Question. Use Bessel’s Interpolation formula find y25? given y20=24, y24=32, y28=35 and y32=40

Question. Use Bessel’s Interpolation formula find y25?  

given y20=24, y24=32, y28=35 and y32=40 

Sol:

  u       x       y         ∆ f(x)        ∆²f(x)        ∆³f( x)

-1      20     24

                                  8

 0      24     32                          -5

                                  3                                 7

 1     28      35                           2

                                  5

2      32      40

     x = a + uh

25  = 24 + u(4)

   u = 25 – 24

             4

 [u = 0.25]

f(u) = { 32 + 35 }  + (0.25 – ½ ) 3

                 2                                2

      

+ 0.25(0.25 – 1) { -5 + 2 }

       2 × 1                 2

+  0.25(0.25 – 1)(0.25 – ½ ) (7)

                3× 2× 1

= 33.5 – 0.75 + 0.25(-0.75) (-3)

                              2           2

     

+ 0.25(-0.75) (-0.25)(7)

                 6

= 33.5 – 0.75 + 0.140625 + 0.05468

= 32.9453