Structs

There are four main UnivariateFunction structs that are part of this package. These are:

• Undefined_Function - An undefined function behaves similarly to "missing" in Julia. Whenever anything is added/multiplied/etc with an undefined function the result is undefined. The integral and derivative of an undefined function is undefined. If an undefined function is evaluated it will return a missing.
• PE_Function - This is the basic function type. It has a form of

\[a \exp(b(x-base_)) (x-base)^d\]

• Sum_Of_Functions - This is an array of PE_Functions. Note that by adding PE_Functions we can replicate any given polynomial. Hence from Weierstrass' approximation theorem we can approximate any continuous function on a bounded domain to any desired level of accuracy (whether this is practical in numerical computing depends on the function being approximated).
• Piecewise_Function - This defines a different UnivariateFunction for each part of the x domain.

It is possible to perform any additions, subtractions, multiplications between any two UnivariateFunctions and between Ints/Floats and any UnivariateFunction. No division is allowed and it is not possible to raise a UnivariateFunction to a negative power. This is to ensure that all Univariatefunctions are analytically integrable and differentiable. This may change in future releases.

Interpolation and Splines

So far this package support the following interpolation schemes:

• Constant interpolation from the left to the right. Such a Piecewise_Function spline can be constructed by the create_constant_interpolation_to_right method.
• Constant interpolation from the right to the left. Such a Piecewise_Function spline can be constructed by the create_constant_interpolation_to_left method.
• Linear interpolation. Such a Piecewise_Function spline can be constructed by the create_linear_interpolation method.

It also supports the following spline (which can also be used for interpolation)

• Schumaker shape preserving spline - Such a Piecewise_Function spline can be constructed by the create_quadratic_spline method.

Approximation and regression

So for this package supports the creation of the following approximation schemes:

• OLS regression. The create_ols_approximation function can create a UnivariateFunction approximating a linear relationship. The degree input to this function can be used to specify the number of higher powers of x to be used in approximating y. For instance if the degree is two then y will be approximated as a linear combination of $x$ and $x^2$ as well as an intercept (if the intercept boolean is true).
• Chebyshev polynomials - This will approximate a function using the Chebyshev basis functions. This approximation function can then be integrated to accomplish Chebyshev–Gauss quadrature.

Date Handling

• All base dates are immediately converted to floats and are not otherwise saved. Thus there is no difference between a PE_Function created with a base as a float and one created with the matching date. This is done to simplify the code. All date conversions is done by finding the year fractions between the date and the global base date of Date(2000,1,1). This particular global base date should not affect anything as long as it is consistent. It is relatively trivial to change it (in the date_conversions.jl file) and recompile however if desired.

Major limitations

• It is not possible to divide by univariate functions or raise them by a negative power.
• When multiplying PE_Functions with different base dates there is often an issue of very high or very low numbers that go outside machine precision. If one were trying to change a PE_Function from base 2010 to 50, this would not generally be possible. This is because to change $a \exp(x-2020)$ to $q \exp(x - 50)$ we need to premultiply the first expression by $\exp(-1950)$ which is a tiny number. In these cases it is better to do the algebra on paper and rewriting the code accordingly as often base changes cancel out on paper. It is also good to change bases as rarely as possible. If different univariate functions use different bases then there is a need to base change when multiplying them which can result in errors. Note that if base changes are segment in the x domain by means of a piecewise function then they should never interact meaning it is ok to use different bases here.
• There is no support for finding optima, roots, fixedpoints etc. If anyone has an idea of how to do it efficiently then please let me know.
• There is no support for finding a function representing the upper/lower envelope of multiple functions. If anyone has an idea of how to do it efficiently then please let me know.