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III B.A./B.Sc. MathematicsPaper IV (Elective -1) - Curriculum

ACHARYA NAGARJUNA UNIVERSITY

CURRICULUM - B.A / B.Sc

MATHEMATICS - PAPER - IV (ELECTIVE - 1)

NUMERICAL ANALYSIS

(Syllabus for the academic years 2010-2011 and onwards)

UNIT - I 20 Hours

Errors in Numerical computations : Numbers and their Accuracy, Errors and their Computation, Absolute, Relative and percentage errors, A general error formula, Error in a series approximation. Solution of Algebraic and Transcendental Equations : The bisection method, The iteration method, The method of false position, Newton- Raphson method, Generalized Newton-Raphson method, Ramanujan's method, Muller's method.

UNIT - II 25 Hours

Interpolation : Errors in polynomial interpolation, Forward differences, Backward differences, Central Differences, Symbolic relations, Detection of errors by use of D. Tables, Differences of a

polynomial, Newton's formulae for interpolation formulae, Gauss's central difference formula, Stirling's

central difference formula, Interpolation with unevenly spaced points, Lagrange's formula, Error in Lagrange's formula, Derivation of governing equations, End conditions, Divided differences and their properties, Newton's general interpolation.

UNIT - III 20 Hours

Curve Fitting : Least-Squares curve fitting procedures, fitting a straight line, nonlinear curve fitting, Curve fitting by a sum of exponentials. Numerical Differentiation and Numerical Integration : Numerical differentiation, Errors in

numerical differentiation, Maximum and minimum values of a tabulated function, Numerical integration,

Trapezoidal rule, Simpson's 1/3-rule, Simpson's 3/8 - rule, Boole's and Weddle's rule.

UNIT - IV 25 Hours

Linear systems of equations : Solution of linear systems - Direct methods, Matrix inversion

method, Gaussian elimination method, Method of factorization, Ill-conditioned linear systems. Iterative

methods : Jacobi's method, Gauss-siedal method. Numerical solution of ordinary differential equations : Introduction, Solution by Taylor's Series, Picard's method of Successive approximations, Euler's method, Modified Euler's method, Runge - Kutta methods, Predictor - Corrector methods, Milne's method. Prescribed Text Book :- Scope as in Introductory methods of Numerical Analysis by S.S. Sastry,

Prentice Hall India (4th edition), Chapter - 1 (1.2, 1.4, 1.5, 1.6); Chapter - 2 (2.2 - 2.7); Chapter - 3 (3.2,

3.3, 3.7.2, 3.9.1, 3.9.2, 3.10.1, 3.10.2); Chapter - 4 (4.2); Chapter - 5 (5.2 - 5.4.5); Chapter - 6 (6.3.2,

6.3.4, 6.3.7, 6.4); Chapter - 7 (7.2 - 7.5, 7.6.2)

Reference Books :-

1. G. Sankar Rao New Age International Publishers, New - Hyderabad.

2. Finite Differences and Numerical Analysis by H.C. Saxena, S. Chand and Company, New

Delhi.90 Hours

1 III B.A./B.Sc. MathematicsPaper IV (Elective -1) - Curriculum

ACHARYA NAGARJUNA UNIVERSITY

CURRICULUM - B.A / B.Sc

MATHEMATICS - PAPER - IV (ELECTIVE - 1)

NUMERICAL ANALYSIS

QUESTION BANK FOR PRACTICALS

UNIT - I

1.i) Which of the following numbers has the greatest precision. a) 4 3201? b) 432? c) 4 320106?

ii) How many digits are to be taken in computing 20 so that the error does not exceed 0 01%?.

2.i) Sum the numbers

0 1532 15 45 0 000354 305 1 8 12 143 3 0 0212 0 643??? ?????,, ,,,, ,and 01734?where

in each of which all the given digits are correct. ii) If uxyz=5 23
/then find relative maximum error in u , given that ΔΔΔxyz===?0001and xyz===1.

3.Find a real root of the equation

fx x x()=--= 3

10 by bisection method.

4.Find a real root of the equation

xx 3

640--= by bisection method.

5.Find a positive root of the equation

xe x =1, which lies between 0 and 1 by bisection method.

6.Find the root of

tanxx+=0 upto two decimal places, which lies between 2 and 21?by bisection method.

7.Find a real root of the equation

xxlog 10

12=? by bisection method.

8.Find a real root of the equation

fx x x()=--= 3

250by the method of false position upto three places

of decimals.

9.Find a real root of the equation

xx 32

20--= by Regula-Falsi method.

10.Find the root of the equation

xe x x =cosusing the Regula Falsi method correct to three decimal places.

11.Find the root of

xx 3

10+-=by iteration method, give that root lies near 1.

12.Find a real root of the equation

cosxx=-31 correct to three decimal places, using iteration method.

13.Find by the iteration method, the root near

38?
, of the equation 27 10 xx-=log correct to four decimal places.

14.Find the real root of the equation

xx 2

520-+=by Newton-Raphson's method.

15.Find by Newton's method, the root of the equation

ex x =4which is near to 2 correct to three places of decimals.

16.Using Newton-Raphson method, establish the iterative formula

xxN x nn n+ 1 1 2 to calculate the square root of N. Hence find the square root of 8. 2 III B.A./B.Sc. MathematicsPaper IV (Elective -1) - Curriculum

17.Using Newton-Raphson method, establish the iterative formula

xxN x nn n+ 12 1 32
to calculate the cube root of N. Hence find the cube root of 12 applying the Newton-Raphson formula twice.

18.Find a double root of the equation

fx x x x()=--+= 32
10 by generalized Newton's method.

19.Find a root of the equation

xe x =1 by Ramanujan's method.

20.Find the root of the equation

yx x x()=--= 3 250
, which lies between 2 and 3 by Muller's method.

21.Show that i)

()()11 1+-?=Δ ii)

E?=Δ

iii) EE

12 12//

iv) 2 v) μδ 22
11 4=+

22.Evaluate i)

23
3 x Ex ii) 2 3 Ex( , the interval of differencing being unity.

23.Prove that i)

uu u u u 32 12
03 0 =+ + +ΔΔ Δ ii) uu u u u 43 22
13 1

24.Find the missing term in the following data.

x :01234 y:139?81

25.Form a table of differences for the function

fx x x()=+- 3

57for x=-1012345,,,,,,and continue

the table to obtain f()6and f()7.

26.Find the function whose first difference is

xxx 32

3512+++, if 1 be the interval of differencing.

27.The population of a country in the decennial census were as under. Estimate the population for the year

1895.
Year ()x: 1891 1901 1911 1921 1931

Population

()y(in thousands) : 46 66 81 93 101

28.From the following find

y value at x=?026. x 010? 015? 020? 025?
030?
yTanx

0 1003?0 1511?02027?0 2553?0 3093?

29.From the following table, find the number of students who obtain less than 56 marks.

Marks :30-40 40-50 50-60 60-70 70-80

No. of students : 31 42 51 35 31

30.Find the cubic polynomial which takes the following values.

x :0123 fx():02110 3 III B.A./B.Sc. MathematicsPaper IV (Elective -1) - Curriculum

31.If l

x represents the number of persons living at age x in a life table, find as accurately as the data will permit the value of l 47
. Given that llll

20 30 40 50

512 439 346 243====,,,.

32.Apply Gauss forward formula to find the value of

u 9 if uuuu 04816

14 24 32 40=== =;;;.

33.Given that

12500 111 803399 12510 111 848111 12520 111 892806=? =? =?;;;

12530 111 937483=?

. Show by Gauss backward formula that

12516 111 874930=?

34.Use Stirling's formula to find

y 28
, given yyyy

20 25 30 35

49225 48316 47236 45926====,,,,

y 40

44306=.

35.Given

yyyy

20 24 28 32

24 32 35 40====,,,,find y

25
by Bessel's formula.

36.By means of Newton's divided difference formula, find the value

f()8and f()15from the following table : x :45710 11 13 fx(): 48 100 294 900 1210 2028

37.Using the Newton's divided difference formula, find a polynomial function satisfying the following data.

x -4-1 025
fx(): 1245 33 5 9 1335

38.Using Lagranges interpolation formula find

yat x=301. x : 300 304 305 307 y:

2 4771?24829?24843?24871?

39.By Lagrange's interpolation formula, find the form of the function given by

x :01234 fx():36111827

40.Using Lagrange's formula, prove that

yyy yyyy

011 31 13

1 21
81
21

2=+- -- -?

41.Find the least square line for the data points

( , ),( , ),( , ),( , ),( , ),( , ),( , )-110091725344350 and (, )61-.

42.Find the least square power function of the form

yax b for the data. i x1 2 3 4 i y3 122135

43.Fit a second degree parabola to the following data :

:x0 1 2 3 4 y:1

18?13?25?63?

4 III B.A./B.Sc. MathematicsPaper IV (Elective -1) - Curriculum

44.Using the given table, find

dy dx and dy dx 2 2 at x=?12. x10?12?14?16?18?20?22? y

27183?3 3201?40552?49530?60496?7 3891?90250?

45.From the following table, find the values of

dy dx and dy dx 2 2 at x=?203 x

196?198?

200?202?042?

y

0 7825?07739?0 7651?0 7563?07473?

46.Find ′?f()06and ′′?f()06from the following table :

x 04? 05? 06? 07? 08? fx()

1 5836?1 7974?20442?23275?26510?

47.Find ′?f()25from the following table :

x 15? 19? 25?
32?

43?59?

fx()

3375?6059?13 625?29 368?73 907?

196 579?

48.Find the maximum value of y, by using data given below :

x 01234
y0 025?
0 225?
16

49.Find

′?f()15and ′′?f()15from the following table. x15?20?25?30?35?40? fx()

3 375?7 000?13 625?24 000?38 875?59 000?

50.Assuming Stirling's formula, show that [][][]

d dxfx fx fx fx fx() ( ) ( ) ( ) ( )= +--- +--2 3111
1222
upto third differences.

51.Evaluate

Idx x=+ 1 01 correct to three decimal places by Trapezoidal rule with h=?025.

52.Evaluate

()43 2 01 xxdx- taking 10 intervals by Trapezoidal rule.

53.Calculate an approximate value of

2/ 0 sin dxxby Trapezoidal rule.

54.By Simpson's

1 3 rule, evaluate 11 12 /xdxwith five ordinates. 5 III B.A./B.Sc. MathematicsPaper IV (Elective -1) - Curriculum

55.Use Simpson's

1 3 rule to prove that log e

7is approximately

19587?

using dx x 17

56.Find the value of the integral

dx x1 2 01 ∫by using Simpson's 1 3 and 3 8 rule. Hence obtain the approximate value of

π in each case.

57.Evaluate

1 1 01 xdx by Boole's rule.

58.Evaluate the integral

edx x 04 by Boole's Rule.

59.Evaluate the integral

logxdx 452
, using Weddle's rule.

60.Integrate numerically

2/ 0 sin dxx by Weddle's rule.

61.Solve the equations

32472 7 352xyz xyz xyz++= ++= ++=;; by matrix inversion method.

62.Solve the equations

xyz x y z xyz++= + + = +-=92 5 7 522 0,,by Cramer's rule.

63.Solve the equations

2 4 12 8 3 2 23 4 11 33xy z x y z x yz-+ = - + = + -=;;by Gauss elimination

method.

64.Solve the system of equations

2 1032318 4916xyz x y z x y z++= + + = + + =,,by Gauss elimina-

tion method.

65.Solve the equations

32472 7 352xyz xyz xyz++= ++= ++=;;by Factorization method.

66.Solve the equations

23 9 2363 28xyz xyz xyz++= ++= ++=,, by LU decomposition method.

67.Solve the following equations by Gauss-Jacobi method :

10 12xyz-+=;xyz-+=10 12;

xy z +- =10 12 correct to 3 decimals.

68.Solve the following equations by Jacobi method :

20 2 17xy z+- =; 3 20 18xyz+-=-;

2 3 20 25xy z-+ =

69.By Gauss-seidel iterative method solve the linear equations

xxx xxx

123 123

10 6 10 6++= ++=; and

xx x 12 3

10 6++ =.

70.Solve the following system by Gauss-Seidel method :

10 2 9 2 20 2 44xyz z yz++= + -=-,,

2 3 10 22xy z

71.Solve the differential equation

yxdxdy+=, with yx() , [,]01 01=? by Taylor series expansion to obtain y for x=?01. 6 III B.A./B.Sc. MathematicsPaper IV (Elective -1) - Curriculum

72.Solve the equation

′=+yxy 2 with y 0

1=when

x=0

73.Given

xyxy dxdy with y=1, when x=0 . Find approximately the value of y for x=?01 by Picard's method.

74.Given

1)0(, 3 =+=yyxdxdy , compute y=?()02by Euler's method taking h=?001

75.Using Euler's method, compute

y()05?for differential equationquotesdbs_dbs9.pdfusesText_15