When using interpolating polynomials to approximate a function, are higher degree polynomials guaranteed to give more accuracy?

Explain what the Runge effect is.

The following results are determined when constructing Newton-Gregory polynomials of various degrees to approximate f(4.8).

1. Calculate the missing entries in the table.
2. Determine the optimal interpolation degree.
3. State the (presumed) best approximation of f(4.8).

                                                             P[n](4.8)      
                 n-th             n-th       n-th Order      Polynomial     Polynomial
    n       Binomial Coeff     Difference       Term        Interpolation    Degree n
  ------------------------------------------------------------------------------------
    0     B(0) =   1.0000000    20.00000     20.00000000     20.00000000         0
    1     B(1) =   1.2500000     1.70000      2.125          22.12500000         1
    2     B(2) =   0.1562500     0.40000                                         2
    3     B(3) =  -0.0390625    -0.70000      0.02734375                         3
    4     B(4) =   0.0170898    -0.10000                     22.21313477         4
    5     B(5) =  -0.0093994     1.00000                                         5
    6     B(6) =   0.0058746    -1.50000                     22.19492347         6
 

ANSWERS:

• The optimal interpolation degree is 3rd degree.
• The (presumed) best approximation of f(4.8) is 22.21484375.
• Note:   If you were to check my binomial coefficients, you would see that they have some truncation error.

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