sicmutils.rational-function — sicmutils 0.22.0

->expression clj/s

(->expression r vars)

Accepts a [[RationalFunction]] r and a sequence of symbols for each indeterminate, and emits the canonical form of the symbolic expression that represents [[RationalFunction]] r.

NOTE: this is the output stage of Rational Function canonical form simplification. The input stage is handled by expression->.

NOTE See analyzer for an instance usable by sicmutils.expression.analyze/make-analyzer.

abs clj/s

(abs r)

If the numerator of r is negative, returns (negate r), else acts as identity.

add clj/s

(add r s)

Returns the sum of rational functions r and s, with appropriate handling of [[RationalFunction]], [[polynomial/Polynomial]] or coefficients of neither type on either side.

analyzer clj/s

Singleton [[a/ICanonicalize]] instance.

arg-scale clj/s

(arg-scale r factors)

Given some [[RationalFunction]] r, returns a new [[RationalFunction]] generated by substituting each indeterminate x_i for f_i * x_i, where f_i is a factor supplied in the factors sequence.

Given a non-[[RationalFunction]], delegates to polynomial/arg-scale.

arg-shift clj/s

(arg-shift r shifts)

Given some [[RationalFunction]] r, returns a new [[RationalFunction]] generated by substituting each indeterminate x_i for s_i + x_i, where s_i is a shift supplied in the shifts sequence.

Given a non-[[RationalFunction]], delegates to polynomial/arg-shift.

arity clj/s

(arity r)

Returns the declared arity of the supplied [[RationalFunction]] or [[polynomial/Polynomial]], or 0 for arguments of other types.

coeff? clj/s

(coeff? x)

Returns true if x is explicitly not an instance of [[RationalFunction]] or [[polynomial/Polynomial]], false if it is.

cube clj/s

(cube r)

Returns the cube of rational function r. Equivalent to (mul r (mul r r)).

div clj/s

(div r s)

Returns the quotient of rational functions r and s, with appropriate handling of [[RationalFunction]], [[polynomial/Polynomial]] or coefficients of neither type on either side.

eq clj/s

(eq this that)

Returns true if the [[RationalFunction]] this is equal to that. If that is a [[RationalFunction]], this and that are equal if they have equal u and v and equal arity. u and v entries are compared using sicmutils.value/=.

If that is non-[[RationalFunction]], eq only returns true if u and v respectively match the [[ratio/numerator]] and [[ratio/denominator]] of that.

evaluate clj/s

(evaluate r xs)

Given some rational function xs and a sequence of arguments with length >= 0 and < the arity of r, returns the result of evaluating the numerator and denominator using xs and re-forming a rational function with the results.

Supplying fewer arguments than the arity will result in a partial evaluation. Supplying too many arguments will error.

expression-> clj/s

(expression-> expr)

(expression-> expr cont)

(expression-> expr cont v-compare)

Converts the supplied symbolic expression expr into Rational Function canonical form (ie, a [[RationalFunction]] instance). expr should be a bare, unwrapped expression built out of Clojure data structures.

Returns the result of calling continuation cont with the [[RationalFunction]] and the list of variables corresponding to each indeterminate in the [[RationalFunction]]. (contdefaults to vector).

The second optional argument v-compare allows you to provide a Comparator between variables. Sorting indeterminates by v-compare will determine the order of the indeterminates in the generated [[RationalFunction]]. The list of variables passed to cont will be sorted using v-compare.

Absorbing an expression with expression-> and emitting it again with ->expression will generate the canonical form of an expression, with respect to the operations in the [[operators-known]] set.

This kind of simplification proceeds purely symbolically over the known Rational Function operations; other operations outside the arithmetic available should be factored out by an expression analyzer (see sicmutils.expression.analyze/make-analyzer) before calling expression->.

NOTE that cont might receive a scalar, fraction or [[polynomial/Polynomial]] instance; both are valid 'rational functions'. The latter as a rational function with a denominator equal to 1, and the former 2 result from non-polynomial numerator and denominator.

NOTE See analyzer for an instance usable by sicmutils.expression.analyze/make-analyzer.

expt clj/s

(expt r n)

Returns a rational function generated by raising the input rational function r to the (integer) power n.

from-points clj/s

(from-points xs)

Given a sequence of points of the form [x, f(x)], returns a rational function that passes through each input point.

gcd clj/s

(gcd r s)

Returns the greatest common divisor of rational functions r and s, with appropriate handling of [[RationalFunction]], [[polynomial/Polynomial]] or coefficients of neither type on either side.

invert clj/s

(invert r)

Given some rational function r, returns the inverse of r, ie, a rational function with numerator and denominator reversed. The returned rational function guarantees a positive denominator.

Acts as generic/invert for non-[[RationalFunction]] inputs.

make clj/s

(make u v)

Given a numerator u and denominator v, attempts to form a [[RationalFunction]] instance by

• cancelling out any common factors between u and v
• normalizing u and v such that v is always positive
• multiplying u and v through by a commo factor, such that neither term contains any rational coefficients

Returns a [[RationalFunction]] instance if either u or v remains a [[polynomial/Polynomial]] after this process; else, returns (g/div u' v'), where u' and v' are the reduced numerator and denominator.

mul clj/s

(mul r s)

Returns the product of rational functions r and s, with appropriate handling of [[RationalFunction]], [[polynomial/Polynomial]] or coefficients of neither type on either side.

negate clj/s

(negate r)

Returns the negation of rational function r, ie, a [[RationalFunction]] with its numerator negated.

Acts as generic/negate for non-[[RationalFunction]] inputs.

negative? clj/s

(negative? r)

Returns true if the numerator of r is polynomial/negative?, false otherwise.

partial-derivative clj/s

(partial-derivative r i)

Given some [[RationalFunction]] or [[polynomial/Polynomial]] r, returns the partial derivative of r with respect to the ith indeterminate. Throws if i is an invalid indeterminate index for r.

For non-polynomial or rational function inputs, returns 0.

partial-derivatives clj/s

(partial-derivatives r)

Returns the sequence of partial derivatives of [[RationalFunction]] (or [[polynomial/Polynomial]]) r with respect to each indeterminate. The returned sequence has length equal to the arity of r.

For non-polynomial or rational function inputs, returns an empty sequence.

rational-function? clj/s

(rational-function? r)

Returns true if the supplied argument is an instance of [[RationalFunction]], false otherwise.

square clj/s

(square r)

Returns the square of rational function r. Equivalent to (mul r r).

sub clj/s

(sub r s)

Returns the difference of rational functions r and s, with appropriate handling of [[RationalFunction]], [[polynomial/Polynomial]] or coefficients of neither type on either side.