Basic Numerical Methods

by

Al Bernstein

Signal Science, LLC

www.signalscience.net

 

Numerical Differentiation

 

A numerical derivative can be derived using the following approach:

 

The Taylor Series for about the point is given by equation (1)

††††††††††††† (1)

If we quantize by setting for ≡ an integer and

then ††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (2)

 

We set then

 

†††††††††††††††††††††††††††††††† (3)

†††††††††††††††††††††††††††† (4)

 

Subtracting equation (4) from equation (3) gives

 

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (5)

 

Equation (5) can be rewritten as

 

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (6)

 

††††† For a general quantized point , equation (6) gives

 

or

 

 

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (7)

 

Equation (7) will work when . If , then is implicitly modeled by a second order polynomial. Higher order derivatives can be derived in the same way.

 

Numerical Integration

 

We start with the Taylor series using equation (1) with , use equation (7) for and use equation (8) for.

 

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (8)

 

Then††††††††††††††††††††††††††††††††††††††††††† (9)

 

 

 

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (10)

 

which is Simpsonís Rule. Higher order integration methods can be derived in a similar way. For a general discussion on this approach for deriving numerical methods see 1

 

 

Electronic Implementation

 

This section shows how to implement a numerical algorithm in a circuit design. For example if we start with Simpsonís Rule equation (10), we can first modify it to be causal. This means that we can shift the indices so that at time t=0 we use the index n (i.e. the system is not operating before time t=0). We do this by changing the indexing (delaying by 1 sample) in equation (10).

 

 

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† (11)

 

 

The block diagram is shown below in Figure 1

 

†††††††††††† Block Diagram

 

 

 

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† Figure 1

 

 

This is shown as a system that can be implemented in the digital domain. The z-1 and z-2 represent delays of 1 and 2 samples respectively. They could be implemented with digital electronics, analog electronics or theoretically other mechanical or electrical systems which provide a delay equivalent to one sample in time and gain elements.

 

References

 

1 Steve E. Koonin, Computational Physics, Addison-Wesley, pp 2-6 (1986)