mth(3) AFNIX Module mth(3)

# NAME¶

mth - standard math module

# STANDARD MATH MODULE¶

The Standard Mathematicalmodule is an original implementation of various mathematical facilities. The module can be divided into several catgeories which include convenient functions, linear algebra and real analysis.

Random number
The mathmodule provides various functions that generate random numbers in different formats.

 Function Description get-random-integer return a random integer number get-random-real return a random real number between 0.0 and 1.0 get-random-relatif return a random relatif number get-random-prime return a random probable prime relatif number

The numbers are generated with the help of the system random generator. Such generator is machine dependant and results can vary from one machine to another.

Primality testing
The mathmodule provides various predicates that test a number for a primality condition. Most of these predicates are intricate and are normally not used except the prime-probable-ppredicate.

 Predicate Description fermat-p Fermat test predicate miller-rabin-p Miller-Rabin test predicate prime-probable-p general purpose prime probable test get-random-prime return a random probable prime relatif number

The fermat-pand miller-rabin-ppredicates return true if the primality condition is verified. These predicate operate with a base number. The prime number to test is the second argument.

Fermat primality testing
The fermat-ppredicate is a simple primality test based on the "little Fermat theorem". A base number greater than 1 and less than the number to test must be given to run the test.

```afnix:mth:fermat-p 2 7
```

In the preceeding example, the number 7 is tested, and the fermat-ppredicate returns true. If a number is prime, it is guaranted to pass the test. The oppositte is not true. For example, 561 is a composite number, but the Fermat test will succeed with the base 2. Numbers that successfully pass the Fermat test but which are composite are called Carmichael numbers. For those numbers, a better test needs to be employed, such like the Miller-Rabin test.

Miller-Rabin primality testing
The miller-rabin-ppredicate is a complex primality test that is more efficient in detecting prime number at the cost of a longer computation. A base number greater than 1 and less than the number to test must be given to run the test.

```afnix:mth:miller-rabin-p 2 561
```

In the preceeding example, the number 561, which is a Carmichael number, is tested, and the miller-rabin-ppredicate returns false. The probability that a number is prime depends on the number of times the test is ran. Numerous studies have been made to determine the optimal number of passes that are needed to declare that a number is prime with a good probability. The prime-probable-ppredicate takes care to run the optimal number of passes.

General primality testing
The prime-probable-ppredicate is a complex primality test that incorporates various primality tests. To make the story short, the prime candidate is first tested with a series of small prime numbers. Then a fast Fermat test is executed. Finally, a series of Miller-Rabin tests are executed. Unlike the other primality tests, this predicate operates with a number only and optionally, the number of test passes. This predicate is the recommended test for the folks who want to test their numbers.

```afnix:mth:prime-probable-p 17863
```

Linear algebra
The mathmodule provides an original and extensive support for linear and non linear algebra. This includes vector, matrix and solvers. Complex methods for non linear operations are also integrated tightly in this module.

Real vector
The mathmodule provides the Rvectorobject which implements the real vector interface Rvi. Such interface provides numerous operators and methods for manipulating vectors as traditionnaly found in linear algebra packages.

 Operator Description == compare two vectors for equality != compare two vectors for difference ?= compare two vectors upto a precision += add a scalar or vector to the vector -= substract a scalar or vector to the vector *= multiply a scalar or vector to the vector /= divide a vector by a scalar

 Method Description set set a vector component by index get get a vector component by index clear clear a vector reset reset a vector get-size get the vector dimension dot compute the dot product with another vector norm compute the vector norm

Creating a vector
A vector is always created by size. A null size is perfectly valid. When a vector is created, it can be filled by setting the components by index.

```# create a simple vector
const rv (afnix:mth:Rvector 3)
# set the components by index
rv:set 0 0.0
rv:set 1 3.0
rv:set 2 4.0
```

Real matrix
The mathmodule provides the Rmatrixobject which implements the real matrix interface Rmi. This interface is designed to operate with the vector interface and can handle sparse or full matrix.

 Operator Description == compare two matrices for equality != compare two matrices for difference ?= compare two matrices upto a precision

 Method Description set set a matrix component by index get get a matrix component by index clear clear a vector get-row-size get the matrix row dimension get-col-size get the matrix column dimension norm compute the matrix norm

# STANDARD MATH REFERENCE¶

Rvi
The Rviclass an abstract class that models the behavior of a real based vector. The class defines the vector length as well as the accessor and mutator methods.

Predicate

rvi-p

Inheritance

Serial

Operators

== -> Boolean (Vector)
The ==operator returns true if the calling object is equal to the vector argument.

!= -> Boolean (Vector)
The ==operator returns true if the calling object is not equal to the vector argument.

?= -> Boolean (Vector)
The ?=operator returns true if the calling object is equal to the vector argument upto a certain precision.

+= -> Vector (Real|Vector)
The +=operator returns the calling vector by adding the argument object. In the first form, the real argument is added to all vector components. In the second form, the vector components are added one by one.

-= -> Vector (Real|Vector)
The -=operator returns the calling vector by substracting the argument object. In the first form, the real argument is substracted to all vector components. In the second form, the vector components are substracted one by one.

*= -> Vector (Real|Vector)
The *=operator returns the calling vector by multiplying the argument object. In the first form, the real argument is multiplied to all vector components. In the second form, the vector components are multiplied one by one.

/= -> Vector (Real)
The /=operator returns the calling vector by dividing the argument object. The vector components are divided by the real argument.

Methods

set -> None (Integer Real)
The setmethod sets a vector component by index.

get -> Real (Integer)
The getmethod gets a vector component by index.

clear -> None (None)
The clearmethod clears a vector. The dimension is not changed.

reset -> None (None)
The resetmethod resets a vector. The size is set to 0.

get-size -> Real (None)
The get-sizemethod returns the vector dimension.

dot -> Real (Vector)
The dotmethod computes the dot product with the vector argument.

norm -> Real (None)
The normmethod computes the vector norm.

permutate -> Vector (Cpi)
The permutatemethod permutates the vector components with the help of a combinatoric permutation object.

reverse -> Vector (Cpi)
The reversemethod reverse (permutate) the vector components with the help of a combinatoric permutation object.

Rvector
The Rvectorclass is the default implementation of the real vector interface.

Predicate

r-vector-p

Inheritance

Rvi

Constructors

Rvector (None)
The Rvectorconstructor creates a default null real vector.

Rvector (Integer)
The Rvectorconstructor creates a real vector those dimension is given as the calling argument.

Functions

get-random-integer -> Integer (none|Integer)
The get-random-integerfunction returns a random integer number. Without argument, the integer range is machine dependent. With one integer argument, the resulting integer number is less than the specified maximum bound.

get-random-real -> Real (none|Boolean)
The get-random-realfunction returns a random real number between 0.0 and 1.0. In the first form, without argument, the random number is between 0.0 and 1.0 with 1.0 included. In the second form, the boolean flag controls whether or not the 1.0 is included in the result. If the argument is false, the 1.0 value is guaranted to be excluded from the result. If the argument is true, the 1.0 is a possible random real value. Calling this function with the argument set to true is equivalent to the first form without argument.

get-random-relatif -> Relatif (Integer|Integer Boolean)
The get-random-relatiffunction returns a n bits random positive relatif number. In the first form, the argument is the number of bits. In the second form, the first argument is the number of bits and the second argument, when true produce an odd number, or an even number when false.

get-random-prime -> Relatif (Integer)
The get-random-primefunction returns a n bits random positive relatif probable prime number. The argument is the number of bits. The prime number is generated by using the Miller-Rabin primality test. As such, the returned number is declared probable prime. The more bits needed, the longer it takes to generate such number.

get-random-bitset -> Bitset (Integer)
The get-random-bitsetfunction returns a n bits random bitset. The argument is the number of bits.

fermat-p -> Boolean (Integer|Relatif Integer|Relatif)
The fermat-ppredicate returns true if the little fermat theorem is validated. The first argument is the base number and the second argument is the prime number to validate.

miller-rabin-p -> Boolean (Integer|Relatif Integer|Relatif)
The miller-rabin-ppredicate returns true if the Miller-Rabin test is validated. The first argument is the base number and the second argument is the prime number to validate.

prime-probable-p -> Boolean (Integer|Relatif [Integer])
The prime-probable-ppredicate returns true if the argument is a probable prime. In the first form, only an integer or relatif number is required. In the second form, the number of iterations is specified as the second argument. By default, the number of iterations is specified to 56.
 2018-07-20 AFNIX