LARCS/manipulation.org

Simple Algebraic Manipulation

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Introduction   nonum

CLOSED: [2016-05-01 Sun 14:33]

As a part of my lisp-based Computer Algebra System, an algebraic manipulation toolkit is required. This will be used to simplify equations, or for that matter solve them. This creates this toolkit, but does not create a complete simplifier or solver. It does this by providing manipulators and automatic rewriters. These together will provide simplification and solving utilities.

TOC   ignore

WORKING Expression Typing [0/5]

  <<classification-storage>>
  <<define-classification>>
  <<check-classification>>
  <<classify-expression>>
  <<possible-classifications>>

WORKING Classification Storage

  (defvar *classifications* '())

WORKING Define Classification

  (defmacro define-classification (name &body body)
    `(progn
       (defun ,(symbolicate name '-classifier) (expression &aux (length (if (listp expression) (length expression) 1)))
         (declare (ignorable length))
         ,@body)
       (pushnew '(,name . ,(symbolicate name '-classifier)) *classifications*)
       ',name))

WORKING Check Classification

  (defun expression-type-p (expression type)
    (if (eq '* type)
        t
        (funcall (cdr (assoc type *classifications*))
                 expression)))

WORKING Classify Expression

  (defun classify (expression)
    (remove-if #'null
               (map 'list #'(lambda (name-and-checker)
                              (let ((name (car name-and-checker))
                                    (checker (cdr name-and-checker)))
                                (if (funcall checker expression)
                                    name
                                    nil)))
                    ,*classifications*)))

WORKING Classifications [0/12]

  <<classify-numbers>>
  <<classify-variables>>
  <<classify-additives>>
  <<classify-subtractives>>
  <<classify-powers>>
  <<classify-exponentials>>
  <<classify-multiplicatives>>
  <<classify-logarithmics>>
  <<classify-rationals>>
  <<classify-polynomial-term>>
  <<classify-polynomials>>
  <<classify-trigonometrics>>

WORKING Numbers

  (define-classification numeric
    (numberp expression))

WORKING Variables

  (define-classification variable
    (symbolp expression))

WORKING Additives

  (define-classification additive
    (when (listp expression)
      (eq '+ (first expression))))

WORKING Subtractive

  (define-classification subtractive
    (when (listp expression)
      (eq '- (first expression))))

WORKING Powers

  (define-classification power
    (when (listp expression)
      (and (eq 'expt (first expression))
         (expression-type-p (second expression) 'variable)
         (expression-type-p (third expression) 'numeric))))

WORKING Exponentials

  (define-classification natural-exponential
    (when (listp expression)
      (and (= 2 length)
         (eq 'exp (first expression)))))

  (define-classification exponential
    (when (listp expression)
      (and (= 3 length)
         (eq 'expt (first expression)))))

WORKING Multiplicatives

  (define-classification multiplicative
    (when (listp expression)
      (eq '* (first expression))))

WORKING Logarithmics

  (define-classification natural-logarithmic
    (when (listp expression)
      (and (= 2 length)
         (eq 'log (first expression)))))

  (define-classification logarithmic
    (when (listp expression)
      (and (= 3 length)
         (eq 'log (first expression)))))

WORKING Rationals

  (define-classification rational
    (when (listp expression)
      (and (= 3 length)
         (eq '/ (first expression)))))

WORKING Polynomial Terms

  (define-classification polynomial-term
    (or (expression-type-p expression 'numeric)
       (expression-type-p expression 'variable)
       (expression-type-p expression 'power)
       (and (expression-type-p expression 'multiplicative)
          (= (length (rest expression)) 2)
          (or (and (expression-type-p (second expression) 'numeric)
                (or (expression-type-p (third expression) 'power)
                   (expression-type-p (third expression) 'variable)))
             (and (expression-type-p (third expression) 'numeric)
                (or (expression-type-p (second expression) 'power)
                   (expression-type-p (second expression) 'variable)))))))

WORKING Polynomials

  (define-classification polynomial
    (when (listp expression)
      (and (or (eq '- (first expression))
            (eq '+ (first expression)))
         (reduce #'(lambda (a b)
                     (and a b))
                 (map 'list
                   #'(lambda (the-expression)
                       (expression-type-p the-expression 'polynomial-term))
                   (rest expression))))))

WORKING Trigonometrics

  (define-classification sin
    (when (listp expression)
      (eq 'sin (first expression))))

  (define-classification cos
    (when (listp expression)
      (eq 'cos (first expression))))

  (define-classification tan
    (when (listp expression)
      (eq 'tan (first expression))))

  (define-classification csc
    (when (listp expression)
      (eq 'csc (first expression))))

  (define-classification sec
    (when (listp expression)
      (eq 'sec (first expression))))

  (define-classification cot
    (when (listp expression)
      (eq 'cot (first expression))))

WORKING Term Collector

Foo

  (defun collect-terms (expression)
    (let ((terms (rest expression)))
      ))

WORKING Polynomial Related Functions

  (defun coefficient (term)
    (when (expression-type-p term 'polynomial-term)
      (cond
        ((expression-type-p term 'variable) 1)
        ((expression-type-p term 'power) 1)
        ((expression-type-p term 'multiplicative) (second term))
        ((expression-type-p term 'numeric) term))))

  (defun term-variable (term)
    (when (expression-type-p term 'polynomial-term)
      (cond
        ((expression-type-p term 'multiplicative) (second (third term)))
        ((expression-type-p term 'power) (second term))
        (t nil))))

  (defun get-power (term)
    (cond
      ((expression-type-p term 'power) (third term))
      ((expression-type-p term 'polynomial-term) (third (third term)))
      (t 0)))

  (defun same-order-p (term-a term-b)
    (= (get-power term-a)
       (get-power term-b)))

  (defun same-variable-p (term-a term-b)
    (eq (term-variable term-a)
        (term-variable term-b)))

  (defun single-term-combinable-p (term-a term-b)
    (and (same-order-p term-a term-b)
       (same-variable-p term-a term-b)))

WORKING Expression Manipulators [0/8]

Foo

  <<misc-manipulator-functions>>
  <<define-expression-manipulator>>
  <<external-manipulator>>
  <<addition-manipulator>>
  <<subtraction-manipulator>>
  <<multiplication-manipulators>>

WORKING Manipulator Miscellaneous Functions

  (defvar *manipulator-map* '())

  (defun gen-args-list (count)
    (let ((letters '(a b c d e f g h i j k l m n o p q r s t u v w x y z)))
      (loop for i from 1 to count
         collect (symbolicate 'expression- (nth (1- i) letters)))))

WORKING Define Expression Manipulator

  (defmacro define-operation (name arity short)
    (check-type name symbol)
    (check-type arity (integer 1 26))
    (check-type short symbol)
    (let* ((args (gen-args-list arity))
           (expression-types (map 'list #'(lambda (x)
                                            (symbolicate x '-type)) args))
           (rules-name (symbolicate '*manipulators- name '*))
           (base-manipulator-name (symbolicate name '-manipulator-))
           (manipulator-define-name (symbolicate 'define- name '-manipulator))
           (is-applicable-name (symbolicate name '-is-applicable-p))
           (get-operations-name (symbolicate 'get- name '-manipulators))
           (type-check-list (let ((i 0))
                              (loop for arg in args
                                 collect (prog1
                                             `(expression-type-p ,arg (nth ,i types))
                                           (incf i))))))
      `(progn
         (push '(,short . ,name) *manipulator-map*)
         (defvar ,rules-name '())
         (defun ,is-applicable-name (types ,@args)
           (and ,@type-check-list))
         (defun ,get-operations-name (,@args)
           (remove-if #'null
                      (map 'list #'(lambda (option)
                                     (let ((types (car option))
                                           (name (cdr option)))
                                       (if (,is-applicable-name types ,@args)
                                           name)))
                           ,rules-name)))
         (defun ,name (,@args)
           (funcall (first (,get-operations-name ,@args))
                    ,@args))
         (defmacro ,manipulator-define-name ((,@expression-types) &body body)
           (let ((manipulator-name (symbolicate ',base-manipulator-name ,@expression-types)))
             `(progn
                (setf ,',rules-name (append ,',rules-name '(((,,@expression-types) . ,manipulator-name))))
                (defun ,manipulator-name ,',args
                  ,@body)))))))

WORKING External Manipulator

  ;; (defun manipulate (action &rest expressions)
  ;;   (case action
  ;;     (+
  ;;      (reduce #'add expressions))
  ;;     (-
  ;;      (reduce #'subtract expressions))
  ;;     (*
  ;;      (reduce #'multiply expressions))
  ;;     (/
  ;;      (reduce #'divide expressions))
  ;;     (sin
  ;;      (reduce #'manip-sin expressions))
  ;;     (cos
  ;;      (reduce #'manip-cos expressions))
  ;;     (tan
  ;;      (reduce #'manip-tan expressions))
  ;;     (expt
  ;;      (reduce #'powers expressions))))

WORKING Addition

Foo

  (define-operation add 2 +)

  (define-add-manipulator (numeric numeric)
    (+ expression-a expression-b))

  (define-add-manipulator (numeric additive)
    (let ((total expression-a)
          (remainder (rest expression-b))
          (non-numeric '()))
      (dolist (element remainder)
        (if (expression-type-p element 'numeric)
            (incf total element)
            (push element non-numeric)))
      (cond
        ((null non-numeric)
         total)
        ((= 0 total)
         `(+ ,@non-numeric))
        (t
         `(+ ,total ,@non-numeric)))))

  (define-add-manipulator (additive additive)
    (let ((total 0)
          (elements (append (rest expression-a)
                            (rest expression-b)))
          (non-numeric '()))
      (dolist (element elements)
        (if (expression-type-p element 'numeric)
            (incf total element)
            (push element non-numeric)))
      (cond
        ((null non-numeric)
         total)
        ((= 0 total)
         `(+ ,@non-numeric))
        (t
         `(+ ,total ,@non-numeric)))))

  (define-add-manipulator (numeric subtractive)
    (let ((total expression-a)
          (the-other (rest expression-b))
          (non-numeric '()))
      (dolist (element the-other)
        (if (expression-type-p element 'numeric)
            (decf total element)
            (push element non-numeric)))
      (cond
        ((null non-numeric)
         total)
        ((= 0 total)
         `(+ ,@non-numeric))
        (t
         `(+ ,total (-,@non-numeric))))))

  (define-add-manipulator (numeric polynomial-term)
    `(+ ,expression-a ,expression-b))

  (define-add-manipulator (polynomial-term polynomial-term)
    (if (single-term-combinable-p expression-a expression-b)
        (let ((new-coefficient (+ (coefficient expression-a)
                                  (coefficient expression-b)))
              (variable (term-variable expression-a))
              (power (get-power expression-a)))
          `(* ,new-coefficient (expt ,variable ,power)))
        `(+ ,expression-a ,expression-b)))

  (define-add-manipulator (* numeric)
    (add expression-b expression-a))

WORKING Subtraction

Foo

  (define-operation subtract 2 -)

  (define-subtract-manipulator (numeric numeric)
    (- expression-a expression-b))

  (define-subtract-manipulator (numeric subtractive)
    (let ((total expression-a)
          (elements (rest expression-b))
          (non-numeric '()))
      (dolist (element elements)
        (if (expression-type-p element 'numeric)
            (decf total element)
            (push element non-numeric)))
      (cond
        ((null non-numeric)
         total)
        ((= 0 total)
         `(- ,@(reverse non-numeric)))
        (t
         `(- ,total ,@(reverse non-numeric))))))

  (define-subtract-manipulator (* numeric)
    (subtract expression-b expression-a))

WORKING Multiplication

Foo

  (define-operation multiply 2 *)

Division

Foo

  (define-operation division 2 /)

Trigonometric

Foo

WORKING Packaging

Foo

  (defpackage #:manipulator
    (:use #:cl)
    (:import-from #:alexandria
                  #:symbolicate)
    (:export #:manipulate))

  (in-package #:manipulator)

  <<determine-expression-type>>

  <<polynomial-related-functions>>

  <<collect-terms>>

  <<expression-manipulation>>