1MA6000 “seen” in-class test preparation questionsAssignment guidelines:Assignment brief:The second MA6000 in-class test will be on Monday 7th March from approx. 15:00-16:30.The test will be closed-book (although calculators will be permitted) and is based on the tenquestions below. To prepare you should practice solutions to all of these questions so thaton the day you will be able to quickly reproduce them, showing that you understand thequestions and the techniques used to solve them, which hopefully means you will be ableto tackle unfamiliar problems in the future. Please note that the test may include some ofthese questions truncated or broken into parts.Module Learning Outcomes Assessed:This assessment (test 2) is designed to assess your ability in the following three modulelearning outcomes:? use finite difference methods for solving PDEs and understand limitations ofnumerical methods.? find analytically the extrema of functions of two or more variables, with and withoutconstraints,? apply appropriate numerical methods to solve unconstrained and constrainedoptimisation problems,Submission deadline:In-class test on Monday 7th March.Feedback Date:You will receive initial feedback electronically in the form of solutions to the questions soonafter the test (by 21th March 2016) and written feedback on your test papers by 28st March2016. Please see Drs Lau and Chen if you would like to ask any questions about thefeedback.This is preparatory work for your second coursework. Your coursework mark will bebased entirely on the in-class timed submission. Please do not hand in any of yourpreparatory work.You can also seek support from MathsAid and Drs Lau and Chen on these questions,however the MathsAid team will have been asked to help you solve problems rather thangive you solutions, so you’re still expected to engage with the mathematics, but please doask questions if you are stuck on a problem.Marking scheme and feedback:You are expected to be able to carry out all necessary working to justify your resultsand discuss them. Marks shown are approximate only – for guidance – 50 markscorresponds roughly with work that should be possible to be completed within 60minutes.More details of how the marks are broken-down will be on the test paper feedback sheet.21. Consider the following elliptic partial differential equation:U U 1000(x y), (x, y) D.xx ? yy? ? ?The region D, the values ofU(x, y)around the boundary and the mesh imposed on theregion are specified in the figure below. You may assume that the grid spacing in bothdirections is 0.1 and that the region is positioned so that the bottom edge lies on x-axis andthe left-hand tip touches the y-axis (thus, for example, mesh point 1 has coordinates (0.2,0.1).Use the central difference method to approximate the above partial differential equationand show that the resulting set of linear equations can be written in the form:?????????????????????????????????????????????????????????5631630 0 0 1 41 0 1 4 10 0 4 1 01 4 0 0 04 1 0 1 054321vvvvv,(15 marks)wherevk, k ?1,…,5are the approximations to the function values at the mesh pointsindicated on the figure. Use Gaussian elimination with back-substitution to solve thelinear equations and work to four decimal places accuracy.(10 marks)Continued…32. Use the Forward Difference method to approximate the solution to the followingparabolic partial differential equation at t=0.1, using?x ?1and?t ? 0.1:0, 0 4, 04222? ? ? ??????x txUtU?U(0,t) ?U(4,t) ? 0, t ? 0,), 0 4.4)(sin4U(x,0) ? (1? 2cos x x ? x ?? ?Work to 4 decimal places accuracy.(15 marks)If the analytical solution to the above problem is:,4sin2( , ) sin 4 x e xt U x t et? ? ????compare it to your numerical solution at the grid points, calculate the errors andcomment.(10 marks)Continued…43. By using the central differences, show that an explicit numerical scheme for solvingthe following hyperbolic equation??2??????2 – ????2??????2 = 0, ?? = 0, 0 = ?? = 1 (D is a constant) (3.1)subject to:??(0,??) = ??(1,??) = 0, ?? = 0,??(??, 0) = ??(??), 0 = ?? = 1 ,????????(??, 0) = ??(??), 0 = ?? = 1 (3.2)is:????,??+1 = ????2 ????-1,?? + 2(1 – ????2)????,?? + ????2 ????+1,?? – ????,??-1 (3.3)where????,?? = ??(????,????)?? =??h, and h and k are step sizes in the x and t directions respectively.(6 marks)The explicit formula (3.3) requires starting values over two time levels. The first set ofstarting values ????,0 = ??(????). We also need ????,1 , which we obtain from ????????(??, 0) = ??(??)by approximating the derivatives using differences.By approximating ????????(??, 0) = ??(??) using central differences, show that????,1 =12(????2 ????-1,0 + 2(1 – ????2)????,0 + ????2 ????+1,0) + ?? ??(????) (3.4)(5 marks)Apply the explicit scheme (3.3), solve the hyperbolic equation (3.1) with D = 1,h=k=0.2, and ??(??) = sin(????) , ??(??) = 0, for 0 = ?? = 1 ?????? 0 = ?? = 0.8 .Work to 3 decimal places accuracy.(14 marks)Continued…54. Investigate the stationary points of the functionf (x, y) x y 6x 12y3 3? ? ? ?and determine their nature.(6 marks)5. The Golden Section Search is to be applied to a unimodal function to find the minimumin the interval [0,2]. If we require the error in the calculated optimum to be not greaterthan 0.005 how many iteration steps ?? do we need to perform? (Here you may assume???? = ?? = 0.382 is the value of the ratio for symmetric interval sub-division.)(4 marks)6. The Fibonacci Search is to be applied to a unimodal function to find the minimum inthe interval [4,6]. If we require the error in the calculated optimum to be not greaterthan 0.01 how many terms of the Fibonacci sequence do we need to use?(4 marks)7. Use Hooke and Jeeves’ method to minimize2 2f (x, y) ? x ? y ? 2x ? 2xy ? y .TakeTb (0,0)1 ?as the initial point andh ?1as the initial step length. Work to fourdecimal place accuracy and stop when the step lengthh ? 0.25.(5 marks)Continued…68. Use Nelder and Mead’s method on function( , ) 2 36 4 2 2 4f x y ? x ? x y ? y ? . Start from thesimplex defined by (1,2), (2,2) and (1,3) and terminate the procedure either after seveniterations or when you need to reduce the size of the simplex by halving the distances fromlx . Do NOT perform more than 7 iterations.(20 marks)You may assume the following for Nelder and Mead’s method:Ifhx ,lxandsxdenote the vertices with the highest, the lowest and the second highestfunction value respectively, then the centroidcxis given by:( )21c l sx ? x ? xReflection:c hx ? 2x ? x 0Expansion:cx ? x ? x 00 2 0Contraction: If( ) ( )0 hf x ? f xthen( )2100 0 cx ? x ? xIf( ) ( )0 hf x ? f xthen( )2100 h cx ? x ? x(If a new point gives a function value equal to a current value then the new point istreated as having a higher function value.)9. State, in vector form, first three terms of the Taylor series for a function of three variablesf (x) ? f (x, y,z)about the point( , , )0 0 0 0x ? x y z(2 marks)Hence, state the necessary conditions for0xto be an optimum off (x)and the secondorder sufficient condition for0xto be a local maximum off (x) .(3 marks)Continued…710. For the function1 2222f (x) ? x1 ? 2x ? 4x ? 4xby using the method of steepest descentapplied with initial guessTx (0,0)(0)?,generate the sequence(1)xand(2)x(5 marks)Hence deduce the minimum off (x)(5 marks)The method of Steepest Descent is used to find a local minimum of a well behavedfunction of several variablesf (x). Show that the method moves in orthogonal steps,i.e. if( )1 k k kkk k kx ? x ? g ? x ? ?f x ? ? ?then the vectork kx ? x ?1is orthogonal to thevectork?2 ? k?1x x. Here( )kkg ? ?f xis the gradient vector off (x)and?kminimises( )kf xk ? ?g .(10 marks)The method is now applied to the function2 2f (x) ? x ? y ? 2x ? 2xy ? ywithTx (0,0) 0 ?as the starting point andh ? 0.75as the initial step-length. The minimumof( )kf xk ? ?gis calculated using Powell’s quadratic interpolation method. Powell’squadratic interpolation method states that given three points( , )l l ? f , ( , )n n ? f ,( , )u u ? f, the turning point( , ) m m ? fof the quadratic through these points, can becalculated from( ) ( ) ( )( )( )( )212 l n u n u l u l nl u l u u n n lmf f ff f? ?? ? ? ?? ? ? ??? ? ?? ? ???? ? ?? ?and that( , ) m m ? fis a minimum if0( )( )( )( ) ( ) ( )?? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?n u u l l nl n u n u l u l nf f f.Illustrate how Powell’s quadratic interpolation method is applied by performing twoiterations. Work to four decimal places accuracy.(15 marks)END OF COURSEWORK