NUMERICAL ANALYSIS (Syllabus for the academic years and onwards)

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1 ACHARYA NAGARJUNA UNIVERSITY CURRICULUM - B.A / B.Sc MATHEMATICS - PAPER - IV (ELECTIVE - ) NUMERICAL ANALYSIS (Syllabus for the academic years - and onwards) Paper IV (Elective -) - Curriculum 9 Hours UNIT - I 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 approimation. 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 UNIT - II 5 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 Hours Curve Fitting : Least-Squares curve fitting procedures, fitting a straight line, nonlinear curve fitting, Curve fitting by a sum of eponentials. Numerical Differentiation and Numerical Integration : Numerical differentiation, Errors in numerical differentiation, Maimum and minimum values of a tabulated function, Numerical integration, Trapezoidal rule, Simpson's /-rule, Simpson's /8 - rule, Boole's and Weddle's rule. UNIT - IV 5 Hours Linear systems of equations : Solution of linear systems - Direct methods, Matri inversion method, Gaussian elimination method, Method of factorization, Ill-conditioned linear systems. Iterative methods : Jacobi's method, Gauss-siedal Numerical solution of ordinary differential equations : Introduction, Solution by Taylor's Series, Picard's method of Successive approimations, Euler's method, Modified Euler's method, Runge - Kutta methods, Predictor - Corrector methods, Milne's Prescribed Tet Book :- Scope as in Introductory methods of Numerical Analysis by S.S. Sastry, Prentice Hall India (4th edition), Chapter - (.,.4,.5,.6); Chapter - (. -.7); Chapter - (.,.,.7.,.9.,.9.,..,..); Chapter - 4 (4.); Chapter - 5 ( ); Chapter - 6 (6.., 6..4, 6..7, 6.4); Chapter - 7 ( , 7.6.) Reference Books :-. G. Sankar Rao New Age International Publishers, New - Hyderabad.. Finite Differences and Numerical Analysis by H.C. Saena, S. Chand and Company, New Delhi.

2 ACHARYA NAGARJUNA UNIVERSITY CURRICULUM - B.A / B.Sc MATHEMATICS - PAPER - IV (ELECTIVE - ) NUMERICAL ANALYSIS QUESTION BANK F PRACTICALS UNIT - I Paper IV (Elective -) - Curriculum. i) Which of the following numbers has the greatest precision. a) 4 b) 4 c) 4 6 ii) How many digits are to be taken in computing so that the error does not eceed %.. i) Sum the numbers 5, 5 45, 54, 5, 8, 4,, 64 and 74where in each of which all the given digits are correct. ii) If u= 5y / z then find relative maimum error in u, given that = y = z = and = y = z =.. Find a real root of the equation f ( ) = = by bisection 4. Find a real root of the equation 6 4= by bisection 5. Find a positive root of the equation e =, which lies between and by bisection 6. Find the root of tan + = upto two decimal places, which lies between and by bisection 7. Find a real root of the equation log = by bisection 8. Find a real root of the equation f ( ) = 5= by the method of false position upto three places of decimals. 9. Find a real root of the equation = by Regula-Falsi. Find the root of the equation e = cos using the Regula Falsi method correct to three decimal places.. Find the root of + = by iteration method, give that root lies near.. Find a real root of the equation cos = correct to three decimal places, using iteration. Find by the iteration method, the root near 8, of the equation log = 7 correct to four decimal places. 4. Find the real root of the equation 5+ = by Newton-Raphson s 5. Find by Newton s method, the root of the equation e = 4which is near to correct to three places of decimals. 6. Using Newton-Raphson method, establish the iterative formula = N n+ + n to calculate the square root of N. Hence find the square root of 8. n

3 Paper IV (Elective -) - Curriculum 7. Using Newton-Raphson method, establish the iterative formula = N n+ + n to calculate the cube root of N. Hence find the cube root of applying the Newton-Raphson formula twice. 8. Find a double root of the equation f ( ) = + = by generalized Newton s 9. Find a root of the equation e = by Ramanujan s. Find the root of the equation y ( ) = 5=, which lies between and by Muller s. Show that i) ( + )( ) = ii) E = iii) / / δ = E = E iv) = = δ v) µ = + δ 4 n. Evaluate i) E ii) E, the interval of differencing being unity.. Prove that i) u = u + u + u + u ii) u = u + u + u + u 4. Find the missing term in the following data. : 4 y: 9? Form a table of differences for the function f ( ) = + 5 7for = 45,,,,,, and continue the table to obtain f ( 6) and f ( 7 ). 6. Find the function whose first difference is , if be the interval of differencing. 7. The population of a country in the decennial census were as under. Estimate the population for the year 895. Year ( ): Population ( y )(in thousands) : From the following find y value at = y = Tan From the following table, find the number of students who obtain less than 56 marks. Marks : No. of students : Find the cubic polynomial which takes the following values. : f ( ):

4 4 Paper IV (Elective -) - Curriculum. If l represents the number of persons living at age in a life table, find as accurately as the data will permit the value of l 47. Given that l = 5, l = 49, l = 46, l = Apply Gauss forward formula to find the value of u 9 if u = 4; u = 4; u = ; u = Given that 5 = 899; 5 = 848; 5 = 8986; 5 = Show by Gauss backward formula that 56 = Use Stirling s formula to find y 8, given y = 495, y = 486, y = 476, y = 4596, y 4 = Given y = 4, y =, y = 5, y = 4,find y 5 by Bessel s formula By means of Newton s divided difference formula, find the value f () 8 and f ( 5) from the following table : : f ( ): Using the Newton s divided difference formula, find a polynomial function satisfying the following data. : 4 5 f ( ): Using Lagranges interpolation formula find y at =. : y: By Lagrange s interpolation formula, find the form of the function given by : 4 f ( ): Using Lagrange s formula, prove that y = ( y+ y ) y y y y 8 ( ) ( ). 4. Find the least square line for the data points (, ),(, 9),( 7, ),(, 5),(, 4),( 4, ),( 5, ) and ( 6, ). 4. Find the least square power function of the form y = a b for the data. i 4 yi 5 4. Fit a second degree parabola to the following data : : 4 y: 8 5 6

5 5 Paper IV (Elective -) - Curriculum 44. Using the given table, find dy and d y at = y From the following table, find the values of dy and d y at = y Find f ( 6and ) f ( 6) from the following table : f ( ) Find f ( 5) from the following table : f ( ) Find the maimum value of y, by using data given below : 4 y Find f ( 5and ) f ( 5) from the following table f ( ) d f f f f f 5. Assuming Stirling s formula, show that [ ( ) ] = [ ( + ) ( ) ] [ ( + ) ( ) ] upto third differences. 5. Evaluate I = correct to three decimal places by Trapezoidal rule with h = Evaluate ( 4 ) taking intervals by Trapezoidal rule. π / 5. Calculate an approimate value of sin by Trapezoidal rule. 54. By Simpson s rule, evaluate / with five ordinates.

6 6 Paper IV (Elective -) - Curriculum 55. Use Simpson s rule to prove that log e 7 is approimately 9587 using Find the value of the integral + by using Simpson s and 8 rule. Hence obtain the approimate value of π in each case. 57. Evaluate by Boole s rule Evaluate the integral e by Boole s Rule Evaluate the integral log, using Weddle s rule. 4 π / 6. Integrate numerically sin by Weddle s rule. 6. Solve the equations + y+ 4z = 7; + y+ z = 7; + y+ 5z = by matri inversion 6. Solve the equations + y+ z = 9, + 5y+ 7z = 5, + y z = by Cramer s rule. 6. Solve the equations y+ 4z = ; 8 y+ z = ; 4+ y z = by Gauss elimination 64. Solve the system of equations + y+ z =, + y+ z = 8, + 4y+ 9z = 6by Gauss elimination 65. Solve the equations + y+ 4z = 7; + y+ z = 7; + y+ 5z = by Factorization 66. Solve the equations + y+ z = 9, + y+ z = 6, + y+ z = 8 by LU decomposition 67. Solve the following equations by Gauss-Jacobi method : y+ z = ; y+ z = ; + y z = correct to decimals. 68. Solve the following equations by Jacobi method : + y z = 7; + y z = 8; y+ z = By Gauss-seidel iterative method solve the linear equations + + = 6; + + = 6 and + + = Solve the following system by Gauss-Seidel method : + y+ z = 9, z+ y z = 44, + y+ z =. dy 7. Solve the differential equation = + y, with y( ) =, [, ] by Taylor series epansion to obtain y for =.

7 7. Solve the equation y = + y with y = when =. 7 Paper IV (Elective -) - Curriculum 7. Given dy y = with y =, when =. Find approimately the value of y for = by Picard s y + dy 74. Given = + y, y() =, compute y = ( ) by Euler s method taking h =. 75. Using Euler s method, compute y( 5 ) for differential equation dy 76. Determine the value of y when = given that y( ) = and y = + y = y, with y = when = Solve dy = y with y( ) = and h = on the interval [, ] using Runge-Kutta fourth order 78. Given dy = y with y( ) =, find y( ) and y( ) Kutta fourth order correct to four decimal places, using Runge- 79. Using Milne method obtain y ( 4) from the given table of tabulated values and y = y y 5 4 f Solve y = e yat = 4and = 5 by Milne s method, given their values at the four points. y 4 9 _

8 Paper IV (Elective -) - Curriculum ACHARYA NAGARJUNA UNIVERSITY B.A / B.Sc. DEGREE EXAMINATION, THEY MODEL PAPER (Eamination at the end of third year, for - and onwards) MATHEMATICS PAPER - IV (ELECTIVE - ) NUMERICAL ANALYSIS Time : Hours Ma. Marks : SECTION - A (6 X 6 = 6 Marks) Answer any SIX questions. Each question carries 6 marks. Describe a general error formula.. Using Newton-Raphson method, establish the iterative formula = N n+ + n to calculate n the cube root of N.. Given u =, u =, u = 8, u =, u4 =, u5 = 8 ; find 5 u. 4. Given y = 4, y =, y = 5, y = 4,find y 5 by Bessel s formula Fit a second degree parabola to the following data : 6. Evaluate I = : 4 y: correct to three decimal places by Trapezoidal rule with h = Solve the equations 5 = 4; = ; = 5 by using Gauss s elimination dy y 8. Given = with y =, when =. Find approimately the value of y for = by y + Picard s 9.(a) 8 SECTION - B (4 X 6 = 64 Marks) Answer ALL questions. Each question carries 6 marks Find a positive root of the equation e =, which lies between and by using bisection Find the smallest root of the equation f ( ) = 6 + 6= by Ramanujan s.(a) Find a real root of the equation f ( ) = 5= by the method of false position upto three places of decimals. Find the root of the equation = cos + correct to three decimal places by fied point iteration

9 .(a) Paper IV (Elective -) - Curriculum State and prove Newton-Gregory formula for farword interpolation with equal intervals. Using the Newton s divided difference formula, find a polynomial function satisfying the following data : : 4 5 f ( ) : (a) State and prove Lagrange s interpolation formula. Using Gauss forward formula find u from the given data : u = 4 5, u5 = 674, u = 57, u = 74, u = 89, u = (a) Find f ( 5) and f ( 5) from the following table f ( ) rule. Hence obtain the approi- Find the value of the integral + by using Simpson s and 8 mate value of π in each case. 4.(a) Find the maimum value of y, by using data given below : 4 y Evaluate the integral log, using Weddle s rule. 4 5.(a) Solve the equations + y+ 4z = 7; + y+ z = 7; + y+ 5z = by Factorization Determine the value of y when = given that y( ) = and y = + y. 6.(a) Solve the following equations by Gauss-Jacobi method : y+ z = ; y+ z = ; + y z = correct to decimals. Given dy = y with y( ) =, find y( ) and y( ) correct to four decimal places using RK fourth order h h h

10 Paper IV (Elective -) - Curriculum AACHARYA NAGARJUNA UNIVERSITY B.A / B.Sc. DEGREE EXAMINATION, PRACTICAL MODEL PAPER (Practical eamination at the end of third year, for - and onwards) MATHEMATICS PAPER - IV (ELECTIVE - ) NUMERICAL ANALYSIS Time : Hours Ma. Marks : Answer ALL questions. Each question carries 7 marks. 4 7 = M (a) Find a positive root of the equation e =, which lies between and by bisection Using Newton-Raphson method, establish the iterative formula = N n+ + n to calculate the square root of N. Hence find the square root of 8. (a) Find the missing term in the following data. : 4 y: 9? 8 If l represents the number of persons living at age in a life table, find as accurately as the data will permit the value of l 47. Given that l = 5, l = 49, l = 46, l = (a) Find f ( 6and ) f ( 6) from the following table : f ( ) Find the value of the integral + by using Simpson s and rule. Hence obtain the approimate 8 value of π in each case. 4 (a) Solve the system of equations + y+ z =, + y+ z = 8, + 4y+ 9z = 6by Gauss elimination dy y Given = with y =, when =. Find approimately the value of y for = by y + Picard s n Written eam : Marks For record : Marks For viva-voce : Marks Total marks : 5 Marks