Introduction to Numerical Analysis I
Introduction to Numerical Analysis I MA 427
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This 5 page Class Notes was uploaded by Braeden Lind on Thursday October 15, 2015. The Class Notes belongs to MA 427 at North Carolina State University taught by K. Sivaramakr in Fall. Since its upload, it has received 6 views. For similar materials see /class/223704/ma-427-north-carolina-state-university in Mathematics (M) at North Carolina State University.
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Date Created: 10/15/15
MA 427 001 Introduction to Numerical Analysis 1 Final Exam Review lnstructor Dr K0er Sivammakm shnan INSTRUCTIONS The questions on the nal will be similar to these review problems Work out each problem in suf cient detail If you have any questions7 please send me an email 1 will post the solutions to selected problems on the course webpage on Friday 1 Exercise Set 1117 Page 6487 Problem 1 Do part 1a only 2 Consider the following second order ODE dzy dy 7 7 lt lt 1 d2xdy 2x 0x71y0 1andy1 1 a Using the central difference formulas f391 fWiel 21 W f i for approximating the 1st and 2nd order derivatives in 1 respectively7 discretize the ODE and write it in a form suitable for solution with the nite difference method b Write down the tridiagonal system of linear equations for the nite difference method in matrix form for the following two cases 1 h 05 and 2 h 0 Write down your MATLAB commands and functions to solve 1 using MAT LAB7s boundary value solver b14740 3 Consider the following boundary value problem 22 d 4 dy wayby 07039631 I007 y11 2 where a and b are constants r U 03 5 the central difference formula for b the 2nd order forward difference a Discretize the second order ODE using a approximating the 2nd derivative in 2 and formula 73f9ci 4 f AAV 141 142 m h D d for approximating the 1st derivative in the boundary condition 0 What s is the system of nonlinear equations when h 05 We will solve the nonlinear system of equations when h 05 using Newton7s method from Section 102 of Burden and Faires Set up the linear system of equations that are solved in each iteration of Newton7s method You do not have solve these equations Write down your MATLAB commands and functions to solve 2 using MAT LAP s boundary value solver b14740 A O V Consider the following rst order ODE dy dy7x2 0 z 187 y01 a Solve by hand with Euler7s explicit method using h 06 b Solve by hand with modi ed Euler7s method using h 06 c Solve by hand using the 4th order Runge Kutta method using h 06 Write the following system of two 2nd order ODEs as a system of four 1st order ODEs dzz 7 dx 7 ydy dmz W 7 7 ma dt PEE E E where 9 m7 and y are given constants Consider the following system of two ODEs dx a ytz 09312 z01 and y005 4 t dy 97 dt a Solve by hand with Euler7s explicit method using h 04 b Solve by hand using the 4th order Runge Kutta method using h 04 Write down your MATLAB commands and functions to solve the following initial value problems in Burden and Faires using MATLAP s ODE solver 0d645 a Exercise Set 597 Page 3237 Problem 2c b Exercise Set 597 Page 3237 Problem 3d 8 p A particular nite difference formula for the rst derivative of a function is f i 6h where the points xi i17i27 and 133 are all equally spaced with step size h What is the order of the trunaction error Give reasons for your answer The centroid of a circular sector is given by 27quot sin0 36 39 Find 6 such that i using a Use bisection method with a 1 and b 2 b Use secant method with 1 1 and x2 1 c Use Newton7s method with 1 1 Carry out the rst two iterations in each case Consider the system of nonlinear equations 4x2 7 y3 3x3 4y2 Use Newton7s method to solve the nonlinear system with z 1 and y 1 Carry out the rst two iterations MACSC 427 001 Introduction to Numerical Analysis l Review questions for midterm exam Instructor Dr K0er Sivammakm shnan INSTRUCTIONS The questions on the midterm exam will similar to these review problems Work out each problem in suf cient detail with your classmates If you have any questions7 please send me an email I will post the solutions to selected problems on the course webpage next Friday 1 F 9 7 U 03 Floating point representation What is the gap between 2 and the rst lEEE single format number larger than 2 What is the gap between 1024 and the rst lEEE single format number larger than 10247 Finite difference formulas for numerical differentiation a Show that the truncation error in the following 3 point forward difference formula fl 2h is 0h2 b Show that the truncation error in the following 3 point forward difference formula is 0h Elements of numerical integration Derive the following formula for Simpson7s rule fltzgtdx where so 17 2 b7 and h I15 1 7 i gum 4M1 mg 12 3f4 2 7 gm bia accuracy of Simpson7s rule Why Show all your steps What is the degree of Composite numerical integration Exercise Set 44 Page 2047 Problem 13b Order of convergence Exercise Set 247 Page 827 Problems 67 87 and 11 Newton7s divided differences Exercise Set 327 Page 1297 Problem 19 7 00 Polynomial interpolation Find the Lagrange and Newton interpolating polynomials for the following data z f96 i2 0 0 1 1 71 Write both polynomials in the form abxcx2 in order to verify that they are identical Halley7s method for solving the nonlinear equation f 0 uses the following iterate formula f x f xn JoM in the n iteration Show that this formula results when Newton7s method is applied to the function 996 zn1 ni f z 39