# Overview

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# Simple Symbolic Mathematics for Scala

This project contains a very simple, and incomplete, symbolic math library in
Scala. It can *differentiate* and *evaluate* simple mathematical expressions.
The library also contains some aspects of an internal DSL: The expressions can
be entered like regular math with `Int` or `Double` numbers, and there is
a ML style *"let" expression*. Here is a short example that demonstrates the
differentiation feature:

import symathm.Expression._ import symathm.ExprOps._ //Create some symbols (unknown variables). val (a, x) = (Sym("a"), Sym("x")) //Create an expression. `~^` denotes exponentiation (power). val expr1 = a * x~^4 + 5 * x~^2 + x~^0.5 //Differentiate the expression with respect to `x`. val dexpr1 = diff(expr1, x) //Print the expression in human readable form. //Prints: "4.0 * a * x ~^ 3.0 + 10.0 * x + 0.5 * x ~^ -0.5;;" pprintln(dexpr1)

The library is not intended to be used seriously. Instead it should demonstrate simple features of Scala that are interesting for programmers that come form traditional object oriented languages; such as: C++, Java, Python, Ruby. The project should especially demonstrate the usefulness of pattern matching. Therefore this library is implemented three times with different programming paradigms, but with identical features and interfaces:

Package symathm : |
Functional, with pattern matching. |

Package symathv : |
Object oriented with Visitor pattern. |

Package symathoo: |
Classical object oriented. |

The three libraries are big enough (500 to 700 lines) to give an impression how working with a real program would be. But they are small and simple enough, to be easily understood. To write the algorithms, and to verify their correctness, only high school math is necessary. In principle the algorithms can be looked up in Wikipedia (http://en.wikipedia.org/wiki/Table_of_derivatives).

## Repository Contents

`src/``symathm/SymbolicMainM.scala`- Symbolic math library, implemented in functional fashion, with pattern
matching.
(Package:
`symathm`) `symathv/SymbolicMainV.scala`- Implementation of the library with the visitor pattern. Object
oriented, but structure similar to pattern matching.
(Package:
`symathv`) `symathoo/SymbolicMainOO.scala`- The symbolic math library implemented in simple object oriented fashion.
(Package:
`symathoo`) `UseTheLibraries.scala`Program that demonstrates the features of the libraries.

The three implementations of the library have identical features and interfaces.

`pattern/``testdsl.scala`- Short example implementation of the libraries' "DSL" features.
`testvisitor.scala`- Short example implementation of the visitor pattern.

`make-compile.sh`- Compile all Scala source files.
`make-scaladoc.sh`- Create API documentation with
`scaladoc`. `make-tests.sh`- Run all tests. Compiles all sources prior to testing.
`.project`- Project file for the Eclipse IDE and its Scala plugin. Lets Eclipse find the Scala files.
`README.rst`- This file.

## Usage

### Getting the Software

Either get the software by cloning the repository with Mercurial:

hg clone https://bitbucket.org/eike_welk/scala-symbolic-algebra-test

Or download (and extract) one of the auto-generated archives from here:

https://bitbucket.org/eike_welk/scala-symbolic-algebra-test/downloads

### Without IDE

Run the script `make-compile.sh` to compile all source files. This might
take a minute or two:

./make-compile.sh

Then run any object with a `main` method. Start with the usage example
`UseTheLibraries`, which explains all of the libraries' (few) features:

scala -classpath bin/ UseTheLibraries

To start Scala's read-eval-print loop, you need to specify the `classpath`
where the compiled files are found (but don't specify an object that should
be run):

scala -classpath bin/

### With Eclipse IDE

The root directory contains a project file (`.project`) for the
Sala IDE for Eclipse. It lets Eclipse find the source files, so that it
can compile them automatically. You can run any file that contains an object
with a `main` method by clicking the *Run* button.

First try out the usage example `UseTheLibraries.scala`.

Sala IDE for Eclipse can be downloaded from: http://scala-ide.org/

## Required Knowledge of Scala

Only little knowledge of Scala is needed to understand the code. A good introductory text on Scala is:

http://www.artima.com/scalazine/articles/steps.html

The text above unfortunately does not cover pattern matching, which is IMHO one of Scala's main attractions. Pattern matching is covered here:

http://www.artima.com/pins1ed/case-classes-and-pattern-matching.html

## Required Software

Either a working *Scala* installation (programs `scalac` and `scala`) on a
Unix-like operating system. (On Windows you have to come up with the right
command to compile the sources and run them yourself.)

Or even better a *IDE with Scala support*, for example the Scala-IDE for
Eclipse.

http://www.scala-ide.org/

## Projects

To compare the characteristics of the different programming paradigms, you can add features to each version of the library.

- Add derivation of the
`Log`node. - Add new nodes, for example
`sin`,`cos`and`tan`. - Add function call node. Maybe this makes an inert
`diff`node superfluous. (See point below.) - Add
`lambda`(function body) node. - Implement an inert
`diff`node. The "a$x" notation is a hack. - Implement a node for a
`for`loop. Write evaluation and differentiation algorithms for it. (I believe differentiating a`for`loop is possible, because older versions of*Maple*could do it.) - Implement an algorithm to distribute factors over sums, and distribute
powers over products. For example:
`(a + b) * c`-->`a*c + b*c`. This is interesting for`eval`: more operators with only numeric arguments can be found, and evaluated. - Implement an algorithm to collect factors and powers. (The opposite of the algorithm above.) It makes formulas look good.
- Maybe add a separate
`simplify`function. - Implement some of the TODOs in the code.

## Architecture

### Data Structures

All important data structures are defined in object `Expression`.

Mathematical formulas are internally represented as nested trees of *nodes*.
They are implemented as *case classes*, syntactical sugar for simple classes
that are intended to work with the `match` statement.
(http://www.artima.com/pins1ed/case-classes-and-pattern-matching.html)

`Expr`: The common base class of all nodes`Num(num: Double)`: A number (floating point)`Sym(name: String)`: A variable (symbol)`Add(summands: List[Expr])`: Addition (n-ary)`Mul(factors: List[Expr])`: Multiplication (n-ary)`Pow(base: Expr, exponent: Expr)`: Exponentiation (operator`~^`)`Log(base: Expr, power: Expr)`: Logarithm`Let(name: String, value: Expr, exprNext: Expr)`: Bind a value to a variable, and put a single expression into the environment, where the new variables are visible.`Asg(lhs: Expr, rhs: Expr)`:`:=`operator, helper object to create`Let`nodes.

`1+a` and `1+a*2` are respectively expressed as:

Add(List(Num(1.0), Sym("a"))) Add(List(Num(1.0), Mul(List(Sym("a"), Num(2.0)))))

#### N-Ary Addition and Multiplication

Addition and multiplication are n-ary, they can have an arbitrary number of
arguments. `1 + a + 2 + 3` and `1 * a * 2 * 3` are respectively
expressed as:

Add(List(Num(1.0), Sym("a"), Num(2.0), Num(3.0))) Mul(List(Num(1.0), Sym("a"), Num(2.0), Num(3.0)))

There are no nodes for subtraction or division. Subtraction is represented
as multiplication with `-1`: (`-a = -1 * a`). Division is expressed as a
power of `-1`: (`1/a = a~^(-1)`). [1].

As there are no subtraction or division operators, `a-x` and `a/x` are
respectively expressed as:

Add(List(Sym("a"), Mul(List(Num(-1.0), Sym("x"))))) Mul(List(Sym("a"), Pow(Sym("x"), Num(-1.0))))

#### Let Expressions

`Let` nodes are similar to assignment statements in imperative programming
languages. They are commands to create a new *environment*, where
a variable is bound to a value. The dependent expression (the third argument of
`Let`) is evaluated in this new environment. The value of a `Let` node is
the value of its dependent expression.

`Let` nodes are created by a little abuse of Scala's liberal syntax
(the DSL): `let (a:=2) in a*a` results in:

Let("a", Num(2.0), Mul(List(Sym("a"), Sym("a"))))

The `Let` node does not create the new environment by itself, it is
interpreted by an algorithm. The `eval` algorithm interprets `Let`
nodes as described above: It creates a new environment that contains the new
variables and also the variables of the old environment. Then `eval`
evaluates the dependent expression in the new environment. The expression
above would be evaluated to `Num(4)`.

The *environment* that stores the bindings between variables and their values,
is (currently) implemented as a `Map[String, Expr]`.

For convenience there are *type* (and *companion object*) `Environment`,
and a call-able object `Env` to create environments.

### DSL

The library contains a modest attempt to implement a domain specific language
(DSL). The implementation of the DSL is in object `Expression`.
Additionally there is a small example program to illustrate the same technique:
`src/pattern/testdsl.scala`.

The common base class of all nodes, `Expr`, contains the usual mathematical
operators: `+ - * / ~^`, and additionally `:=`. (`~^` is the
exponentiation operator.) Each operator returns a part of the tree. The `+`
operator, for example, returns an `Add` node.

There are implicit conversion functions (`int2Num`, `double2Num`) in
`Expression`, that convert `Int` and `Double` objects into `Num`
nodes. This way numbers and nodes can be freely mixed.

`Let` nodes can be somewhat elegantly created with the call-able helper
object `let`. When called with multiple assignments, `let` creates nested
`Let` nodes. The syntax is:

let (a := 2, x := 3) in a + x

Which returns:

Let("a", Num(2.0), Let("x", Num(3.0), Add(List(Sym("a"), Sym("x")))))

The `eval` algorithm would evaluate this expression to `Num(5)`.

### Algorithms

The high-level algorithms are implemented in object `ExprOps` (in all
implementations of the library). All algorithms traverse a tree of nodes in
a recursive way, and create a new tree as the result.

There are currently two algorithms:

`eval`This algorithm behaves like an interpreter of a programming language.

Differently to a traditional programming language there are no "unknown variable" errors. The algorithm replaces known variables (symbols) by their values, but unknown variables are left unchanged.

`eval`performs the usual arithmetic operations on numbers. Expressions that contain numbers and unknown variables are simplified as much as possible. The n-ary multiplication and addition nodes simplify this task:`1 + a + 2`can easily be simplified to`3 + a`, by partitioning the summands into numbers and other nodes.`diff`Differentiate expressions.

The algorithm can differentiate

`Let`nodes; differentiating:let (a:=f) in g

with respect to

`x`, basically yields:let (a:=f, a$x:=diff(f, x)) in diff(g, x)

[1] | The ideas for n-ary operators (additon, multiplication), and
for the ommission of subtraction and division nodes, were taken from the
computer algebra program Maxima. It is intended to simplify the
algorithms. |