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M251_Quiz2_Spring20072008

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Unformatted text preview: AMERICAN UNIVERSITY OF BEIRUT  Faculty of Arts and Sciences ‐ Mathematics Department.      Test II ­ MATH 251  Closed Book – 75 mn  SPRING 2007 – 2008              STUDENT NAME                                                                                                                          .    ID NUMBER                                                                                                                                   .          Problem 1                                      /   14      .  Problem 2                                       /   14    .  Problem 3                                       /   22    .    TOTAL                                            / 50        1  TABLE 1: Problems 1 and 2    Consider  the  following  table  of  data  for  the  function  f(x).  All  computations  shall  be  carried  out with 8 significant figures.                                    2  #  1  ­  a  –  Use  the  Central  Difference  formula  to  approximate  f  ‘(0.5).  Improve  this  result  using  Richardson’s  extrapolation  of  the  1st  and  2nd  orders.  For  that  purpose,  give  the  formulae that provide  , then fill out the table that follows.                       (10 points)                                    3  b  –Using  table  1,  calculate  the  third  derivative  f  “’(1.000)  ,  using  the  Backward  Difference  Approximation.                      (4 points)  f “’ (x)      f ‘” (1.000)            #  2  –  a‐  Using  table  1,  write  Newton’s  interpolating  polynomial  of  degree  3,  that  would  best approximate f(0.4). Find  f(0.4).                                                                                           (4 points)                                  4  b – Find the equation of the quadratic spline s(x), on the interval [0.125, 0.5], then draw  its graph. Draw clearly the tangent to the curve at each node. (Let z1= s’ (x1) = 0)      (10 points)                                                    5  # 3 ­ Consider the following 4 by 4 square matrix:    a­ Apply  on  this  matrix  Gauss  Elimination  with  the  scaled  partial  pivoting  strategy,  showing  the  status  of  the  4  by  4  matrix  after  each  reduction.  (Each  pivot  row  and  the  corresponding  multipliers  should  be  identified  and  circled).    Specify  the  index  vector IV after each reduction.                                                                                 (10 points)                                                                                                             Reduction 1:  Modified matrix A including multipliers         scales                                        6                                                                                      7  b­ Specify the elements of the matrices P, L and U that satisfy the LU decomposition of the  matrix A.                                                                                                                                    (4 points)                  c­ Use the matrices P, L and U, to find the determinant of A.                                                                                                                                                 (2 points)                            8  d­ Use the matrices P, L and U, to find the  2nd  column (ONLY) of the inverse of A.                                     (6 points)                9  ...
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