## The Method of Undetermined Coefficients

Suppose we want to estimate an integral of a function between two limits with a weighted sum of values of the function at different points:

One way of doing this is to ensure that forcoefficients, the approximation is exact for a polynomial of degree

For example, Simpson's rule in it's simplest form uses three coefficients. The x values below are easily space.

Suppose

Takein succession.

We solve the simultaneous equations

(2)

(3)

(4)

(3)+(4) givethen (3) givesand finally (1) givesSubstitution of these values into (1) gives the general form with arbitraru limits for the simplest form of Simpson's rule with only three coefficients.