MCS 471 Practice Problems for Interpolation with Some Answers
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CAUTION: These problems are just for practice and are not to be handed in.
         Some of these old exam problems and answers may not be relevant to
         your current course.   Caveat Usor!

      In the following problems, use "EXAM PRECISION: chop to "d" 
significant digits only when you write an intermediate or final answer down
and continue calculations with those numbers recorded.
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     Background Reading: Gerald and Wheatley, Chapter 3 Topics

3.2 Lagrangian Polynomials
3.3 Divided Differences
3.4 Evenly Spaced Data (Newton-Gregory Polynomial Form)

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     0. For problems in the Gerald and Wheatley text, 
try chapter 3 exercises (pp. 300-307, 5th ed.)
     #1,2,3,4,11,12,13,14,24,26,28,29,30,31,32.
 
     1.   Find  an  approximation  to the value f(0.25) using an interpolating
polynomial passing through the   3   points:    (x(i),f(x(i))=  (0.1,0.07972),
(0.2,0.1591), (0.3,0.2376),   using  d=4 digit chopping.  
     (Ans.:  0.1984)
 
     2.   Inverse  quadratic  interpolation:   Suppose  the  inverse  function
x=g(y)=y*exp(y)  is given instead of the direct function  y=f(y).
     (a)  Using    y1=1,   y2=2,   y3=3     find    x1,    x2,    x3     (i.e.
xi=x(i)=y(i)*exp(y(i)))  to d=3 digits.  (note:  exp(x) on some calculators is
keyed as "x,expx" while it is "x,inv,lnx" on others.)
     (b)  Use the (inverse quadratic) interpolating  polynominal  p2(x)  through
(xi,yi)=(x(i),y(i)) for i=1 to 3 from  (a) to approximate  y=f(10)  (i.e.  the
solution    of    10=y*exp(y)     )     to     d=3    digits.    
     (Ans.:    (a) x-vector=(2.71,14.7,60.2);   
               (b)  f(10) =1.64+or-0.01 depending on the chops.)
 
     3.  Compute  pn(n+1),  where pn(x) = "p-sub-n-of-x",  for  n=0,1,2,3,4 if
pn(k)=k/(k+1)  for k=0 to n.   
     (Partial Ans.:  p2(3)=0.500).
 
     4.   How  many multiplications/divisions and additions/subtractions as a
function of  n  are needed to compute  the sum of x**x from k=0 to n  by   (a)
direct   (slow)   sum   of   powers  counting  exponentiations  as  equivalent
multiplications and  (b)  by  Newton's  (Horner's)  rule  of  fast  polynomial
evaluation?.   (Hint:  sum of k for  k=1 to m is m*(m+1)/2).  
     (Ans.:  mults. = n; adds. = n)
 
     5.   Approximate   v(2.738)  using an interpolatory polynomial that best
fits the data,
          x:  2.600  2.700  2.800  2.900
          v:  6.815  7.944  9.299  10.93        .
Here  d=4.   
     (Final Ans.: v(2.738)=(4c) 8.429 or 8.428 with extra intermediate chops).
 
     6.   Approximate j3(5.137) by interpolating the data:
          x:  5.000  5.200  5.600
      j3(x):  .3648  .3265  .2298
Here d =4.  
     (Final Ans.:  j3(5.137)=(4c) .3392 or .3390 with extra chops).

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