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Math251_Final_Spring2007-08

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Unformatted text preview: AMERICAN UNIVERSITY OF BEIRUT  Faculty of Arts and Sciences ‐ Mathematics Department.      MATH 251  Final Exam (2hrs)  SPRING 2007 – 2008              STUDENT NAME                                                                                                                          .    ID NUMBER                                                                                                                                   .          Problem 1                                      /   19      .  Problem 2                                       /   22    .  Problem 3                                       /   15    .  Problem 4                                       /   27    .  Problem 5                                       /   17    .    TOTAL                                            / 100    1  1 .  (a) ‐ Loss of significant figures may result in the computation of the functions     for certain values of x. Explain and specify these values then propose alternative functions  that would remedy the loss of significant figures.  Note that :                                      (b)  ‐  Let  f(x)  =  g(x)  –  h(x).  Using  the  results  in  (a),  find  a  way  to  calculate  accurately  f(x)  for  values of x close to zero. Find then  .                                                                        (3 points)              (10 points)   2  (c)  –Solving  the  quadratic  equation    ,  using  a  machine  that  carries  only  8  decimal  digits  causes  a  problem.  Investigate  this  example,  observe  the  difficulty,  and  propose a remedy to approximate the 2 solutions  .                                            (6 points)                           2 .  The Reciprocal of the cubic root of a positive number    (i.e.  an iterative formula that does not use division by the iterate     i  – Establish this formula by applying Newton’s method to some appropriate function f(x).    Determine  also  any  necessary  restrictions  on  the  choice  of  the  initial  value    of  the  iterative procedure.                                                                                                                    (9  points)                            3   ), can be computed by  ii – Let a = 2. Plot the graph of the function f, and give a graphic justification of the necessity  of this restriction on  .                                                                                                                 (4 points)                                iii  –  For  a  suitable  choice  of                              4  ,  approximate    by  computing  the  first  3  iterates   by Newton’s formula.                                                                                             (5 points)  iv – Prove that Newton’s method is quadratic.                                                                                                                       (4 points)                          3.  Based  on  a  set  of  data   i. Give  the  formula  for  the  composite  Trapezoid  Rule  T(h)  to  approximate  the  integral  I, and derive the expression of  I – T(h) in terms of the powers of h.  (8 points)                ,  where  ,  and  nh=2,  let    5  ii. Obtain  an  upper  bound  on  the  absolute  error  when  we  compute    by   means  of the composite Trapezoid Rule using  51  equally spaced points.  (7 points)                            4. Based on the following table of values for a function f(x)     i 0 1 2 3 4 5 6 7 8 x ( i) 0 .0 0 0 0 .1 2 5 0 .2 5 0 0 .3 7 5 0 .5 0 0 0 .6 2 5 0 .7 5 0 0 .8 7 5 1 .0 0 0 f ( x ( i) ) 1 .0 0 0 0 0 0 0 0 .9 8 4 4 9 6 4 0 .9 3 9 4 1 3 1 0 .8 6 8 8 1 5 1 0 .7 7 8 8 0 0 8 0 .6 7 6 6 3 3 8 0 .5 6 9 7 8 2 8 0 .4 6 5 0 4 3 2 0 .3 6 7 8 7 9 4     6    Consider the approximation the integral  composite trapezoid rule T(h).  i. Derive  Romberg  subsequent  integration  formulae,  and obtain expressions of the errors  ,   and  ,  .  (12 points)                                              7  , using Romberg integration based on the  ii. Prove that the Simpson’s composite rule S(h) is equivalent to  R1 (h).  (5 points)                        iii. Using  the  table  and  the  results  above,  fill  the  empty  slots  in  the  following  table.     (10 points)       h  1  0.5  0.25  0.125  M(h)              T(h)                        8  R1(h)              R2(h)              R3(h)                                                                      9  5. (a) – Fill in the missing statements in the following Matlab program  (8 points)  function p = Neville(X, Y, x)  % Input arguments: a vector X representing the distinct xi data  %                                  : a vector Y representing the corresponding yi = f(xi )data  %                                  : x is a vector of numbers at which we want to interpolate  % Output arguments: p = p(x) the value of the interpolation polynomial at the X data  % Assume length (X) = length (Y) > 0  % Place the base statement of the recursion  n = length(X);  if n = 1       ……………………………………………………………………………………………………………………….  else  % Place the statements of the recursion        X1 = X(1 : n­1) ; Y1 = Y(1 : n­1);        X2 = X(2 : n) ; Y2 = Y(2 : n);        p2 = ………………………………………………………………………………………………………………… ;        p =    ………………………………………………………………………………………………………………….;  end                                p1 = ………………………………………………………………………………………………………………… ;                                       10  (b)  –  Show  the  various  steps  of  the  execution  process  of  this  program,  when  n  =  3  ,  in  the  form of a diagram:                                                                                                                           (9 points)        p1 = ……….                                                                                           p2 = ……….                                                        11            12  ...
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