LAXMI INSTITUTE OF TECHNOLOGY, SARIGAM · 20 Newton’s Backward Interpolation Formula 11/01/2019...

LAXMI INSTITUTE OF TECHNOLOGY, SARIGAMLAXMI INSTITUTE OF TECHNOLOGY, SARIGAM

COURSE FILECOURSE FILE

Faculty NameFaculty Name : Hiral Desai: Hiral Desai

Subject NameSubject Name : Complex Variables and Numerical Methods : Complex Variables and Numerical Methods

Subject CodeSubject Code : 2141905: 2141905

Course NameCourse Name : B.E.: B.E.

Semester : 4Semester : 4

SessionSession : December-April_2018-19 : December-April_2018-19

DEPARTMENT OF AUTOMOBILE ENGINEERINGDEPARTMENT OF AUTOMOBILE ENGINEERING

Laxmi Institute of Technology, SarigamApproved by AICTE, New Delhi; Affiliated to Gujarat Technological University, Ahmedabad

LECTURE PLANDepartment: AUTOMOBILEName of Subject In charge: HIRAL DESAI

Subject Name: CVNM Subject Code: 2141905Academic Year: 2018-19 Semester: 4TH

Lecture

No.Topics To Be Covered

PlannedDate

ActualDate

Remark/Teaching Aids/

Books

Unit-01:System of linear equations

01Solutions of a system Of Linear Equations

18/12/2018

02Numerical Methods For Solutions of a System of Linear Equations

18/12/2018

03 Gauss-Elimination Method 19/12/2018

04Gauss-Elimination Method with Partial Pivoting

20/12/2018

05 Gauss-Jordan Method 21/12/201806 Gauss-Jacobi Method 26/12/201807 Gauss-Siedel Method 27/12/2018

Unit-02:Roots of Algebraic and Transcendental Equations08 Bisection method 28/12/2018

09 Regula Falsi Method 29/12/2018

10 Newton-Raphson Method 01/01/2019

11 Secant Method 01/01/2019

Unit-03: Eigen values by Power Method12 Eigen Values and Eigen Vectors 02/01/2019

13 Power Method For Smallest Eigen Value 03/01/2019

14 Examples 04/01/2019

15 Power Method For Largest Eigen Value 05/01/2019

16 Examples 08/01/2019


Unit-04:Interpolation17 Finite Differences 08/01/2019

18 Different Operators And their Relations 09/01/2019

19 Newton’s Forward Interpolation Formula 10/01/2019

20 Newton’s Backward Interpolation Formula 11/01/2019

21 Central Difference Interpolation 15/01/201922 Gauss’s Forward Interpolation Formula 15/01/201923 Gauss’s Backward Interpolation Formula 16/01/201924 Stirling’s Formula 17/01/201925 Interpolation with Unequal Intervals 18/01/201926 Lagrange’s Interpolation Formula 19/01/201927 Divided Differences 22/01/201928 Newton’s Divided Difference Formula 22/01/201929 Inverse Interpolation 22/01/2019

Unit-05:Numerical Integration30 Newton-Cotes Quadrature Formula 23/01/201931 Trapezoidal Rule 24/01/201932 Simpson’s 1/3 Rule 25/01/201933 Simpson’s 3/8 Rule 05/02/201934 Gaussian Quadrature Formula 05/02/2019

Unit-06:Numerical Solution of Ordinary Differential Equations35 Euler’s Method 06/02/201936 Modified Euler’s Method 07/02/2019

37First order Runge-Kutta Method 08/02/2019

38

Second order and Third Order Runge-Kutta method

19/02/2019


Unit-07:Complex number and function

39Basic definition and Notation of Complex Numbers

19/02/2019

40 De Moivre’s Theorem 20/02/2019

41 Roots of a complex number 21/02/2019

42 Hyperbolic functions and their properties 22/02/2019

43Logarithmic function and Complex Exponent Function

26/02/2019

44 Analytic functions 26/02/2019

45 Cauchy-Riemann Equations 27/02/2019

46Harmonic Functions, Harmonic Conjugate functions

28/02/2019

Unit-08:Complex Integration

47Curves, Line Integral(contour integral) and its properties,

01/03/2019

48 Cauchy- Goursat Theorem, 02/03/2019

49 Cauchy Integral Formula, 05/03/2019

50 Maximum Modulus Theorems 05/03/2019

Unit-09: Power series51 Convergence of Power series 06/03/2019

52 Taylor’s Series 07/03/2019

53 Laurent series 08/03/2019

54 Zeros of analytic functions 12/03/2019

55 Singularities of analytic functions 13/03/2019

56Classification of Singularities of analytic function

14/03/2019

57 Cauchy’s Residue Theorem 15/03/2019


58 Argument Theorem 16/03/2019

59 Rouche’s Theorem 19/03/2019

Unit-10 Application of contour Integration60 Definite Real Integration around semicircle 20/03/2019

61 Definite Real Integration around contour C 22/03/2019

62Definite Real Integration around contour having poles

02/04/2019

63 Definite Real Integration having finite poles 03/04/2019

Unit-11 Conformal mappings and its Applications64 Conformal Mapping 04/04/2019

65 Some Standard Transformations 09/04/2019

66 Some Special Transformations 10/04/2019

67 Schwarz-Christoffel Transformation 12/04/2019

68 Bilinear Transformation 16/04/2019

Total Planned Lectures: 68Total Actual Lectures:

HIRAL DESAI

Subject In charge Head of Department

ASSIGNMENT: 1

Name of Subject :CVNM

Subject Code : 2141905

Unit: 1 COMPLEX NUMBERS

Sr.No.

Name of Question Remarks

11 2 1 2( , ) ( ) ( )?Arg z z Arg z Arg z Justify.

2Find the principal argument Argz when

2

1 3z

i

.

3 Solve the equation 2 (2 3) 5 0z i z i .

4Prove that 1tan log

2

i i zz

i z

5If

12cos ,x

x Prove that (i)

12cosn

nx n

x (ii)

2

2 1

1 cos

cos( 1)

n

n

x n

x x n

6 If and are the roots of the equation 2 2 2 0x x ,find the value of n n .

Hence, deduce that 8 8 32 .7

Separate real and imaginary parts of (i)sinh z (ii) 1 3cos

4

i

(iii) 1sin ie

8 For the principal branch, show that 3( ) 3Log i Logi

9 Prove that ii is wholly real and find its principal value. Also show that the values of iiform a geometric progression.

10 Find the real and imaginary parts of 1tanh ( )x iy

ASSIGNMENT: 2

Name of Subject : CVNM


Unit: 2 ANALYTIC FUNCTIONS

Sr.No.


1 Sketch the following sets and determine which are domains?

(I ) Im 1z (ii) 0 arg4

z

2 Is the set 1 2 2z i a domain?

3 Show that 2

( )f z z is continuous but not differentiable, at each point in the plane.

4 Show that the function ( )f z z is nowhere differentiable.

5 Discuss the differentiability of 2 2( )f z x iy .

6 Examine the analyticity ofsinh z .

7 Show that ( )f z xy is not differentiable at origin although C-R equations are

satisfied.8 If ( )f z u iv is analytic in the domain D then prove that

2 22 2

2 2Re ( ) 2 ( )f z f z

x y

9 Find the Analytic function ( )f z u iv if 3 3u x xy

10 Find the Analytic function u iv ,if 2 2( )( 4 )u v x y x xy y

ASSIGNMENT: 3



Unit: 3 COMPLEX INTEGRATION

Sr.No.


1Find an upper bound for the absolute value of the integral

2

C

z dz , where C is the

straight-line segment from 0 to 1 i .2

Find an upper bound for the absolute value of the integral 1

z

C

edz

z Ñ , where C is 4z

3Evaluate

22

0

i

z dz

along the line2

xy .

4Evaluate

C

zdz , where C is along the sides of the triangle having vertices 0,1,z i .

5Evaluate

z

C

edz

z iÑ , where : 1 1C z .

6Prove that 2

C

dzi

z a

and ( ) 0n

C

z a dz (n is an integer and 1)n where C is

the Circle z a r .

7Evaluate 2

3 6

( )C

dzz i z i

, where : 2C z .

ASSIGNMENT: 4



Unit: 4 POWER SERIES

Sr.No.


1 Find the radii of convergence and region of convergence of the following:

(1)1 2 1

n

nn

z

(2)1

! nn

n

nz

n

(3)

1

( 1)( 2 )

nn

n

z in

2Expand ( )

( 1)( 3)

zf z

z z

as a Taylor’s series about 0z

3Expand

1( )

( 1)( 2)f z

z z

in Laurent’s series in the following regions;

(1) 1Z (2) 1 2z (3) 2z

4Expand 2

sin( )

z zf z

z

at 0z ,classify the singular point 0z .

5Find the residues of the function

2

2 2

2( )

( 1) ( 4)

z zf z

z z

at each of its poles.

6Evaluate 2

5 7

2 3C

zdz

z z

Ñ ,where C is 2 2z .

7Using the residue theorem, evaluate 3

z

C

e zdz

z z

, where :

2C z

8Evaluate 2

2 6

4C

zdz

z

Ñ where C is 2z i

9Find the value of the integral

2

2

3 2

( 1)( 9)C

zdz

z z

taken counter clockwise around the

circle : 2 2C z

10Evaluate

4

2( 1)( )

zdz

z z i where 2 2: 9 4 36C x y by using the residue theorem.

11 Use Rouche’s theorem to determine the number of zeros of the polynomial6 4 35 2z z z z inside the circle 1z .

ASSIGNMENT: 5



Unit: 5 APPLICATIONS OF CONTOUR INTEGRATION

Sr.No.


1Evaluate a real integral

2

20

1

(2 cos )d

using residue.

2Evaluate

2

0

1 cos

5 4cosd

using contour integration.

3Evaluate

2

2 2( 1)( 4)

xdx

x x


4Evaluate

2

2 2( 1)

xdx

x


5Evaluate 4

02 21

dx

x

.

6Using the theory of residues, evaluate 2

cos

1

xdx

x

7

Evaluate a real integral 20

sin

9

x xdx

x

using residue.

ASSIGNMENT: 6



Unit: 6 INTERPOLATION

Sr.No.


1 Using Newton’s Forward interpolation formula , find the value of (218)f .x 100 150 200 250 300 350 400

( )f x 10.63 13.03 15.04 16.81 18.42 19.90 21.27

2 From the following table, estimate the number of students who obtained marks between 40 and 45:Marks 30-40 40-50 50-60 60-70 70-80

Number of students

31 42 51 35 31

3 Construct Newton’s forward interpolation polynomial for the following data:x 4 6 8 10

y 1 3 8 16

4 From the following table, find P when 142t C o and 175t C o using appropriate Newton’s interpolation formula.

Temperature t Co 140 150 160 170 180

Pressure P 3685 4845 6302 8076 10225

5 The population of a town is given below. Estimate the population for the year 1895 and1930 using suitable interpolation.Year x 1891 1901 1911 1921 1931

Population y 46 66 81 93 101

6 Use Gauss’s Forward interpolation formula to find (3.3)y from the following table: x 1 2 3 4 5

y 15.3 15.1 15 14.5 14

7 Using Gauss’s Backward interpolation formula, find the population for the year 1936given thatYear x 1901 1911 1921 1931 1941 1951

Population inthousands y

12 15 20 27 39 52

8 Using Stirling’s formula, estimate the value of tan16o .x 0o 5o 10o 15o 20o 25o 30o

tany x 0 0.0875 0.1763 0.2679 0.3640 0.4663 0.5774

9 Compute (9.2)f by using Lagrange’s interpolation method from the following data:x 9 9.5 11

( )f x 2..1972 2.2513 2.3979

10 Find a second degree polynomial passing through the points (0,0), (1,1) (2, 20)andusing Lagrange’s interpolation.

11 Find the second degree divided difference for the argument 1, 2,5 7x and for the

function 2( )f x x 12 Construct the divided difference for the data given below:

x -4 -1 0 2 5

( )f x 1245 33 5 9 1335

13 Using Newton’s Divided Difference interpolation, compute the value of (6)f from thetable given below:x 1 2 7 8

( )f x 1 5 5 4

14 Using Newton’s Divided Difference formula, Find a polynomial and also, find

( 1) (6)f andf .x 1 2 4 7

( )f x 10 15 67 430

ASSIGNMENT: 7



Unit: 7 NUMERICAL INTEGRATION

Sr.No.


1 Given the data below, find the isothermal work done on the gas if it is compressed

from 1 22v L to 2 2v L . Use 2

1

v

v

W pdv .By Trapezoidal Rule.

,v L 2 7 12 17 22

,p atm 12.20 3.49 2.049 1.44 1.11

2Calculate

1

0

2 xe dx with n=10 using the trapezoidal rule.

3Evaluate

6

0

1

1dxx taking h=1 using Simpson’s 1

3 rule. Hence, obtain an approximate

value of log7.

4Evaluate the integral

62

2

32

(1 )x dx

by Simpson’s 13 rule with taking 6 sub-intervals.

5Using Simpson’s 1

3 rule, find 2

0.6

0

xe dx by taking n=6

6Evaluate

3

0

1

1dxx with n=6 by using Simpson’s 3

8 rule, and hence, calculate log2.

7Evaluate

5.2

4

log xdx using the trapezoidal rule and Simpson’s 38 rule, take h=0.2.

8Evaluate

62

2

32(1 )x dx

by the Gaussian formula for n=3.

9Evaluate

21

0

xe dx by using the Gaussian Quadrature formula with n=3.

ASSIGNMENT: 8



Unit: 8 ROOTS OF ALGEBRAIC AND TRANSCENDENTALEQUATIONS

Sr.No.


1 Find a root of 3 5 3 0x x by the Bisection method correct up to four decimalplaces.

2 Find the approximate solution of 3 1 0x x correct to three decimal places.

3 Solve cosx x by the bisection method correct to two decimal places.

4 Find a positive root of 3 4 1 0x x correct up to three decimal places by regula falsi method.

5 Find a smallest root of an equation 0xx e correct to three significant digits by regula falsi method.

6 Find a real root of 10log 1.2 0x x correct upto three decimal places by Newton-Raphson method.

7 Find a real positive root of the equation sin cos 0x x x which is near x correct up to four significant digits by Newton-Raphson method.

8 Derive iteration formula for N and hence find (i) 28 (ii) 65 (iii) 3

9 Find a positive solution of 2sin 0x x correct up to three decimal places starting from 0 2x and 1 1.9x by Secant method.

10 Solve 1 0xxe correct up to three decimal places between o and 1 by Secant method.

Laxmi Institute of Technology , SarigamApproved by AICTE, New Delhi; Affiliated to Gujarat

Technological University, AhmedabadAcademic Year 2018-19

Centre Code: 086 Exam : Mid-I Marks:20Branch: AUTOMOBILE Semester:4TH Sub Code:2141905Sub: CVNM Date:10/01/2019 Time:9:00 TO 10:00 AM

Q-1(A) Solve by Gauss-Elimination Method:2 3;2 3 3 10;3 2 13x y z x y z x y z .

03

(B) Derive iterative formula to find N . Use this formula to find

28 .

03

(C) Using the Power Method, find the largest eigen value of the2 1 0

1 2 1

0 1 2

A

07

OR(C) Solve by Gauss-Seidel Method:

10 6; 10 6; 10 6x y z x y z x y z 07

Q-2(A) Find the root of the equation 2 4 10 0x x , Correct up to three decimal places by using Bisection Method.

03

(B) The table below gives the values of function tany x . Obtainthe value of tan(0.40) . Using Newton’s Backward interpolation.x 0.10 0.15 0.20 0.25 0.30

tany x 0.1003 0.1511 0.2027 0.2533 0.3093

04

OR(B) Construct Newton’s Forward interpolation polynomial for the

following data:x 4 6 8 10

y 1 3 8 16

04

LAXMI INSTITUTE OF TECHNOLOGY, SARIGAM · 20 Newton’s Backward Interpolation Formula 11/01/2019...

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