Answer to Quiz 7 Fri 4 Mar 2005

1. Consider (0,e^0), (1,e^1), (2,e^2) as given interpolation data.
   Give the polynomial interpolating through these 3 points in the form of Lagrange. Evaluate this form at 1.5.

   Answer:

             (x-1)*(x-2)       (x-0)*(x-2)       (x-0)*(x-1)
      p(x) = ----------- e^0 + ----------- e^1 + ----------- e^2
             (0-1)*(0-2)       (1-0)*(1-2)       (2-0)*(2-1)
   
             1 1     1         1  3     1       1 3 1
           = - - (-1)- e^0 + ---- - (-1)- e^1 + - - - e^2
   	  2 2     2       (-1) 2     2       2 2 2
   
               1   3     3
           = - - + - e + - e^2  is approximately 4.685E+0
   	    8   4     8
   

2. Set up the linear system for the natural spline for sin(x) over [0,pi], using 4 intervals of equal length. Do not solve the system.

   Answer:

      We divide [0,pi] in 4 equal subintervals: h = pi/4.  
   
      The unknowns in the linear system are S1,S2,S3,
      as S0 = 0 and S4 = 0 for a natural spline.
   
      [ pi     pi/4   0    ] [ S1 ]     [ sin[pi/4,pi/2]   - sin[0,pi/4]      ]
      [ pi/4   pi     pi/4 ] [ S2 ] = 6 [ sin[pi/2,3*pi/4] - sin[pi/4,pi/2]   ]
      [  0     pi/4   pi   ] [ S3 ]     [ sin[3*pi/4,pi]   - sin[pi/2,3*pi/4] ]
   
                          sin(pi/4) - sin(0)          4  ( sqrt(2)     )
      sin[0,pi/4]      = --------------------      = --- ( ------- - 0 )
                              pi/4  -  0              pi (    2        )
   
                          sin(pi/2) - sin(pi/4)       4  (     sqrt(2) )
      sin[pi/4,pi/2]   = -----------------------   = --- ( 1 - ------- )
                              pi/2  -  pi/4           pi (        2    )
   
                          sin(3*pi/4) - sin(pi/2)     4  ( sqrt(2)     )
      sin[pi/2,3*pi/4] = ------------------------- = --- ( ------- - 1 )
                              3*pi/4  -  pi/2         pi (    2        )
   
                          sin(pi) - sin(3*pi/4)       4  (     sqrt(2) )
      sin[3*pi/4,pi]   = -----------------------   = --- ( 0 - ------- )
                              pi  -  3*pi/4           pi (        2    )
   
      So the righthand side vector equals
   
         24  [ 1 - sqrt(2) ]
        ---- [ sqrt(2) - 2 ]
         pi  [ 1 - sqrt(2) ]