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Functions Smalltalk

~ Error,Gammma,Beta

~ Smalltalk Server

POLYNOMIALS

ERROR FUNCTION

ERROR FUNCTION
Error Function - General Implementation

Note: The implementation of Error Function, Gamma Function, Beta Function is shown here using development tools, for a real production version, please check Smalltalk Server Applications

The error function is the integral of the normal distribution. The error function is used in statistics to evaluate the probability of finding a measurement larger than a given value when the measurements are distributed according to a normal distribution.

Because it is the integral of the normal distribution, the error function, erf(x) (a method name we created to evaluate this function), gives the probability of finding a lower value than x when the values are distributed according to a normal distribution with a mean of zero and a standard deviation of 1.

We wrote an error function following Gaussian distribution procedures and standard deviation.

This probability is expressed by the following integral:

erf (x) = 1 / √ 2π ∫^x_-∞ e^-t²/2 dt.

The result of the error function lies between 0 and 1. A Gaussian distribution with average µ and standard deviation ß, can be given by:

Prob (|x - µ| / ß ≤ t ) = 2 . erf (t) - 1, for t ≥ 0.

IMPLEMENTATION

We do not have a mining application to show the use of our error function erf(x), so we borrowed a sample taken from Smalltalk author [D.Besset, Phd] to illustrate its implementation.

In medical sciences, centile is the value of the error function expressed as a percentage. For example, obstetricians consider whether the weight at birth of the first-born child is located below the 10th centile or above the 90th centile to assess a risk factor for a second pregnancy.

The following facts are available:

[Weight at birth of first child: 2.85 kg. ], [Duration of first pregnancy: 39 weeks. ], [Measurement over a representative sample of all births yielding healthy babies have an average of: 3.39 kg. ], [Standard deviation is: 0.44 kg.]

The idea here is finding the probability of a second child having a weight at birth less than that of the woman's first child.

Prob (weight ≤ 2.85 kg) = erf ( [2.85 - 3.39] / 0.44)

Script for Error Function Evaluation

The following code provides the set up to execute the error function. It is presented as a window class in Smalltalk (shown below) and executed on a Smalltalk workspace window.

Class DhbPolynomial was used to hold related methods(not shown).
The method code valueOfError is straightforward and mimics the example shown above.

RESULTS follow and are shown on a Smalltalk transcript window.

RESULTS

Based on these results and according to current practice, this second pregnancy does not require special attention.

Other Implementations...

Object-Oriented Implementation

Graphics and Animation

Sample Applications

Ore Extraction Optimization

Vectors and Matrices

Complex Numbers and Functions

Ordinary Differential Equations - Euler Method

Ordinary Differential Equations 2nd-Order Runge-Kutta

Ordinary Differential Equations 4th-Order Runge-Kutta

Higher Order Differential Equations

Nonlinear Systems

Numerical Integration

Numerical Differentiation

GAMMA FUNCTION

GAMMA FUNCTION
Gamma Function - General Implementation

The definition of the gamma function is a little more complex. This is because the gamma function uses the number e unlike more familiar functions such as polynomials or trigonometric functions.

the gamma function is defined as the improper integral of another function.

The gamma function is often used to compute the normalization value factor of several probability density functions and also to compute the beta function which will be addressed in the next tab. The gamma function is defined by the following integral, (also called Euler's integral):

(x) = ∫^∞₀ t^x e^-t dt.

from this equation, a recurrence formula can be derived:

γ (x + 1) = x . γ (x).

The value of the gamma function can be computed for special values of x:

γ (1) = 1
γ (2) = 1

The most precise approximation for the gamma function is given by a formula discovered by Lanczos[1964]:

γ (x) ≈ e^x+5/2 ( x + 5/2) √2π/x ( c₀ + ∑_n=1⁶ c_n/x + n + ε).

IMPLEMENTATION

In practice one can see from the leading term of Lanczos' formula that the gamma function raises faster than the exponential. Thus, evaluating gamma functions for numbers larger than a few hundred will exceed the capacity of the floating-number representation on most machines.

In this case, it is best to use the logarithm of the gamma function whenever it is used in quotients involving large numbers.

Script for Gamma Function Evaluation

The following code provides the set up to execute the gamma function. It is presented as a window class in Smalltalk (shown below) and executed on a Smalltalk workspace window.

The class DhbPolynomial was used to hold related methods(not shown).
The method code valueOfGamma is rather straightforward and simple to illustrate the power of this function.
Included are computations for the gamma function and a logarithm gamma function.

RESULTS follow and are shown on a Smalltalk transcript window.

RESULTS

POLYNOMIALS
Polynomials - General Implementation

Polynomials are quite important because they are often used in approximating functions, which are determined by experimental measurements in the absence on any theory on the nature of the function.

A polynomial is a special mathematical function whose value is computed as follows:

P(x) = ∑ⁿ_k a_kx^k

In this equation, n is called the degree of the polynomial. A second-order polynomial:

x² - 3x + 2

has degree 2. It represents a parabola crossing the x-axis at points 1 and 2 and having a minimum at x = 2/3. This second-order polynomial, is represented by the array {2, -3, 1}.

In the equations above, the numbers a₀,...,s_n are called the coefficients of the polynomial. Thus, a polynomial can be represented by the array {a₀,...,a_n}. For example, the second-order polynomial described above, is represented by the array {2, -3, 1}.

The best approach to evaluate a polynomial consists of factoring out x before the evaluation of each term, such as:

(x) = a₀ + x {a₁ + x[a₂ + x(a₃]}

Evaluating this expression requires only multiplications, instead of raising the variable to an integral power at each term as shown in the sum before. The resulting algorithm is quite straightforward to implement.

on polynomial derivatives...

The derivative of a polynomial with n coefficients is another polynomial, with n - 1 coefficients derived from the coefficients of the original polynomial, as follows:

a'_k = (k + 1)a_{k + 1} for k = 0,...,n-1.

on polynomial integrals...

The integral of a polynomial with n coefficients is another polynomial, with n + 1 coefficients derived from the coefficients of the original polynomial, as follows:

a^- = a_k-1 / k for k = 1,...,n+1.

Adding or subtracting two polynomials yields a polynomial whose degree is the maximum of the degrees of the two polynomials. Multiplying two polynomials yields a polynomial whose degree is the product of the degrees of the two polynomials.

IMPLEMENTATION

Being a special case of a function a class KmpPolynomial was created to hold related methods for execution.
The code first creates an instance of the class by giving the coefficients of the polynomial.
Code for derivative, integral and multiplication are included as examples of polynomial computation.

Script for Polynomial Evaluation

The following code provides the set up to execute the polynomial. It is presented as a window class in Smalltalk (shown below) and executed on a Smalltalk workspace window. The RESULTS follow and are shown on a Smalltalk transcript window.