7 Built-in Prefix Operators

In the following subsections are descriptions of the most useful prefix operators built into REDUCE that are not defined in other sections (such as substitution operators). Some are fully defined internally as procedures; others are more nearly abstract operators, with only some of their properties known to the system.

In many cases, an operator is described by a prototypical header line as follows. Each formal parameter is given a name and followed by its allowed type. The names of classes referred to in the definition are printed in lower case, and parameter names in upper case. If a parameter type is not commonly used, it may be a specific set enclosed in brackets { … }. Operators that accept formal parameter lists of arbitrary length have the parameter and type class enclosed in square brackets indicating that zero or more occurrences of that argument are permitted. Optional parameters and their type classes are enclosed in angle brackets.

7.1 Numerical Operators

REDUCE includes a number of functions that are analogs of those found in most numerical systems. With numerical arguments, such functions return the expected result. However, they may also be called with non-numerical arguments. In such cases, except where noted, the system attempts to simplify the expression as far as it can. In such cases, a residual expression involving the original operator usually remains. These operators are as follows:

AbstractTensors.absFunction
abs(r)

abs returns the absolute value of its single argument, if that argument has a numerical value. A non-numerical argument is returned as an absolute value, with an overall numerical coefficient taken outside the absolute value operator. For example:

julia> Algebra.abs(-3/4)
0.75

julia> Algebra.abs(:(2a))
:(2 * abs(a))

julia> Algebra.abs(im)
1.0

julia> Algebra.abs(:(-x))
:(abs(x))
source
Reduce.Algebra.ceilingFunction
ceiling(r)

This operator returns the ceiling (i.e., the least integer greater than the given argument) if its single argument has a numerical value. A non-numerical argument is returned as an expression in the original operator. For example:

julia> Algebra.ceiling(-5/4)
-1

julia> Algebra.ceiling(:(-a))
:(ceiling(-a))
source
AbstractTensors.conjFunction
conj(r)

This operator returns the ceiling (i.e., the least integer greater than the given argument) if its single argument has a numerical value. A non-numerical argument is returned as an expression in the original operator. For example:

julia> Algebra.conj(1+im)
1 - 1im

julia> Algebra.conj(:(a+im*b))
:(repart(a) - ((impart(a) + repart(b)) * im + impart(b)))
source
Reduce.Algebra.factorialFunction
factorial(r)

If the single argument of factorial evaluates to a non-negative integer, its factorial is returned. Otherwise an expression involving factorial is returned. For example:

julia> Algebra.factorial(5)
120

julia> Algebra.factorial(:a)
:(factorial(a))
source
Reduce.Algebra.fixFunction
fix(r)

This operator returns the fixed value (i.e., the integer part of the given argument) if its single argument has a numerical value. A non-numerical argument is returned as an expression in the original operator. For example:

julia> Algebra.fix(-5/4)
-1

julia> Algebra.fix(:a)
:(fix(a))
source
Reduce.Algebra.floorFunction
floor(r)

This operator returns the floor (i.e., the greatest integer less than the given argument) if its single argument has a numerical value. A non-numerical argument is returned as an expression in the original operator. For example:

julia> Algebra.floor(-5/4)
-2.0

julia> Algebra.floor(:a)
:(floor(a))
source
Reduce.Algebra.impartFunction
impart(r)

This operator returns the imaginary part of an expression, if that argument has an numerical value. A non-numerical argument is returned as an expression in the operators repart and impart. For example:

julia> Algebra.impart(1+im)
1

julia> Algebra.impart(:(a+im*b))
:(impart(a) + repart(b))
source
Reduce.Algebra.maxFunction
max(r...)

max can take an arbitrary number of expressions as their arguments. If all arguments evaluate to numerical values, the maximum of the argument list is returned. If any argument is non-numeric, an appropriately reduced expression is returned. For example:

julia> Algebra.max(2,-3,4,5)
5

julia> Algebra.max(:a,2,3)
:(max(3, a))

max of an empty list returns 0.

source
Reduce.Algebra.minFunction
min(r...)

min can take an arbitrary number of expressions as their arguments. If all arguments evaluate to numerical values, the minimum of the argument list is returned. If any argument is non-numeric, an appropriately reduced expression is returned. For example:

julia> Algebra.min(2,-2)
-2

julia> Algebra.min(:x)
:x

min of an empty list returns 0.

source
Reduce.Algebra.nextprimeFunction
nextprime(r)

nextprime returns the next prime greater than its integer argument, using a probabilistic algorithm. A type error occurs if the value of the argument is not an integer. For example:

julia> Algebra.nextprime(5)
7

julia> Algebra.nextprime(-2)
2

julia> Algebra.nextprime(-7)
-5

julia> Algebra.nextprime(1000000)
1000003

whereas Algebra.nextprime(:a) gives a type error.

source
Reduce.Algebra.randomFunction
random(r)

random(n) returns a random number $r$ in the range $0 ≤ r < n$. A type error occurs if the value of the argument is not a positive integer in algebraic mode, or positive number in symbolic mode. For example:

julia> Algebra.random(5)
3

julia> Algebra.random(1000)
191

whereas Algebra.random(:a) gives a type error.

source
Reduce.Algebra.random_new_seedFunction
random_new_seed(r)

random_new_seed(n) reseeds the random number generator to a sequence determined by the integer argument n. It can be used to ensure that a repeatable pseudo-random sequence will be delivered regardless of any previous use of random, or can be called early in a run with an argument derived from something variable (such as the time of day) to arrange that different runs of a REDUCE program will use different random sequences. When a fresh copy of REDUCE is first created it is as if random_new_seed(1) has been obeyed.

A type error occurs if the value of the argument is not a positive integer.

source
Reduce.Algebra.repartFunction
repart(r)

This returns the real part of an expression, if that argument has an numerical value. A non-numerical argument is returned as an expression in the operators repart and impart. For example:

julia> Algebra.repart(1+im)
1

julia> Algebra.repart(:(a+im*b))
:(repart(a) - impart(b))
source
Reduce.Algebra.roundFunction
round(r)

This operator returns the rounded value (i.e, the nearest integer) of its single argument if that argument has a numerical value. A non-numeric argument is returned as an expression in the original operator. For example:

julia> Algebra.round(-5/4)
-1.0

julia> Algebra.round(:a)
:(round(a))
source
Reduce.Algebra.signFunction
sign(r)

sign tries to evaluate the sign of its argument. If this is possible sign returns one of 1, 0 or -1. Otherwise, the result is the original form or a simplified variant. For example:

julia> Algebra.sign(-5)
-1

julia> Algebra.sign(:(-a^2*b))
:(-(sign(b)))

Note that even powers of formal expressions are assumed to be positive only as long as the switch complex is off.

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7.2 Mathematical Functions

REDUCE knows that the following represent mathematical functions that can take arbitrary scalar expressions as their argument(s):

acos acosh acot acoth acsc acsch asec asech asin  
asinh atan atanh atan2 beta ci cos cosh cot coth csc  
csch dilog ei exp gamma hypot ibeta igamma ln log  
logb log10 sec sech si sin sinh sqrt tan tanh  
airy_ai airy_aiprime airy_bi airy_biprime  
besseli besselj besselk bessely  
hankel1 hankel2 kummerm kummeru lommel1 lommel2  
struveh struvel whittakerm whittakeru  
polygamma psi zeta  
solidharmonicy sphericalharmonicy

where log is the natural logarithm (and equivalent to ln), and logb has two arguments of which the second is the logarithmic base.

The derivatives of all these functions are also known to the system.

REDUCE knows various elementary identities and properties of these functions. For example:

 cos(-x) = cos(x)              sin(-x) = - sin (x)  
 cos(n*pi) = (-1)^n            sin(n*pi) = 0  
 log(e)  = 1                   e^(i*pi/2) = i  
 log(1)  = 0                   e^(i*pi) = -1  
 log(e^x) = x                  e^(3*i*pi/2) = -i

Beside these identities, there are a lot of simplifications for elementary functions defined in the REDUCE system as rulelists. In order to view these, the showrules operator can be used, e.g.

reduce> showrules tan;  
 
{tan(~n*arbint(~i)*pi + ~(~ x)) => tan(x) when fixp(n),  
 
 tan(~x)  
 
  => trigquot(sin(x),cos(x)) when knowledge_about(sin,x,tan),  
 
      ~x + ~(~ k)*pi  
 tan(----------------)  
            ~d  
 
             x                  k                     k     1  
  =>  - cot(---) + i*pi*impart(---)) when abs(repart(---))=---,  
             d                  d                     d     2  
 
      ~(~ w) + ~(~ k)*pi           w      k                k  
 tan(--------------------) => tan(--- + (--- - fix(repart(---)))*pi)  
            ~(~ d)                 d      d                d  
 
                              k  
 when whereexp({rp => repart(---)},bool-eval(ratnump(rp) and abs(rp)>=1)),  
                              d  
 
 tan(atan(~x)) => x,  
 
                             2  
 df(tan(~x),~x) => 1 + tan(x) }  

For further simplification, especially of expressions involving trigonometric functions, see the TRIGSIMP package (chapter 16.72) documentation.

Functions not listed above may be defined in the special functions package SPECFN.

The user can add further rules for the reduction of expressions involving these operators by using the let command.

In many cases it is desirable to expand product arguments of logarithms, or collect a sum of logarithms into a single logarithm. Since these are inverse operations, it is not possible to provide rules for doing both at the same time and preserve the REDUCE concept of idempotent evaluation. As an alternative, REDUCE provides two switches expandlogs and combinelogs to carry out these operations. Both are off by default, and are subject to the value of the switch precise. This switch is on by default and prevents modifications that may be false in a complex domain. Thus to expand log(3*y) into a sum of logs, one can say

julia> Algebra.on(:expandlogs);

julia> Algebra.log(:(3*y))

whereas to expand log(x*y) into a sum of logs, one needs to say

julia> Algebra.off(:precise); Algebra.on(:expandlogs);

julia> Algebra.log(:(x*y))

To combine this sum into a single log:

julia> Algebra.off(:precise); Algebra.on(:combinelogs);

julia> Alebra.:+(log(:x),log(:y))

These switches affect the logarithmic functions log10 (base 10) and logb (arbitrary base) as well.

At the present time, it is possible to have both switches on at once, which could lead to infinite recursion. However, an expression is switched from one form to the other in this case. Users should not rely on this behavior, since it may change in the next release.

The current version of REDUCE does a poor job of simplifying surds. In particular, expressions involving the product of variables raised to non-integer powers do not usually have their powers combined internally, even though they are printed as if those powers were combined. For example, the expression

reduce> x^(1/3)*x^(1/6)

will print as

sqrt(x)

but will have an internal form containing the two exponentiated terms. If you now subtract sqrt(x) from this expression, you will not get zero. Instead, the confusing form

sqrt(x) - sqrt(x)

will result. To combine such exponentiated terms, the switch combineexpt should be turned on.

The square root function can be input using the name sqrt, or the power operation ^(1/2). On output, unsimplified square roots are normally represented by the operator sqrt rather than a fractional power. With the default system switch settings, the argument of a square root is first simplified, and any divisors of the expression that are perfect squares taken outside the square root argument. The remaining expression is left under the square root. Thus the expression

julia> Algebra.sqrt(:(-8a^2*b))

becomes

:(2 * sqrt(b) * sqrt(2) * a * im)

Note that such simplifications can cause trouble if A is eventually given a value that is a negative number. If it is important that the positive property of the square root and higher even roots always be preserved, the switch PRECISE should be set on (the default value). This causes any non-numerical factors taken out of surds to be represented by their absolute value form. With PRECISE on then, the above example would become

:(2 * sqrt(-2b) * abs(a))

However, this is incorrect in the complex domain, where the $\sqrt{x^2}$ is not identical to $|x|$. To avoid the above simplification, the switch precise_complex should be set on (default is off). For example:

julia> Algebra.on(:precise_complex); Algebra.sqrt(:(-8a^2*b))

yields the output

:(2 * sqrt(-2 * a ^ 2 * b))

The statement that REDUCE knows very little about these functions applies only in the mathematically exact off rounded mode. If rounded is on, any of the functions

acos acosh acot acoth acsc acsch asec asech asin  
asinh atan atanh atan2 cos cosh cot coth csc csch  
exp hypot ibeta igamma ln log logb log10 psi sec  
sech sin sinh sqrt tan tanh

when given a numerical argument has its value calculated to the current degree of floating point precision. In addition, real (non-integer valued) powers of numbers will also be evaluated.

If the complex switch is turned on in addition to rounded, these functions will also calculate a real or complex result, again to the current degree of floating point precision, if given complex arguments. For example,

julia> @rounded @complex 2.3^(5.6im)
:(-0.0480793490914 - 0.998843519372im)

julia> @rounded @complex cos(2+3im)
:(-4.18962569097 - 9.10922789376im)

7.3 Bernoulli Numbers and Euler Numbers

Reduce.Algebra.bernoulliFunction
bernoulli(n)

The unary operator bernoulli provides notation and computation for Bernoulli numbers. bernoulli(n) evaluates to the nth Bernoulli number; all of the odd Bernoulli numbers, except bernoulli(1), are zero.

The algorithms are based upon those by Herbert Wilf, presented by Sandra Fillebrown [?]. If the rounded switch is off, the algorithms are exactly those; if it is on, some further rounding may be done to prevent computation of redundant digits. Hence, these functions are particularly fast when used to approximate the Bernoulli numbers in rounded mode.

source

Euler numbers are computed by the unary operator Euler, which return the nth Euler number. The computation is derived directly from Pascal’s triangle of binomial coefficients.

7.4 Fibonacci Numbers and Fibonacci Polynomials

Reduce.Algebra.fibonacciFunction
fibonacci(n)

The unary operator fibonacci provides notation and computation for Fibonacci numbers. fibonacci(n) evaluates to the nth Fibonacci number. If n is a positive or negative integer, it will be evaluated following the definition:

\[F_0 = 0; F_1 = 1; F_n = F_{n-1} + F_{n-2}\]
source
Reduce.Algebra.fibonaccipFunction
fibonacci(n,x)

Fibonacci Polynomials are computed by the binary operator fibonaccip. fibonaccip(n,x) returns the nth Fibonacci polynomial in the variable x. If n is a positive or negative integer, it will be evaluated following the definition:

\[F_0(x) = 0; F_1(x) = 1; F_n(x) = xF_{n-1}(x) + F_{n-2}(x)\]
source

7.5 Motzkin numbers

Reduce.Algebra.motzkinFunction
motzkin(n)

A Motzkin number $M_n$ (named after Theodore Motzkin) is the number of different ways of drawing non-intersecting chords on a circle between n points. For a non-negative integer n, the operator motzkin(n) returns the nth Motzkin number, according to the recursion formula

\[M_0 = 1; M_1 = 1; M_{n+1} = \frac{2n+3}{n+3}M_n + \frac{3n}{n+3}M_{n-1}.\]
source

7.6 CHANGEVAR Operator

Author: G. Üçoluk.

Reduce.Algebra.changevarFunction
changevar(depvars,newvars,eqlist,diffeq)

The operator changevar does a variable transformation in a set of differential equations. Syntax:

changevar(⟨depvars⟩,⟨newvars⟩,⟨eqlist⟩,⟨diffeq⟩)

⟨diffeq⟩ is either a single differential equation or a list of differential equations, ⟨depvars⟩ are the dependent variables to be substituted, ⟨newvars⟩ are the new depend variables, and ⟨eqlist⟩ is a list of equations of the form ⟨depvar⟩=⟨expression⟩ where ⟨expression⟩ is some function in the new dependent variables.

The three lists ⟨depvars⟩, ⟨newvars⟩, and ⟨eqlist⟩ must be of the same length. If there is only one variable to be substituted, then it can be given instead of the list. The same applies to the list of differential equations, i.e., the following two commands are equivalent

Algebra.operator(:u)
Algebra.changevar(:u,:y,:(x==e^y),:(df(u(x),x) - log(x)))
Algebra.changevar((:u,),(:y,),(:(x=e^y),),(:(df(u(x),x) - log(x)),))

except for one difference: the first command returns the transformed differential equation, the second one a list with a single element.

source

The switch dispjacobian governs the display the entries of the inverse Jacobian, it is off per default.

The mathematics behind the change of independent variable(s) in differential equations is quite straightforward. It is basically the application of the chain rule. If the dependent variable of the differential equation is $F$ , the independent variables are $x_i$ and the new independent variables are $u_i$ (where $i=1…n$) then the first derivatives are: $\frac{-\partial F}{\partial x_i} = \frac{\partial F}{\partial u_j} \frac{\partial u_j}{\partial x_i}$

We assumed Einstein’s summation convention. Here the problem is to calculate the $∂u_j∕∂x_i$ terms if the change of variables is given by $x_i = f_i (u_1 ,...,u_n ).$

The first thought might be solving the above given equations for $u_j$ and then differentiating them with respect to $x_i$, then again making use of the equations above, substituting new variables for the old ones in the calculated derivatives. This is not always a preferable way to proceed. Mainly because the functions $f_i$ may not always be easily invertible. Another approach that makes use of the Jacobian is better. Consider the above given equations which relate the old variables to the new ones. Let us differentiate them:

\[\frac{\partial x_j}{\partial x_i} = \frac{\partial f_j}{\partial x_i}\]
\[\delta_{ij} = \frac{\partial f_j \partial u_k}{\partial u_k \partial x_i}\]

The first derivative is nothing but the $(j,k)$ th entry of the Jacobian matrix.

So if we speak in matrix language $1 = J ⋅ D$ where we defined the Jacobian $J_{ij} = \frac{\partial f_i}{\partial u_j}$ and the matrix of the derivatives we wanted to obtain as $D_{ij} = \frac{\partial u_i}{\partial x_j}.$ If the Jacobian has a non-vanishing determinant then it is invertible and we are able to write from the matrix equation above: $D = J^{-1}$ so finally we have what we want $\frac{\partial u_i}{\partial x_j} = [ J^{-1} ]_{ij}$

The higher derivatives are obtained by the successive application of the chain rule and using the definitions of the old variables in terms of the new ones. It can be easily verified that the only derivatives that are needed to be calculated are the first order ones which are obtained above.

7.6.1 CHANGEVAR example: The 2-dim. Laplace Equation

The 2-dimensional Laplace equation in cartesian coordinates is: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ Now assume we want to obtain the polar coordinate form of Laplace equation. The change of variables is: $x = r\cos θ, y = r\sin θ$ The solution using changevar is as follows

Algebra.changevar((:u,),(:r,:θ),(:(x=r*cos(θ)),:(y=r*sin(θ))),
            (:(df(u(x,y),x,2)+df(u(x,y),y,2)),) )

Here we could omit the list parenthesis in the first and last arguments (because those lists have only one member) and the list parenthesis in the third argument (because they are optional), but you cannot leave off the list parenthesis in the second argument. So one could equivalently write

Algebra.changevar(:u,(:r,:θ),:(x==r*cos(θ)),:(y==r*sin(θ)),  
             :(df(u(x,y),x,2)+df(u(x,y),y,2)) )

If you have tried out the above example, you will notice that the denominator contains a $\cos^2θ + \sin^2θ$ which is actually equal to 1. This has of course nothing to do with changevar. One has to be overcome these pattern matching problems by the conventional methods REDUCE provides (a rule, for example, will fix it).

Secondly you will notice that your u(x,y) operator has changed to u(r,θ) in the result. Nothing magical about this. That is just what we do with pencil and paper. u(r,θ) represents the the transformed dependent variable.

7.6.2 Another CHANGEVAR example: An Euler Equation

Consider a differential equation which is of Euler type, for instance: $x^3y ′′′ - 3x^2y′′ + 6xy ′ - 6y = 0$ where prime denotes differentiation with respect to $x$. As is well known, Euler type of equations are solved by a change of variable: $x = e^u.$ So our call to changevar reads as follows:

Algebra.changevar(:y, :u, :(x==e^u), :(x^3*df(y(x),x,3)-  
             3*x^2*df(y(x),x,2)+6*x*df(y(x),x)-6*y(x)))

and returns the result

:(((11 * df(y(u), u) - 6 * y(u)) - 6 * df(y(u), u, 2)) + df(y(u), u, 3))

7.7 CONTINUED_FRACTION Operator

Not initially supported by Reduce.jl parser, see upstream docs for more information.

7.8 DF Operator

Reduce.Algebra.dfFunction
df(r...)

The operator df is used to represent partial differentiation with respect to one or more variables. It is used with the syntax:

R"df(⟨EXPRN:algebraic⟩[,⟨VAR:kernel⟩ <,⟨NUM:integer⟩ >])"

The first argument is the expression to be differentiated. The remaining arguments specify the differentiation variables and the number of times they are applied.

The number num may be omitted if it is 1. For example,

reduce> df(y,x)

reduce> df(y,x,2)

reduce> df(y,x1,2,x2,x3,2)

The evaluation of df(y,x) proceeds as follows: first, the values of y and x are found. Let us assume that x has no assigned value, so its value is x. Each term or other part of the value of y that contains the variable x is differentiated by the standard rules. If z is another variable, not x itself, then its derivative with respect to x is taken to be 0, unless z has previously been declared to depend on x, in which case the derivative is reported as the symbol df(z,x).

source

7.8.1 Switches influencing differentiation

Consider df(u,x,y,z). If none of x,y,z are equal to u then the order of differentiation is commuted into a canonical form, unless the switch nocommutedf is turned on (default is off). if at least one of x,y,z is equal to u then the order of differentiation is not commuted and the derivative is not simplified to zero, unless the switch commutedf is turned on. It is off by default.

If commutedf is off and the switch simpnoncomdf is on then simplify as follows:

df(u,x,u)        ->  df(u,x,2) / df(u,x)  
df(u,x,n,u)      ->  df(u,x,n+1) / df(u,x)

provided u depends only on the one variable x. This simplification removes the non-commutative aspect of the derivative.

If the switch expanddf is turned on then REDUCE uses the chain rule to expand symbolic derivatives of indirectly dependent variables provided the result is unambiguous, i.e. provided there is no direct dependence. It is off by default. Thus, for example, given

julia> Algebra.depend(:f,:u,:v); Algebra.depend((:u,:v),:x)

julia> Algebra.on(:expanddf)

julia> Algebra.df(:f,:x)
:(df(f, u) * df(u, x) + df(f, v) * df(v, x))

whereas after

julia> Algebra.depend(:f,:x)

df(:f,:x) does not expand at all (since the result would be ambiguous and the algorithm would loop).

Turning on the switch allowdfint allows “differentiation under the integral sign”, i.e.

df(int(y, x), v) -> int(df(y, v), x)

if this results in a simplification. If the switch dfint is also turned on then this happens regardless of whether the result simplifies. Both switches are off by default.

7.8.2 Adding Differentiation Rules

The let statement can be used to introduce rules for differentiation of user-defined operators. Its general form is

R"for all ⟨var1⟩,…,⟨varn⟩ let df(⟨operator⟩⟨varlist⟩,⟨vari⟩)=⟨expression⟩"

where ⟨varlist⟩ ::= (⟨var1⟩,…,⟨varn⟩), and ⟨var1⟩,…,⟨varn⟩ are the dummy variable arguments of ⟨operator⟩.

An analogous form applies to infix operators.

Examples:

R"for all x let df(tan x,x)= 1 + tan(x)^2"

(This is how the tan differentiation rule appears in the REDUCE source.)

R"for all x,y let df(f(x,y),x)=2*f(x,y),  df(f(x,y),y)=x*f(x,y)"

Notice that all dummy arguments of the relevant operator must be declared arbitrary by the for all command, and that rules may be supplied for operators with any number of arguments. If no differentiation rule appears for an argument in an operator, the differentiation routines will return as result an expression in terms of df. For example, if the rule for the differentiation with respect to the second argument of f is not supplied, the evaluation of df(f(x,z),z) would leave this expression unchanged. (No depend declaration is needed here, since f(x,z) obviously “depends on” z.)

Once such a rule has been defined for a given operator, any future differentiation rules for that operator must be defined with the same number of arguments for that operator, otherwise we get the error message

ERROR: Reduce: 
Incompatible df rule argument length for <operator>

7.9 INT Operator

Reduce.Algebra.intFunction
int(r...)

int is an operator in REDUCE for indefinite integration using a combination of the Risch-Norman algorithm and pattern matching. It is used with the syntax:

R"int(⟨EXPRN:algebraic⟩,⟨VAR:kernel⟩)"

This will return correctly the indefinite integral for expressions comprising polynomials, log functions, exponential functions and tan and atan. The arbitrary constant is not represented. If the integral cannot be done in closed terms, it returns a formal integral for the answer in one of two ways:

  1. It returns the input, int(…,…) unchanged.
  2. It returns an expression involving ints of some other functions (sometimes more complicated than the original one, unfortunately).

Rational functions can be integrated when the denominator is factorizable by the program. In addition it will attempt to integrate expressions involving error functions, dilogarithms and other trigonometric expressions. In these cases it might not always succeed in finding the solution, even if one exists.

Examples:

julia> Algebra.int(:(log(x)),:x)
:((log(x) - 1) * x)

julia> Algebra.int(:(e^x),:x)
:(e ^ x)

The program checks that the second argument is a variable and gives an error if it is not.

Note: If the int operator is called with 4 arguments, REDUCE will implicitly call the definite integration package (DEFINT) and this package will interpret the third and fourth arguments as the lower and upper limit of integration, respectively. For details, consult the documentation on the DEFINT package.

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7.9.1 Options

The switch trint when on will trace the operation of the algorithm. It produces a great deal of output in a somewhat illegible form, and is not of much interest to the general user. It is normally off.

The switch trintsubst when on will trace the heuristic attempts to solve the integral by substitution. It is normally off.

If the switch failhard is on the algorithm will terminate with an error if the integral cannot be done in closed terms, rather than return a formal integration form. failhard is normally off.

The switch nolnr suppresses the use of the linear properties of integration in cases when the integral cannot be found in closed terms. It is normally off.

The switch nointsubst disables the heuristic attempts to solve the integral by substitution. It is normally off.

7.9.2 Advanced Use

If a function appears in the integrand that is not one of the functions exp, erf, tan, atan, log, dilog then the algorithm will make an attempt to integrate the argument if it can, differentiate it and reach a known function. However the answer cannot be guaranteed in this case. If a function is known to be algebraically independent of this set it can be flagged transcendental by

R"flag(’(trilog),’transcendental)"

in which case this function will be added to the permitted field descriptors for a genuine decision procedure. If this is done the user is responsible for the mathematical correctness of his actions.

The standard version does not deal with algebraic extensions. Thus integration of expressions involving square roots and other like things can lead to trouble. A contributed package that supports integration of functions involving square roots is available, however (ALGINT, chapter 16.1). In addition there is a definite integration package, DEFINT( chapter 16.18).

7.9.3 References

A. C. Norman & P. M. A. Moore, “Implementing the New Risch Algorithm”, Proc. 4th International Symposium on Advanced Comp. Methods in Theor. Phys., CNRS, Marseilles, 1977.

S. J. Harrington, “A New Symbolic Integration System in Reduce”, Comp. Journ. 22 (1979) 2.

A. C. Norman & J. H. Davenport, “Symbolic Integration — The Dust Settles?”, Proc. EUROSAM 79, Lecture Notes in Computer Science 72, Springer-Verlag, Berlin Heidelberg New York (1979) 398-407.

7.10 LENGTH Operator

Reduce.Algebra.lengthFunction
length(r)

length is a generic operator for finding the length of various objects in the system. The meaning depends on the type of the object. In particular, the length of an algebraic expression is the number of additive top-level terms its expanded representation.

Examples:

julia> length(:(a+b))
2

julia> length(2)
1

Other objects that support a length operator include arrays, lists and matrices. The explicit meaning in these cases is included in the description of these objects.

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7.11 MAP Operator

Reduce.Algebra.mapFunction
map(fnc,obj)

The map operator applies a uniform evaluation pattern to all members of a composite structure: a matrix, a list, or the arguments of an operator expression. The evaluation pattern can be a unary procedure, an operator, or an algebraic expression with one free variable.

It is used with the syntax:

   map(FNC:function,OBJ:object)

Here OBJ is a list, a matrix or an operator expression. FNC can be one of the following:

  1. the name of an operator with a single argument: the operator is evaluated once with each element of OBJ as its single argument;
  2. an algebraic expression with exactly one free variable, i.e. a variable preceded by the tilde symbol. The expression is evaluated for each element of OBJ, with the element substituted for the free variable;
  3. a replacement rule of the form var => rep where var is a variable (a kernel without a subscript) and rep is an expression that contains var. The replacement expression rep is evaluated for each element of OBJ with the element substituted for var. The variable var may be optionally preceded by a tilde.

The rule form for FNC is needed when more than one free variable occurs.

Examples:

julia> Algebra.map(:abs, (1,-2,:a,:(-a)))
(1, 2, :(abs(a)), :(abs(a)))

julia> Algebra.map(:(int(~w,x)), [:(x^2) :(x^5); :(x^4) :(x^5)])
2×2 Array{Any,2}:
 :(x ^ 3 // 3)  :(x ^ 6 // 6)
 :(x ^ 5 // 5)  :(x ^ 6 // 6)

 julia> Algebra.map(:(~w*6), :(x^2/3 == y^3/2 -1))
:(2 * x ^ 2 = 3 * (y ^ 3 - 2))

You can use map in nested expressions. However, you cannot apply map to a non-composite object, e.g. an identifier or a number.

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7.12 MKID Operator

Reduce.Algebra.mkidFunction
mkid(u,v)

In many applications, it is useful to create a set of identifiers for naming objects in a consistent manner. In most cases, it is sufficient to create such names from two components. The operator mkid is provided for this purpose. Its syntax is:

R"mkid(U:id,V:id|non-negative integer)"

for example

julia> Algebra.mkid(:a,3)
:a3

julia> Algebra.mkid(:apple,:s)
:apples

while mkid(:(a+b),2) gives an error.

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The set statement can be used to give a value to the identifiers created by mkid, for example

julia> Algebra.set(Algebra.mkid(:a,3),2)

will give a3 the value 2. Similarly, the unset statement can be used to remove the value from these identifiers, for example

julia> Algebra.unset(Algebra.mkid(:a,3))

7.13 The Pochhammer Notation

Reduce.Algebra.pochhammerFunction
pochhammer(a,k)

The Pochhammer notation $(a)_k$ (also called Pochhammer’s symbol) is supported by the binary operator pochhammer(a,k). For a non-negative integer k, it is defined as (http://dlmf.nist.gov/5.2.iii)

$(a)_0 = 1,$

$(a)_k = a(a + 1)(a + 2)⋅⋅⋅(a + k - 1).$

For $a ⁄= 0,±1,±2,…$, this is equivalent to

$(a)k = \frac{\Gamma (a+-k-)}{\Gamma (a)}$

With rounded off, this expression is evaluated numerically if a and k are both integral, and otherwise may be simplified where appropriate. The simplification rules are based upon algorithms supplied by Wolfram Koepf.

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7.14 PF Operator

Reduce.Algebra.pfFunction
pf(expr,var)

R"pf(⟨exp⟩,⟨var⟩)" transforms the expression ⟨exp⟩ into a list of partial fractions with respect to the main variable, ⟨var⟩. pf does a complete partial fraction decomposition, and as the algorithms used are fairly unsophisticated (factorization and the extended Euclidean algorithm), the code may be unacceptably slow in complicated cases.

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Example: Given R"2/((x+1)^2*(x+2))" in the workspace, R"pf(ws,x)" gives the result

    2      - 2         2
{-------,-------,--------------}
  x + 2   x + 1    2
                  x  + 2*x + 1

If you want the denominators in factored form, use off(:exp). Thus, with R"2/((x+1)^2*(x+2))" in the workspace, the commands R"off(exp); pf(ws,x)" give the result

    2      - 2       2
{-------,-------,----------}
  x + 2   x + 1          2
                  (x + 1)

To recombine the terms, for each… sum can be used. So with the above list in the workspace, R"for each j in ws sum j" returns the result

        2
------------------
                2
 (x + 2)*(x + 1)

Alternatively, one can use the operations on lists to extract any desired term.

7.15 SELECT Operator

Reduce.Algebra.selectFunction
select(fnc,lst)

The select operator extracts from a list, or from the arguments of an n–ary operator, elements corresponding to a boolean predicate. It is used with the syntax:

select(⟨FNC:function⟩,⟨LST:list⟩)

FNC can be one of the following forms:

  1. the name of an operator with a single argument: the operator is evaluated once on each element of LST;
  2. an algebraic expression with exactly one free variable, i.e. a variable preceded by the tilde symbol. The expression is evaluated for each element of ⟨LST⟩, with the element substituted for the free variable;
  3. a replacement rule of the form ⟨var⟩ => ⟨rep⟩ where ⟨var⟩ is a variable (a kernel without subscript) and ⟨rep⟩ is an expression that contains ⟨var⟩. ⟨rep⟩ is evaluated for each element of LST with the element substituted for ⟨var⟩. ⟨var⟩ may be optionally preceded by a tilde.

The rule form for FNC is needed when more than one free variable occurs.

The result of evaluating FNC is interpreted as a boolean value corresponding to the conventions of REDUCE. These values are composed with the leading operator of the input expression.

Examples:

julia> Algebra.select(:(~w>0), (1,-1,2,-3,3))
(1, 2, 3)
    select(evenp deg(~w,y),part((x+y)^5,0):=list)  
           -> {X^5 ,10*X^3*Y^2 ,5*X*Y^4}  
    select(evenp deg(~w,x),2x^2+3x^3+4x^4) -> 4X^4 + 2X^2
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7.16 SOLVE Operator

Reduce.Algebra.solveFunction
solve(r...)

solve is an operator for solving one or more simultaneous algebraic equations. It is used with the syntax:

R"SOLVE(⟨EXPRN:algebraic⟩[,⟨VAR:kernel⟩∣,⟨VARLIST:list of kernels⟩])"

exprn is of the form ⟨expression⟩ or {⟨expression1⟩,⟨expression2⟩, …}. Each expression is an algebraic equation, or is the difference of the two sides of the equation. The second argument is either a kernel or a list of kernels representing the unknowns in the system. This argument may be omitted if the number of distinct, non-constant, top-level kernels equals the number of unknowns, in which case these kernels are presumed to be the unknowns.

For one equation, solve recursively uses factorization and decomposition, together with the known inverses of log, sin, cos, ^, acos, asin, and linear, quadratic, cubic, quartic, or binomial factors. Solutions of equations built with exponentials or logarithms are often expressed in terms of Lambert’s W function. This function is (partially) implemented in the special functions package.

Linear equations are solved by the multi-step elimination method due to Bareiss, unless the switch cramer is on, in which case Cramer’s method is used. The Bareiss method is usually more efficient unless the system is large and dense.

Non-linear equations are solved using the Groebner basis package (chapter 16.28). Users should note that this can be quite a time consuming process.

Examples:

Algebra.solve(:(log(sin(x+3))^5 == 8),:x)
Algebra.solve(:(a*log(sin(x+3))^5 - b), :(sin(x+3)))
Algebra.solve((:(a*x+y==3),:(y=-2)),(:x,:y))

solve returns a list of solutions. If there is one unknown, each solution is an equation for the unknown. If a complete solution was found, the unknown will appear by itself on the left-hand side of the equation. On the other hand, if the solve package could not find a solution, the “solution” will be an equation for the unknown in terms of the operator root_of. If there are several unknowns, each solution will be a list of equations for the unknowns. For example,

julia> Algebra.solve(:(x^2==1),:x)
(:(x = 1), :(x = -1))

julia> Algebra.solve(:(x^7-x^6+x^2==1),:x)
(:(x = root_of(x_ ^ 6 + x_ + 1, x_, tag_1)), :(x = 1))

julia> Algebra.solve((:(x+3y==7),:(y-x==1)),(:x,:y))
(:(x = 1), :(y = 2))

The tag argument is used to uniquely identify those particular solutions. ```

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Reduce.Algebra.root_multiplicitiesFunction
root_multiplicities()

Solution multiplicities are stored in the global variable root_multiplicities rather than the solution list. The value of this variable is a list of the multiplicities of the solutions for the last call of solve. For example,

julia> Algebra.solve(:(x^2==2x-1),:x); Algebra.root_multiplicities()

gives the results

(:(x = 1),)
 
(2,)
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Reduce.multiplicitiesFunction
multiplicities(::Bool)

If you want the multiplicities explicitly displayed, the switch multiplicities can be turned on. For example

julia> Algebra.on(:multiplicities); Algebra.solve(:(x^2==2x-1),:x)

yields the result

(:(x = 1), :(x = 1))
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7.16.1 Handling of Undetermined Solutions

When solve cannot find a solution to an equation, it normally returns an equation for the relevant indeterminates in terms of the operator root_of. For example, the expression

julia> Algebra.solve(:(cos(x)+log(x)),:x)

returns the result

(:(x = root_of(cos(x_) + log(x_), x_, tag_1)),)

An expression with a top-level root_of operator is implicitly a list with an unknown number of elements (since we don’t always know how many solutions an equation has). If a substitution is made into such an expression, closed form solutions can emerge. If this occurs, the root_of construct is replaced by an operator one_of. At this point it is of course possible to transform the result of the original solve operator expression into a standard solve solution. To effect this, the operator expand_cases can be used.

The following example shows the use of these facilities:

julia> Algebra.solve(:(-a*x^3+a*x^2+x^4-x^3-4*x^2+4),:x)
(:(x = root_of((a * x_ ^ 2 - x_ ^ 3) + 4x_ + 4, x_, tag_2)), :(x = 1))

julia> Algebra.sub(:a=-1,ans)
(:(x=one_of((2, -1, -2), tag_2)), :(x=1))
 
julia> Algebra.expand_cases(ans)
(:(x=2), :(x=-1), :(x=-2), :(x=1))

7.16.2 Solutions of Equations Involving Cubics and Quartics

Since roots of cubics and quartics can often be very messy, a switch fullroots is available, that, when off (the default), will prevent the production of a result in closed form. The root_of construct will be used in this case instead.

In constructing the solutions of cubics and quartics, trigonometrical forms are used where appropriate. This option is under the control of a switch trigform, which is normally on.

The following example illustrates the use of these facilities:

julia> Algebra.rlet(:xx => :(solve(x^3+x+1,x)))
 
julia> rcall(:xx)
(:(x = root_of(x_ ^ 3 + x_ + 1, x_, tag_1)),)

julia> Algebra.on(:fullroots)

julia> collect(rcall(:xx))
3-element Array{Expr,1}:
 :(x = -((sqrt(3) * cosh(asinh((3 * sqrt(3)) // 2) // 3) * im - sinh(asinh((3 * sqrt(3)) // 2) // 3))) // sqrt(3))
 :(x = (sqrt(3) * cosh(asinh((3 * sqrt(3)) // 2) // 3) * im + sinh(asinh((3 * sqrt(3)) // 2) // 3)) // sqrt(3))
 :(x = (-2 * sinh(asinh((3 * sqrt(3)) // 2) // 3)) // sqrt(3))

julia> Algebra.off(:trigform)
 
julia> collect(rcall(:xx))
3-element Array{Expr,1}:
 :(x = -(((sqrt(31) - 3 * sqrt(3)) ^ (2 / 3) * (sqrt(3) * im + 1) + 2 ^ (2 / 3) * (sqrt(3) * im - 1))) / (2 * (sqrt(31) - 3 * sqrt(3)) ^ (1 / 3) * 6 ^ (1 / 3) * 3 ^ (1 / 6)))
 :(x = (2 ^ (2 / 3) * (sqrt(3) * im + 1) + (sqrt(31) - 3 * sqrt(3)) ^ (2 / 3) * (sqrt(3) * im - 1)) / (2 * (sqrt(31) - 3 * sqrt(3)) ^ (1 / 3) * 6 ^ (1 / 3) * 3 ^ (1 / 6)))   
 :(x = ((sqrt(31) - 3 * sqrt(3)) ^ (2 / 3) - 2 ^ (2 / 3)) / ((sqrt(31) - 3 * sqrt(3)) ^ (1 / 3) * 6 ^ (1 / 3) * 3 ^ (1 / 6))) 

7.16.3 Other Options

If solvesingular is on (the default setting), degenerate systems such as x+y=0, 2x+2y=0 will be solved by introducing appropriate arbitrary constants. The consistent singular equation 0=0 or equations involving functions with multiple inverses may introduce unique new indeterminant kernels arbcomplex(j), or arbint(j), ($j=1,2,...$), representing arbitrary complex or integer numbers respectively. To automatically select the principal branches, do off(:allbranch). To avoid the introduction of new indeterminant kernels do off(:arbvars) – then no equations are generated for the free variables and their original names are used to express the solution forms. To suppress solutions of consistent singular equations do off(:solvesingular).

To incorporate additional inverse functions do, for example:

R"put(’sinh,’inverse,’asinh)"
R"put(’asinh,’inverse,’sinh)"

together with any desired simplification rules such as

R"for all x let sinh(asinh(x))=x, asinh(sinh(x))=x"

For completeness, functions with non-unique inverses should be treated as ^, sin, and cos are in the solve module source.

Arguments of asin and acos are not checked to ensure that the absolute value of the real part does not exceed 1; and arguments of log are not checked to ensure that the absolute value of the imaginary part does not exceed π; but checks (perhaps involving user response for non-numerical arguments) could be introduced using let statements for these operators.

7.16.4 Parameters and Variable Dependency

Reduce.Algebra.requirementsFunction
requirements()

The proper design of a variable sequence supplied as a second argument to solve is important for the structure of the solution of an equation system. Any unknown in the system not in this list is considered totally free. E.g. the call

Algebra.solve((:(x==2z),:(z==2y)),(:z,))

produces an empty list as a result because there is no function z = z(x,y) which fulfills both equations for arbitrary x and y values. In such a case the share variable requirements displays a set of restrictions for the parameters of the system:

julia> Algebra.requirements()
(:(x - 4y),)

The non-existence of a formal solution is caused by a contradiction which disappears only if the parameters of the initial system are set such that all members of the requirements list take the value zero. For a linear system the set is complete: a solution of the requirements list makes the initial system solvable. E.g. in the above case a substitution $x = 4y$ makes the equation set consistent. For a non-linear system only one inconsistency is detected. If such a system has more than one inconsistency, you must reduce them one after the other. 1 The set shows you also the dependency among the parameters: here one of $x$ and $y$ is free and a formal solution of the system can be computed by adding it to the variable list of solve. The requirement set is not unique – there may be other such sets.

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Reduce.Algebra.assumptionsFunction
assumptions()

A system with parameters may have a formal solution, e.g.

julia> Algebra.solve((:(x==a*z+1),:(0==b*z-y)),(:z,:x))
(:(z = y // b), :(x = (a * y + b) // b))

which is not valid for all possible values of the parameters. The variable assumptions contains then a list of restrictions: the solutions are valid only as long as none of these expressions vanishes. Any zero of one of them represents a special case that is not covered by the formal solution. In the above case the value is

julia> Algebra.assumptions()
(:b,)

which excludes formally the case $b = 0$; obviously this special parameter value makes the system singular. The set of assumptions is complete for both, linear and non–linear systems.

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solve rearranges the variable sequence to reduce the (expected) computing time. This behavior is controlled by the switch varopt, which is on by default. If it is turned off, the supplied variable sequence is used or the system kernel ordering is taken if the variable list is omitted. The effect is demonstrated by an example:

julia> @rcall s=(y^3+3x=0,x^2+y^2=1);
 
julia> Algebra.solve(:s,(:y,:x)) |> collect
2-element Array{Expr,1}:
 :(y = root_of((y_ ^ 6 + 9 * y_ ^ 2) - 9, y_, tag_2))
 :(x = -(y ^ 3) // 3)    

julia> Algebra.off(:varopt); Algebra.solve(:s,(:y,:x)) |> collect
2-element Array{Expr,1}:
 :(y = (-(((x ^ 4 - 2 * x ^ 2) + 10)) * x) // 3)                     
 :(x = root_of(((x_ ^ 6 - 3 * x_ ^ 4) + 12 * x_ ^ 2) - 1, x_, tag_3))

In the first case, solve forms the solution as a set of pairs $(y_i,x(y_i))$ because the degree of x is higher – such a rearrangement makes the internal computation of the Gröbner basis generally faster. For the second case the explicitly given variable sequence is used such that the solution has now the form (x_i,y(x_i)). Controlling the variable sequence is especially important if the system has one or more free variables. As an alternative to turning off varopt, a partial dependency among the variables can be declared using the depend statement: solve then rearranges the variable sequence but keeps any variable ahead of those on which it depends.

julia> Algebra.on(:varopt)

julia> @rcall s=(a^3+b,b^2+c);

julia> Algebra.solve(:s,(:a,:b,:c))
(:(a = arbcomplex(1)), :(b = -(a ^ 3)), :(c = -(a ^ 6))) 

julia> Algebra.depend(:a,:c); Algebra.depend(:b,:c)

julia> Algebra.solve(:s,(:a,:b,:c))
3-element Array{Expr,1}:
 :(c = arbcomplex(2))                  
 :(a = root_of(a_ ^ 6 + c, a_, tag_3))
 :(b = -(a ^ 3)) 

Here solve is forced to put $c$ after $a$ and after $b$, but there is no obstacle to interchanging $a$ and $b$.

7.17 Even and Odd Operators

Reduce.Algebra.evenFunction
even(r...)

An operator can be declared to be even in its first argument by the declarations even. Expressions involving an operator declared in this manner are transformed if the first argument contains a minus sign. Any other arguments are not affected. For example, the declaration

julia> Algebra.even(:f1)

means that

        f1(-a)    ->    f1(a)  
        f1(-a,-b) ->    f1(a,-b)  
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Reduce.Algebra.oddFunction
odd(r...)

An operator can be declared to be odd in its first argument by the declarations odd. Expressions involving an operator declared in this manner are transformed if the first argument contains a minus sign. Any other arguments are not affected. In addition, if say f is declared odd, then f(0) is replaced by zero unless f is also declared non zero by the declaration nonzero. For example, the declarations

julia> Algebra.odd(:f2)

means that

        f2(-a)    ->   -f2(a)  
        f2(0)     ->    0

To inhibit the last transformation, say nonzero(:f2).

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7.18 Linear Operators

Reduce.Algebra.linearFunction
linear(r...)

An operator can be declared to be linear in its first argument over powers of its second argument. If an operator f is so declared, f of any sum is broken up into sums of fs, and any factors that are not powers of the variable are taken outside. This means that f must have (at least) two arguments. In addition, the second argument must be an identifier (or more generally a kernel), not an expression.

Example: If f were declared linear, then

f(a*x^5+b*x+c,x) ->  f(x^5,x)*a + f(x,x)*b + f(1,x)*c

More precisely, not only will the variable and its powers remain within the scope of the f operator, but so will any variable and its powers that had been declared to depend on the prescribed variable; and so would any expression that contains that variable or a dependent variable on any level, e.g. cos(sin(x)).

To declare operators f and g to be linear operators, use:

julia> Algebra.linear(:f,:g)

The analysis is done of the first argument with respect to the second; any other arguments are ignored. It uses the following rules of evaluation:

f(0) 		-> 0
f(-y,x) 	-> -f(y,x)
f(y+z,x) 	-> f(y,x)+f(z,x)
f(y*z,x) 	-> z*f(y,x)   	if z does not depend on x
f(y/z,x) 	-> f(y,x)/z	if z does not depend on x
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To summarize, y “depends” on the indeterminate x in the above if either of the following hold:

  1. y is an expression that contains x at any level as a variable, e.g.: cos(sin(x))
  2. Any variable in the expression y has been declared dependent on x by use of the declaration depend.

The use of such linear operators can be seen in the paper Fox, J.A. and A. C. Hearn, “Analytic Computation of Some Integrals in Fourth Order Quantum Electrodynamics” Journ. Comp. Phys. 14 (1974) 301-317, which contains a complete listing of a program for definite integration of some expressions that arise in fourth order quantum electrodynamics.

7.19 Non-Commuting Operators

Reduce.Algebra.noncomFunction
noncom(r...)

An operator can be declared to be non-commutative under multiplication by the declaration noncom.

Example: After the declaration

julia> Algebra.noncom(:u,:v);

the expressions u(x)*u(y)-u(y)*u(x) and u(x)*v(y)-v(y)*u(x) will remain unchanged on simplification, and in particular will not simplify to zero.

Note that it is the operator (u and v in the above example) and not the variable that has the non-commutative property.

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The let statement may be used to introduce rules of evaluation for such operators. In particular, the boolean operator ordp is useful for introducing an ordering on such expressions.

Example: The rule

R"for all x,y such that x neq y and ordp(x,y) let u(x)*u(y)= u(y)*u(x)+comm(x,y)"

would introduce the commutator of u(x) and u(y) for all x and y. Note that since ordp(x,x) is true, the equality check is necessary in the degenerate case to avoid a circular loop in the rule.

7.20 Symmetric and Antisymmetric Operators

Reduce.Algebra.symmetricFunction
symmetric(r...)

An operator can be declared to be symmetric with respect to its arguments by the declaration symmetric. For example

julia> Algebra.symmetric(:u,:v);

means that any expression involving the top level operators u or v will have its arguments reordered to conform to the internal order used by REDUCE. The user can change this order for kernels by the command korder. For example, u(x,v(1,2)) would become u(v(2,1),x), since numbers are ordered in decreasing order, and expressions are ordered in decreasing order of complexity.

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Reduce.Algebra.antisymmetricFunction
antisymmetric(r...)

the declaration antisymmetric declares an operator antisymmetric. For example,

julia> Algebra.antisymmetric(:l,:m);

means that any expression involving the top level operators l or m will have its arguments reordered to conform to the internal order of the system, and the sign of the expression changed if there are an odd number of argument interchanges necessary to bring about the new order.

For example, l(x,m(1,2)) would become -l(-m(2,1),x) since one interchange occurs with each operator. An expression like l(x,x) would also be replaced by 0.

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7.21 Declaring New Prefix Operators

Reduce.Algebra.operatorFunction
operator(r...)

The user may add new prefix operators to the system by using the declaration operator. For example:

julia> Algebra.operator(:h,:g1,:arctan)

adds the prefix operators h, g1 and arctan to the system.

This allows symbols like h(w), h(x,y,z), g1(p+q), arctan(u/v) to be used in expressions, but no meaning or properties of the operator are implied. The same operator symbol can be used equally well as a 0-, 1-, 2-, 3-, etc.-place operator.

To give a meaning to an operator symbol, or express some of its properties, let statements can be used, or the operator can be given a definition as a procedure.

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If the user forgets to declare an identifier as an operator, the system will prompt the user to do so in interactive mode, or do it automatically in non-interactive mode. A diagnostic message will also be printed if an identifier is declared operator more than once.

Operators once declared are global in scope, and so can then be referenced anywhere in the program. In other words, a declaration within a block (or a procedure) does not limit the scope of the operator to that block, nor does the operator go away on exiting the block (use clear instead for this purpose).

7.22 Declaring New Infix Operators

Reduce.Algebra.infixFunction
infix(r...)

Users can add new infix operators by using the declarations infix and precedence. For example,

julia> Algebra.infix(:mm)

The declaration infix(:mm) would allow one to use the symbol mm as an infix operator: R"a mm b" instead of R"mm(a,b)".

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Reduce.Algebra.precedenceFunction
precedence(a,b)

Users can add new infix operators by using the declarations infix and precedence. For example,

julia> Algebra.precedence(:mm,:-)

The declaration precedence(:mm,:-) says that mm should be inserted into the infix operator precedence list just after the - operator. This gives it higher precedence than - and lower precedence than * . Thus R"a - b mm c - d" means R"a - (b mm c) - d", while R"a * b mm c * d" means R"(a * b) mm (c * d)".

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Both infix and prefix operators have no transformation properties unless let statements or procedure declarations are used to assign a meaning.

We should note here that infix operators so defined are always binary: R"a mm b mm c" means R"(a mm b) mm c".

7.23 Creating/Removing Variable Dependency

Reduce.Algebra.dependFunction
depend(r...)

There are several facilities in REDUCE, such as the differentiation operator and the linear operator facility, that can utilize knowledge of the dependency between various variables, or kernels. Such dependency may be expressed by the command depend. This takes an arbitrary number of arguments and sets up a dependency of the first argument on the remaining arguments. For example,

julia> Algebra.depend(:x,:y,:z)

says that x is dependent on both y and z.

julia> Algebra.depend(:z,:(cos(x)),:y)

says that z is dependent on cos(x) and y.

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Reduce.Algebra.nodependFunction
nodepend(r...)

Dependencies introduced by depend can be removed by nodepend. The arguments of this are the same as for depend. For example, given the above dependencies,

julia> Algebra.nodepend(:z,:(cos(x)))

says that z is no longer dependent on cos(x), although it remains dependent on y.

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