Diffe. Eqs.
Initial Value & Boundary Value Problems
Laplace & Poisson Equations
Heat Conduction
Wave Equations

6. Solution of Non-linear Simultaneous Equations (Newton-Raphson Method)

7. This Method is extension of N-R Method for Non-Linear Equation
Method for Simultaneous equations:
Suppose we are given the following Simultaneous equations
…(1)
Why simultaneous..?

8. We consider that be the first approximation
And be the real roots
It means, we must have :
….(2)

9. Now we expand the relation, we had in eq
Now we expand the relation, we had in eq. (2) Using Taylor Series Expansion (3) (similarly for ) Now since h and k are very small, we can neglect the terms containing (or )and higher order terms…

10. Now, eq.(2), eq.(3) and assumption suggests
..(4b)

11. Or simply:
…5(a)
..5(b)
Let us say

12. Cramer’s Rule for or the solution of a system of linear equation

13. We have from Eqs. 5(a) & 5(b)
Or in matrix form

14. Now let us call
And hence we can have (from Cramer’s rule)

15. This gives the next approximation, as
Using these values we can proceed further for next approximation (better value)
This completes the method.

16. Solve the given simultaneous equations
(up to two iterations)

17. Q.2

18. Solution of Q.2

20. And hence we have the second approximation as
We repeat the process for next approximation

21. Tridiagonal system of equations (Thomas algorithm)

22. [Upper Triangular & Lower Triangular Matrices]
In applications, many times we can have system of equations which gives a coefficient matrix of special structure, majority of zeros .  Sparse matrices 
[Upper Triangular & Lower Triangular Matrices]
What, If we have a matrix such that except the principal diagonal and its upper and lower diagonal, all the elements are zero….???

23. Such a matrix is called TRIDIAGONAL MATRIX

24. Something which looks like…….

25. Tridiagonal Systems aij=0 if |i-j| > 1 Tridiagonal Systems:
The non-zero elements are in the main diagonal, super diagonal and subdiagonal.
aij=0 if |i-j| > 1
CISE301_Topic3

26. Tridiagonal System of Equations
Mathematically….
An  n×n  matrix A is called a tridiagonal matrix if     whenever 
The system of equations which gives rise to a tridiagonal coefficient matrix is called …..
Tridiagonal System of Equations

27. A tridiagonal system for n unknowns may be written as
…(1)
B.V.P. with second order ODE, Heat conduction equations…. Generate equations like (1)

28. Now if we put i = 1, 2…n in eq. (1), we get. 2(a). 2(b). 2(c) …

29. Which in matrix form can be written as

30. Thomas algorithm (named after L. H
Thomas algorithm (named after L. H. Thomas), is a simplified form of Gauss elimination method
Suppose we have a system like
And we have to find all the unknowns…

31. It means first we shall get and then by back substitution, we will get
Then Thomas Algo. Gives the values of unknown in the order last to first..

32. Where
and

33. Q.1 Solve the following system of equations

34. In matrix form
CISE301_Topic3

36. Numerical Integration
Section: 2
Finite Difference
Interpolation
Numerical Integration

37. Let we have ..(1) which is tabulated for equally spaced values..
Finite Differences
Let we have (1) which is tabulated for equally spaced values..

38. And returns the following values It means

39. Then we can define the following Differences
Forward Difference DELTA
Backward Difference Nabla
Central Difference delta (lower case)

40. Forward Difference
First Forward Difference
We define the FORWARD DIFFERENCE OPERATOR, denoted by   as
The expression   gives the FIRST FORWARD DIFFERENCE of  and the operator   is called the FIRST FORWARD DIFFERENCE OPERATOR. 

41. Notation (First Forward Differences)

42. Second Forward Differences are given by

43. And hence in general the pth Forward Diff.
..(2)

44. Forward Difference Table

45. Forward Diff. Terms

46. Backward differences are defined and denoted by following

47. Following the concept of Forward Diff
Following the concept of Forward Diff. Make a table for Backward Difference up to 5th Difference

48. Central Differences
The central Diff. operator is defined by the following relation
Define second third and other central differences
And make a central diff. table

49. Write the forward Diff. table if…

50. Other operators
Shift Operator E
Average Operator

51. Shift Operator is the operation of increasing of argument x by h
and hence defined by
The inverse shift operator is defined by

52. We have defined the following Differences
Forward Difference DELTA
Backward Difference Nabla
Central Difference delta
Shift Operator E
Average Operator

53. Relations among the operators

54. Proof of (i)

55. Interpolation
Suppose we have following values for y=f(x)

56. Interpolation is the technique of estimating the value of a function for any intermediate value of the independent variable
The process of computing the value of the function outside the range is called Extrapolation

57. Interpolation
Equal spaced Unequal Differences
Forward Interpolation Newton’s Divided Diff.
(Newton’s Fwd. Int.)
Backward Interpolation Lagrange’s Interpolation
(Newton’s Bwd. Int.)
Central Diff. Interpolation
Gauss Forward
Gauss Backward
Stirling Formula
Bessel’s Formula
Everett’s Formula

58. Newton’s Forward Interpolation
Suppose the function is tabulated for equally spaced values.. And it takes the following values

59. Suppose we have to calculate f(x) at x=x0+ph
(p is any real number), then we have
Now we expand the term by Binomial theorem

60. Newton’s Forward Interpolation Formula
The above blue colored equation (1) is
Newton’s Forward Interpolation Formula
Because it contains y0 and forward differences of y0

61. Example
Estimate f (3.17)from the data using Newton Forward Interpolation. x:
f(x):

62. Solution

63. Solution

64. Newton’s Backward Interpolation
Suppose the function is tabulated for equally spaced values.. And it takes the following values

65. Suppose we have to calculate f(x) at x=xn+ph
(p is any real number, may be -ve), then we have
Now we expand the term by Binomial theorem

66. NEWTON GREGORY BACKWARD INTERPOLATION FORMULA

67. Example
Estimate f(42) from the following data using newton backward interpolation. x:
f(x):

68. Solution
Here xn = 45, h = 5, x = 42
and p = - 0.6

69. Solution

70. Central Difference Formula
Gauss Forward
Gauss Backward
Stirling Formula
Bessel’s Formula
Everett’s Formula

72. If y=f(x) is any function, which takes values
and returns corresponding values
Then the difference table can be written as

73. Central Difference table Table-1

74. 1. GAUSS FORWARD INTERPOLATION FORMULA OR …..(A)

75. Derivation of Gauss Fwd. Int. Formula
We have Newton’s Forward Int. Formula
We have following relations (from table)

77. Note:
Gauss forward formula contains odd differences just below the central line and even differences on the central line
Formula is useful to interpolate the value of y for 0<p<1, measured forwardly

78. Example
Find f(30) from the following table values using Gauss forward difference formula: x: 21 25 29 33 37
F(x):

79. Solution
f (30) =

80. 2. Gauss Backward interpolation formula
Again starting with Newton’s Forward Int. Formula

81. this is Gauss Backward Formula
Also we can have
And similarly other differences..
Putting these values we finally get
this is Gauss Backward Formula

82. Note:
Gauss Backward formula contains odd differences just above the central line and even differences on the central line
Formula is useful to interpolate the value of y for p lying between -1 & 0

83. Example
Estimate cos 51 42' by Gauss backward interpolation from the following data:
x:  51 52 53 54
cos x:

84. Solution

85. Solution
P(51.7) =

86. 3. STIRLING’S FORMULA
This formula gives the average of the values obtained by Gauss forward and backward interpolation formulae. For using this formula we should have – ½ < p< ½.
We can get very good estimates if - ¼ < p < ¼.
The formula is:

87. Stirling Formula (Derivation) [Mean of Gauss Fwd. & Gauss Bkwd]
Eq. (3) is Stirling Formula

88. Note:
Stirling Formula contains means of odd differences just above and below the central line and even differences on central line

89. Example
Using Sterling Formula estimate f (1.63) from the following table: x:
f(x):
Solution
X0 = 1.60, x = 1.63, h = 0.1

90. Solution
Difference table

91. 4. BESSELS’ INTERPOLATION FORMULA
We have Gauss Fwd. Interpolation Formula as
Now we use following relations

92. We get the following Bessel’s Formula

93. BESSELS’ INTERPOLATION FORMULA/other way
This formula involves means of even difference on and below the central line and odd difference below the line.
The formula is

94. BESSELS’ INTERPOLATION FORMULA
Note
When p = 1/2 , the terms containing odd differences vanish.
Then we get the formula in a more simple form:

95. Example Use Bessels’ Formula to find (46.24)1/3 from the
following table of x1/3.

96. Solution x0 = 45, x = 46.24, h = 4 p = 0.31 Difference table is:
Applying Bessels’ formula, we get
(46.24)1/3 =

97. 5. Laplace-Everett’s Formula
Gauss’s forward interpolation formula is We eliminate the odd differences in above equation by using the relations

98. And then to change the terms with negative sign, putting , we obtain
This is known as Laplace-Everett’s formula.

99. Choice of an interpolation formula
The following rules will be found useful:
To find a tabulated value near the beginning of the table, use Newton’s forward formula.
To find a value near the end of the table, use Newton’s backward formula.
To find an interpolated value near the centre of the table, use either Stirling’s or Bessel’s or Everett’s formula.

100. Choice of central difference
If interpolation is required for p lying between
and , prefer Stirling’s formula.
If interpolation is desired for p lying between
and , use Bessel’s or Everett’s formula.
Gauss’s Backward interpolation formula is used to interpolate the value of “y” for a negative value of p lying between -1 and 0.

101. Gauss’s forward interpolation formula is used to interpolate the value of “y” for p (0<p<1) measured forwardly from the origin.

102. Bessel’s and Everett’s central difference
interpolation formula can be obtained by each
other by suitable rearrangements.