1. Numerical
Differentiation
,I. INTRODUCTION
This chapter deals with numerical approximations of
derivatives. The first questions that comes up to
mind is: why do we need to approximate derivatives
at all? After all, we do know how to analytically
differentiate every function. Nevertheless, there are
several reasons as of why we still need to
approximate derivatives:
2
,I. INTRODUCTION
▪ Even if there exists an underlying function that we need to
differentiate, we might know its values only at a sampled data
set without knowing the function itself.
▪ There are some cases where it may not be obvious that an
underlying function exists and all that we have is a discrete
data set. We may still be interested in studying changes in the
data, which are related, of course, to derivatives.
3
, I. INTRODUCTION
▪ There are times in which exact formulas are available but they are very
complicated to the point that an exact computation of the derivative requires a
lot of function evaluations. It might be significantly simpler to approximate the
derivative instead of computing its exact value.
▪ When approximating solutions to ordinary (or partial) differential equations, we
typically represent the solution as a discrete approximation that is defined on a
grid. Since we then have to evaluate derivatives at the grid points, we need to
be able to come up with methods for approximating the derivatives at these
points, and again, this will typically be done using only values that are defined
on a lattice. The underlying function itself (which in this cased is the solution of
the equation) is unknown.
4
Differentiation
,I. INTRODUCTION
This chapter deals with numerical approximations of
derivatives. The first questions that comes up to
mind is: why do we need to approximate derivatives
at all? After all, we do know how to analytically
differentiate every function. Nevertheless, there are
several reasons as of why we still need to
approximate derivatives:
2
,I. INTRODUCTION
▪ Even if there exists an underlying function that we need to
differentiate, we might know its values only at a sampled data
set without knowing the function itself.
▪ There are some cases where it may not be obvious that an
underlying function exists and all that we have is a discrete
data set. We may still be interested in studying changes in the
data, which are related, of course, to derivatives.
3
, I. INTRODUCTION
▪ There are times in which exact formulas are available but they are very
complicated to the point that an exact computation of the derivative requires a
lot of function evaluations. It might be significantly simpler to approximate the
derivative instead of computing its exact value.
▪ When approximating solutions to ordinary (or partial) differential equations, we
typically represent the solution as a discrete approximation that is defined on a
grid. Since we then have to evaluate derivatives at the grid points, we need to
be able to come up with methods for approximating the derivatives at these
points, and again, this will typically be done using only values that are defined
on a lattice. The underlying function itself (which in this cased is the solution of
the equation) is unknown.
4
