About Scalar and Vector

Few Words

In my design of this calculator (although I am afraid it is evolving into language)

I made a decesion about making its types like the mathmatical numbering types

1- Scalar

2- Vector

3- Matrix

4- Tensor   (LATER LATER I HOPE)

The Scalar in Qs is  a normal  number with a unit.

(remember the idea of this calculator is that all numbers are quantities)

so Scalar in my programming model for now is  holding a quantity   which is number+unit

and ofcourse Vector is an array of Scalars

the great thing I’ve discovered is that there are two Vector types

Vector Types

a) Euclidean Vector : This is only a three component vector

This Vector is always expressing the point in the space

It has the famous Cross Product Multiplication   (implemented in Qs as ‘x’ letter)

b) Normal Vector : This is an array holding a lot

This Vector is any number of components and can have Dot Product and

I am dealing with this types as a whole entity, I am not taking the approach of element by element calculation

because I am not convinced again about this approach :S

Multiplication Operators

So after great thinking and a lot of fancy shadows in my tiny room I made a three multiplication operators

1- Ordinary multiplication  ’*’

2- Dot Product                          ’.’

3- Cross Product                      ’x’    <== YES it is the X letter  [Who cares??!] my language after all

Power Operators

and to make it worse I made a distinction between the three Power operators which are attached by meaning by their associated Multiplication operators

1- Ordinary Power              ’^’

2- Dot Power                        ’^.’               Which is different completely from the matlab ‘.^’ element power operator

3- Cross Power                   ‘^x’

Matrix Exponential

So you Remember about the sequence implementation

with sequence implementation I can make functions like sin, cos, and my famous Exponential sequence

Qs> e[n](x) ..> x^n/n!

then wrapping the sequence into a function call

Qs> e(x) = e[0++50](x)

This functions is working great in scalars

have you tried it with matrix ??

I tried it and succeed look into this wikipedia page Matrix Exponential (by the way you should donate to wikipedia)

so if you passed a matrix to the function it will be treated as a whole matrix not element by element like matlab for example.

and this was an advantage from my model.

Power Dot

another funny thing is the ‘^.’ Power Dot when applying it to the vector.

by the way in any Power operation in Qs I get internally the first term to be Identity

for example in scalar it is 1<1>

Vector: {1<1> 1<1> … }   with the number of the components

Matrix Identity : Which is  diagonal ones  in a matrix with the same dimensions

so when applying the Power dot to vector

first item  or zero repetion is the identity term, then by the power number I multiply the vector in the identity for repeat one and etc.

the funny thing is that dot product in vector gives a scalar value then scalar is multiplied by vector again to give a vector value then a scalar

so when you make a power dot to  an even exponent the result is vector, and when you make an odd exponent it will be scalar.

pretty strange

Relation Operators

Relation operators are <, >, <=, >=, ==, !=

ofcourse it is well known for scalars

but what about Vector and Scalar

I solved this issue by comparing the norm of the vector with the norm of the scalar

review Vector Norm so The comparison is based on Norm which is Absolute in Scalar and Magnitude in Vector.

Finally

As you can see implementing the Scalar, Vector , and Matrix as a first citizen entities in Qs is overwhelming

and a lot of challenges appeared.

however I hope that next post to be about my new innovative idea of the Comparison statement itself

see you

Namespaces

ok I mentioned before that I’ve implemented namespaces in my quantity system

Why I’ve made this ??

because simply I wanted to add the possibility of adding multi functions in a form of external libraries.

Namespace Operator or Separator

1. in C++ the name space is two colons :: like the famous std::vector
2. in C# the namespace is a dot . like  System.Console.WriteLine
3. in xml the namespace is single colon :

so I’ve choose the xml namespace operator because I find it shorter than C++ namespace and more understandable than the dot of C#.

because I’ve always considered the dot .  to separate between the object and its members

Built in Functions

like any respectable language you have some built in library for mathmatical functions

you can find the functions and constants to be used

I’ve defined Math namespace to hold some (but not all) functions

Math:Sin, and Math:Sinh

Math:Cos, and Math:Cosh

Math:Log, and Math:Log10

and Math:PI  as a constant

 

User Defined Functions

you can define your user defined function in any namespace you desire

just write the namespace:function  and suddenly you have a new namespace if it wasn’t there before

Qs> G:V(x,y) = x(y)   # function that take other function as a parameter

Qs> g:v(Math:Sin, 8 )  # passing built in sin function with 8

 

and as you can notice the functions and namespaces are all case insensitive.

 

User Defined Functions (External Modules)

and yes you can define your functions outside the quantity system

but the feature is not completed yet

however please try

Qs> Music:Play(c d e f g a b c>)

if you hear the music from your speaker then you are a windows guy

and you can save this as a midi file too

Qs> Music:SaveMidi(c d e f g a b c>, tt.mid)

please find the result file into the program directory.

 

 

About DLR and other funny things.

DLR

(Dynamic Runtime Library) is microsoft latest technology that will prove itself as long as .NET struggle to survive.

however this is an old news, you must heard about IronPython, and IronRuby.

 

My Quantity System Calculator is build over the DLR expressions. This has permitted me to do a numerous things in my calculator.

I added functions, new sequence concept, and lately implemented NameSpaces.

the latest DLR version is 0.92 and you can get it from http://dlr.codeplex.com

 

Funny Things

if you download the last Quantity system sources from the source panel in code plex

and run this code

 

Qs> Sin[n](x) ..> ((-1)^n*x^(2*n+1))/(2*n+1)! #Sin sequence

Qs> Sin(x) = Sin[0++50](x) #Sin function

 

Qs> Math:Sin(8)

DimensionlessQuantity: 0.989358246623382 <1>

Qs> Sin(8)

DimensionlessQuantity: 0.989358246623403 <1>

can you notice the difference between the last 3 digits from right

the first function from the Math:Sin(8) is a direct calling for the .NET Math.Sin  function

and the second Sin(8) is a call for my sequence-series implementation.

well I have double checked by the windows calculator and the result was the same as the .NET library.

Windows Calculator Sin(8) in radian mode: 0.98935824662338177780812359824529

So which one is the right approximation

let’s try to call the SIN series with less items

 

Qs> Sin[0++20](8)

DimensionlessQuantity: 0.989358246623414 <1>

ok also have an error

let’s try it for many more items.

 

Qs> Sin[0++160](8)

DimensionlessQuantity: 0.989358246623403 <1>

we are back to the same result

damn it what should i trust more??  my sequence-series implementation or the .NET Sin implementation, and what if this was obtained from the FPU ???

 

 

I really don’t know.

 

 

 

Quantities! are they all dimensional uniques??

Small Talk
Is it really important to have a unit associated to value.
I really don’t know the answer

I assumed it is important to know the dimension of the quantity.
* To be able to sum, and subtract different units but same dimension quantities.
* To be able to predict that the outcome of the division and multiplication process of quantities.
* To deduce how can I reach specific quantity from another quantity.

Article Begins
So let us focus on the important objective of the library and its calculator.
The main objective was to replace the Dimensionless Variable Concept into Quantity Concept

Reviewing the SI model there are 7 basic base units that form all of our Quantities experience in Life

do I really have to list them ?? I think I should

1- kg (kilogram) [Mass]
2- m (metre) [Length]
3- s (second) [Time]
4- A (Ampere) [Electric Current]
5- K (Kelvin) [Temperature]
6- mol (mole) [Amount of Substance]
7- cd (candela) [Luminous Intensity]

these are 7 base quantities with 7 base units
(I feel they made it 7 so we can tie our self to the 7 number like 7 days and more legendary 7s)

if you encounter a fluid course in your study you will encounter the dimensional analysis course that teach you MLT system

the lecture will teach you about the dimensionality of the quantity
so Length is equal to M=0 L=1 T=0 or for simple writing M0L1T0

Speed or Velocity: M0L1T-1
Acceleration : M0L1T-2

Based on the dimensional Analysis course any quantity in universe have a unique dimensions

and we have special quantities that have M0L0T0 which means Dimensionless quantity

Types of Dimensionless quantities are many
- Mach number.
- Reynolds number.
- Affinity Laws.

also based on professors philosophy Dimensionless numbers are a high class numbers because they always give a lot of information.

for example if I say I am traveling to Alexandria from Cairo
and you asked me about the distance I covered from the beginning of the trip!!
If I told you I passed 100 km from the distance you will wonder and may feel that information is not enough.
But if I told you that I passed half of the distance to Alexandria
you can deduce many things.
because you know roughly when I begin traveling.
you know the maximum speed between towns.
you can deduce when I should get there and in what hour.

practically you can sense and understand whats going on and what will happen.
or simply REALIZE

so dimensionless numbers are important for our life

Different Quantities and The same dimensions

BACK to quantities again

Force quantity are taken from the classical newton equations and it is Mass multiplied by Acceleration

F = kg * m/s^2

this is awesome so Force as a derived quantity = M1L1T-2

(NOTE: in dimensional analysis there is another approach to define dimensions by FLT system due to the fact that we can’t get mass without knowing the gravity of the location we are on now however let us stick with MLT)

BUT can we REALLY say in honest words that all quantities have a different unique dimensions ??

{NO}

and let me list them as I can remember for now:

1- (Work) vs (Torque):
———————————————–

The two concepts are referring to the same units which are Joule, calorie, or BTU

Work = Force * Length

The same as Torque

Torque = Force * Length

yes you can say that they are different quantities but look at them in units they are the same units N.m

2- (Dimensionless) vs (Angle)
——————————
Angle is measured by Length of circumference to the length of radius.
it will be dimensionless

3- (Angle) vs (Solid Angle)
—————————-
as angle units, steradian for solid angle
Solid angle is Area to Area

4-(Frequency) vs (Angular Velocity)
————————————
Frequency is equal to M0L0T-1 or 1/s i.e. (1/Time)
Angular Velocity = rad/s

and rad = 1 because it is dimensionless.

so summing frequency with angular velocity should be logically correct (because they resemble the same dimensions) WHICH is WRONG in literature.

do you remember the equation of Omega = 2 * Pi * f
where
Omega: Angular Velocity
f: Frequency.

5- (Luminous Intensity) vs (Luminous Flux)
——————————————–
The first quantity is lm/sr and the second one is (cd.sr)=lm

Units can differentiate between the two quantities

but the unit is dimensionless and in SI standards you can replace either rad and sterad with which the dimensionless unit.

Conclusion:

1. These quantities will prevent any implementation to differentiate between quantities by dimensions and in return will prevent the intelligence of knowing the outcome quantity of an operation
2. You can’t take the dimensional analysis approach in differentiating between quantities.

So in return most of the implementation of dimensionality of variables took the shorter approach of supporting only units side.

but where is the quantity ???

can the unit approach differentiate between Torque as N.m and Work as N.m

if you implemented the J (Joule) unit you will have to invent a unit for Torque also, so you can make a distinction between units.

But how about the rest of quantities.

In my next post I’ll introduce to you the hypothetical solution of this dilemma and how I solved this problem in my Quantity System implementation.

Thank you
Ahmed Sadek

Introduction

Introduction

This blog is about my last software Quantity System Framework

my Name is Ahmed Sadek

and I am a mechanical engineer who know how to program computers since MSX

my personal web site is http://www.lostparticles.net which hold also my personal blog.

because I have a special interrest in programming and simulating the mechanical engineering problems into software.

I found that we really didn’t have a programming language that support unit conversion between the same quantity.

my first attempt to make this library was about 5 years ago.

I was making a progam to compute the heat gain through building using ASHRAE 97 method of TFM

back there I created a simple concept and tried to make it succeed.

(I FAILED) and you will know later WHY ??

I’ve created this blog especially for writing about this piece of software this library

and my future plans about it.

as my ideas are now burning in my head this blog will serve as a source for ideas related to the scientific applications that will be built around this library.

Being a thermodynamic fan I know what is meant by Effeciency, Second Effeciency, and Availability.

I wanted to create this library (and later the calculator it self) for calculating effeciently not to make another program which let me decide what I should do.

(I have another things to decide about it)

The library and its calculator serve more than one goal (at least in my opinion)

First: young people who are un-aware about units and its conversion tricks

Second: Scientists who want to get their job done without making un-noticed mistakes.

The quantity system is the first to differentiate between work and Toruqe.

this is a really an important sentence.

if you looked into the previous implementations for the same idea.

you will find them lacking the dimensional approach

although they know about the concept and all those pretty things about Pukingham theory and dimensional analyis

but they simply choosed to ignore the dimensional approach because the lake of differentiating between Torque and Work

All other attempts before this one was ignoring this problem by employing the concept of units

without thinking about dimensions.

so their concept is “Unit” as the begining and ending to the problem, and by following the SI standards about units.

however this lead to a serious problem.

The problem about Work which is Force times Length N.m is the same as Torque Which is Force times Length also N.m.

So simple you CAN’t differentiate between Work and Torque by using only units

Another problem about angle

Have you ever looked into the Angle quantity

The Angle is dimensionless quantity which is Length Divided By Length.

And to make things even more worse

Solid Angle Quantity which Area divided by Area is also dimensionless quantity

and Angle unit is Radian

and Solid Angle units is Steradian

How can you differentiate between Angle and Solid Angle ??

telling me they have different units

let me ask again they are dimensionless

how will your program behave about that.

And these problems especially what made me fail before.

this problem also is the stone that render all other attempts as a weak implementation

But I insisted on one Concept

I have to think in quantities not in units

I solved the problem I’ll tell you about it next post.

And that leaded to another surprise for me which was the last thing to break my barriers in starting this blog.

This is a long post I think

but I think later posts will be more interresting for all of the readers

I have a bulk of ideas that are ready to implement.

I have this faith in making every thing open either information or implementation.

and I really wish to enhance our lives

Thank you

Ahmed Sadek