Racket

Symbolic algebraic expressions🔗ℹ

Johannes Tax

package: symalg

This library provides functions to parse and manipulate symbolic algebraic expressions. These expression can be constants, variables, arithmetic operations and exponentiations. Additionally trigonometric functions and logarithms are supported.

1 Example🔗ℹ

> (define expr (parse-infix "3*x^2 - 4*x + cos(x)"))
> (define expr-deriv (simplify
                       (differentiate expr)))
> expr-deriv
(add
(list
  (num -4)
  (mul (list (num 6) (sym 'x)))
  (mul (list (num -1) (sin_ (sym 'x))))))
> (infix expr-deriv)
"-4 + 6 * x + -1 * sin(x)"
> (sexpr expr-deriv)
'(+ -4 (* 6 x) (* -1 (sin x)))
> (latex expr-deriv)
"-4 + 6 x -\\sin(x)"
> (define f (evaluate expr-deriv))
> (f 3)
13.858879991940134

2 API Reference🔗ℹ

procedure
(parse-sexpr s) → symalg-expr?
s : any/c

Parses an s-expression and returns a corresponding symbolic algebraic expression. s can be an expression expr of the following form:

expr :

number?

| symbol?

| e

| pi

| (+ expr ...+)

| (- expr ...+)

| (* expr ...+)

| (/ expr expr)

| (expt expr expr)

| (log expr expr)

| (sin expr)

| (cos expr)

| (tan expr)

procedure
(parse-infix s) → symalg-expr?
s : string?

Parses a string containing an infix expression and returns a corresponding symbolic algebraic expression. An infix expression can be an expression expr of the following form:

expr :

number?

| symbol?

| e

| pi

| (expr)

| expr + expr

| expr - expr

| expr * expr

| expr / expr

| expr ^ expr

| log(expr, expr)

| sin(expr)

| cos(expr)

| tan(expr)

procedure
(symalg-expr? e) → boolean?
e : any/c

This predicate checks, if the argument denotes a symbolic algebraic expression that can be processed by the functions below.

procedure
(simplify e) → symalg-expr?
e : symalg-expr?

Returns a simplified form of e. Simplification is mostly based on Joel S. Cohen’s Computer Algebra and Symbolic Computation.

Some examples:

> (infix (simplify (parse-infix "x+x")))
"2 * x"
> (infix (simplify (parse-infix "x^0")))
"1"
> (infix (simplify (parse-infix "2*x^2 + 4*x^2 + 5 - 6")))
"-1 + 6 * (x)^(2)"
> (infix (simplify (parse-infix "2*x^2 / x")))
"2 * x"
> (infix (simplify (parse-infix "2^x^4")))
"(2)^(4 * x)"

procedure
(differentiate e) → symalg-expr?
e : symalg-expr?

The function differentiate computes the first derivation of a given symbolic algebraic expression.

Take into account that the resulting expression is not simplified automatically, a further call to simplify is necessary to bring it into a canonical form:

> (define expr (parse-infix "2*x^2 - x"))
> (infix (differentiate expr))
"(x)^(2) * 1 * 2 * (x)^(-1) + 0 * log(x) * 2 + (x)^(2) * 0 + 1 * -1 + x * 0"
> (infix (simplify (differentiate expr)))
"-1 + 4 * x"

procedure
(evaluate e) → (f number? ...+)
e : symalg-expr?

Returns a function that evaluates the given symbolic algebraic expression. Parameters of the returned function are bound to variables (unbound symbols) of the symbolic algebraic expression in alphabetical order.

procedure
(infix e) → string?
e : symalg-expr?

Returns the infix representation of a symbolic algebraic expression.

procedure
(latex e) → string?
e : symalg-expr?

Returns the LaTeX math mode representation of a symbolic algebraic expression.

procedure
(sexpr e) → any/c
e : symalg-expr?

Returns the s-expression representation of a symbolic algebraic expression.