NUMERICAL ANALYSIS
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Numerical Integration and Di?erentiation Here the objective is clear; we knowthat many functions are impossible tointegrate analyticallysowewanttohaveanaccuratewayofdoingthisnumerically We would also like to have some idea and control over the accuracy of the results Integration Thewayhowwecannumericallyevaluate RB A y(x)dxis to choose a
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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
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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 apolynomial, 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 innumerical 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 inversionmethod, 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) - CurriculumACHARYA 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()=--= 310 by bisection method.
4.Find a real root of the equation
xx 3640--= 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 1012=? by bisection method.
8.Find a real root of the equation
fx x x()=--= 3250by the method of false position upto three places
of decimals.9.Find a real root of the equation
xx 3220--= 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 310+-=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 2520-+=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) - Curriculum17.Using Newton-Raphson method, establish the iterative formula
xxN x nn n+ 12 1 32to 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()=--+= 3210 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) EE12 12//
iv) 2 v) μδ 2211 4=+
22.Evaluate i)
233 x Ex ii) 2 3 Ex( , the interval of differencing being unity.
23.Prove that i)
uu u u u 32 1203 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?8125.Form a table of differences for the function
fx x x()=+- 357for x=-1012345,,,,,,and continue
the table to obtain f()6and f()7.26.Find the function whose first difference is
xxx 323512+++, 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 10128.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) - Curriculum31.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 0481614 24 32 40=== =;;;.
33.Given that
12500 111 803399 12510 111 848111 12520 111 892806=? =? =?;;;
12530 111 937483=?
. Show by Gauss backward formula that12516 111 874930=?
34.Use Stirling's formula to find
y 28, given yyyy
20 25 30 35
49225 48316 47236 45926====,,,,
y 4044306=.
35.Given
yyyy20 24 28 32
24 32 35 40====,,,,find y
25by 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 202837.Using the Newton's divided difference formula, find a polynomial function satisfying the following data.
x -4-1 025fx(): 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():3611182740.Using Lagrange's formula, prove that
yyy yyyy011 31 13
1 2181
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 12213543.Fit a second degree parabola to the following data :
:x0 1 2 3 4 y:118?13?25?63?
4 III B.A./B.Sc. MathematicsPaper IV (Elective -1) - Curriculum44.Using the given table, find
dy dx and dy dx 2 2 at x=?12. x10?12?14?16?18?20?22? y27183?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 x196?198?
200?202?042?
y0 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 01234y0 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 31111222
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) - Curriculum55.Use Simpson's
1 3 rule to prove that log e7is approximately
19587?
using dx x 1756.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 ofπ 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 xxx123 123
10 6 10 6++= ++=; and
xx x 12 310 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) - Curriculum72.Solve the equation
′=+yxy 2 with y 01=when
x=073.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=?00175.Using Euler's method, compute
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