Developing a matrix arithmetic parser using JavaScript

Question

Developing a matrix arithmetic parser using JavaScript

Currently, I am in the process of developing a program that can solve matrix equations. My main focus right now is on making sure the parser functions correctly. However, I am feeling lost and unsure of where to begin.

In my program, I utilize an array of strings to keep track of both matrices and operators. For instance, if we take the equation

[[0, 0],[4, 5]] + [[4, 8], [12, 6]]

it would be organized as:

variables = ["$", "+", "$"]
matrices = [[[0, 0],[4, 5]], [[4, 8], [12, 6]]

Here, each "$" symbol represents a distinct matrix stored in a separate array. Additionally, I aim to encapsulate matrices within different expressions like so:

det([[0, 0],[4, 5]]) + inv([[0, 0],[4, 5]] + [[4, 8], [12, 6]])

where "det" and "inv" signify determinant and inverse operations respectively.

If represented in code, the above expression would look like this:

variables = ["d", "e", "t", "(", "$", ")", "+", "i", "n", "v", "(", "$", "+", "$", ")"]
matrices = [ [[0, 0],[4, 5]], [[0, 0],[4, 5]],  [[4, 8], [12, 6]] ]

My initial thought is to utilize context-free grammars for constructing a parse tree, as regular expressions may not be suitable due to the presence of parentheses... (?). Another idea I have is to convert the expressions into postfix notation and test its feasibility.

javascript parsing

Answer 1

Answer №1

Constructing a parse tree with the help of a context-free grammar seems like the most effective method in my opinion. (If you're not familiar with context-free grammars and don't have access to a suitable textbook, consider checking out the Wikipedia article on the topic.)

If you search around, you'll discover numerous instances of how to formulate a context-free grammar for arithmetic expressions. (The provided Wikipedia link showcases an example arithmetic grammar.) It's true that most of these examples are geared towards basic calculators rather than matrix arithmetic. However, this distinction doesn't significantly impact the approach.

The key understanding for developing parsers is that syntax remains consistent regardless of the nature of the calculated primitive elements – be it integers, floating-point numbers, complex numbers, or arrays. The variance primarily lies in the syntax for representing a literal object. Furthermore, the format of operators doesn't affect the underlying grammar structure, allowing for a uniform style even when evaluating boolean expressions using operators such as and, or, and not.

In scenarios involving a standard numeric calculator, literal objects (numbers) typically emerge from the lexer effortlessly due to their lack of internal complexity. Conversely, matrices possess internal structuring, making regex-based lexers less efficient for parsing structured data. A more effective strategy involves treating a matrix literal as a sequence of tokens comprising numbers alongside symbols like [, ], and ,, and integrating corresponding productions into your context-free grammar for matrix parsing.

In essence, a matrix can be viewed as a collection of vectors separated by commas, with each vector containing a comma-separated list of numbers. This structure is syntactically analogous to any other comma-separated or delimited list variant, irrespective of the specific delimiter used. Similar production requirements also apply when parsing argument lists within function call expressions.

A fundamental form of a context-free grammar for a delimited list typically follows this schema, replacing X with the actual element type:

/* Parses lists containing one or more elements */
list_of_X: X
         | list_of_X ',' X

In cases where empty lists are permissible (less common with matrices but prevalent in function calls), you can combine the above pattern with another concept:

optional_list_of_X: %empty
                  | list_of_X

Your grammar would then involve rules like:

expression: matrix
          | /* ... all other expression forms */

matrix: '[' list_of_vector ']'
vector: '[' list_of_number ']'

For illustration purposes, I've utilized Bison syntax since it's commonly employed for writing context-free grammars. Keep in mind that every parser generator comes with its unique conventions and quirks.

Answer 2

Constructing a parse tree with the help of a context-free grammar seems like the most effective method in my opinion. (If you're not familiar with context-free grammars and don't have access to a suitable textbook, consider checking out the Wikipedia article on the topic.)

If you search around, you'll discover numerous instances of how to formulate a context-free grammar for arithmetic expressions. (The provided Wikipedia link showcases an example arithmetic grammar.) It's true that most of these examples are geared towards basic calculators rather than matrix arithmetic. However, this distinction doesn't significantly impact the approach.

The key understanding for developing parsers is that syntax remains consistent regardless of the nature of the calculated primitive elements – be it integers, floating-point numbers, complex numbers, or arrays. The variance primarily lies in the syntax for representing a literal object. Furthermore, the format of operators doesn't affect the underlying grammar structure, allowing for a uniform style even when evaluating boolean expressions using operators such as and, or, and not.

In scenarios involving a standard numeric calculator, literal objects (numbers) typically emerge from the lexer effortlessly due to their lack of internal complexity. Conversely, matrices possess internal structuring, making regex-based lexers less efficient for parsing structured data. A more effective strategy involves treating a matrix literal as a sequence of tokens comprising numbers alongside symbols like [, ], and ,, and integrating corresponding productions into your context-free grammar for matrix parsing.

In essence, a matrix can be viewed as a collection of vectors separated by commas, with each vector containing a comma-separated list of numbers. This structure is syntactically analogous to any other comma-separated or delimited list variant, irrespective of the specific delimiter used. Similar production requirements also apply when parsing argument lists within function call expressions.

A fundamental form of a context-free grammar for a delimited list typically follows this schema, replacing X with the actual element type:

/* Parses lists containing one or more elements */
list_of_X: X
         | list_of_X ',' X

In cases where empty lists are permissible (less common with matrices but prevalent in function calls), you can combine the above pattern with another concept:

optional_list_of_X: %empty
                  | list_of_X

Your grammar would then involve rules like:

expression: matrix
          | /* ... all other expression forms */

matrix: '[' list_of_vector ']'
vector: '[' list_of_number ']'

For illustration purposes, I've utilized Bison syntax since it's commonly employed for writing context-free grammars. Keep in mind that every parser generator comes with its unique conventions and quirks.

Developing a matrix arithmetic parser using JavaScript

Answer №1

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