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Functional operators

These operators can help the user to program in the style of functional programming languages such as Miranda or Haskell.

infix :(item, list)

prepend item to list, or concatenate strings

param item

an item to be prepended to a list

param list

a list

param string1

a string

param string2

a string

The first form prepends "item" as the first entry to the list "list". The second form concatenates the strings "string1" and "string2".

Example
In> a:b:c:{}
Out> {a,b,c};
In> "This":"Is":"A":"String"
Out> "ThisIsAString";

Concat, ConcatStrings

infix @(fn, arglist)

apply a function

param fn

function to apply

param arglist

single argument, or a list of arguments

This function is a shorthand for Apply. It applies the function "fn" to the argument(s) in "arglist" and returns the result. The first parameter "fn" can either be a string containing the name of a function or a pure function.

Example
In> "Sin" @ a
Out> Sin(a);
In> {{a},Sin(a)} @ a
Out> Sin(a);
In> "f" @ {a,b}
Out> f(a,b);

Apply

infix /@(fn, list)

apply a function to all entries in a list

param fn

function to apply

param list

list of arguments

This function is a shorthand for {MapSingle}. It successively applies the function "fn" to all the entries in "list" and returns a list contains the results. The parameter "fn" can either be a string containing the name of a function or a pure function.

Example
In> "Sin" /@ {a,b}
Out> {Sin(a),Sin(b)};
In> {{a},Sin(a)*a} /@ {a,b}
Out> {Sin(a)*a,Sin(b)*b};

MapSingle, Map, MapArgs

infix .. (n, m)

construct a list of consecutive integers

param n

integer. the first entry in the list

param m

integer, the last entry in the list

This command returns the list {{n, n+1, n+2, ..., m}}. If {m} is smaller than {n}, the empty list is returned. Note that the {..} operator should be surrounded by spaces to keep the parser happy, if "n" is a number. So one should write "{1 .. 4}" instead of "{1..4}".

NFunction("newname","funcname", {arglist})

make wrapper for numeric functions

param "newname"

name of new function

param "funcname"

name of an existing function

param arglist

symbolic list of arguments

This function will define a function named "newname" with the same arguments as an existing function named "funcname". The new function will evaluate and return the expression "funcname(arglist)" only when all items in the argument list {arglist} are numbers, and return unevaluated otherwise. This can be useful when plotting functions defined through other Yacas routines that cannot return unevaluated. If the numerical calculation does not return a number (for example, it might return the atom {nan}, "not a number", for some arguments), then the new function will return {Undefined}.

Example
In> f(x) := N(Sin(x));
Out> True;
In> NFunction("f1", "f", {x});
Out> True;
In> f1(a);
Out> f1(a);
In> f1(0);
Out> 0;

Suppose we need to define a complicated function {t(x)} which cannot be evaluated unless {x} is a number:

In> t(x) := If(x<=0.5, 2*x, 2*(1-x));
Out> True;
In> t(0.2);
Out> 0.4;
In> t(x);
In function "If" :
bad argument number 1 (counting from 1)
CommandLine(1) : Invalid argument

Then, we can use {NFunction()} to define a wrapper {t1(x)} around {t(x)} which will not try to evaluate {t(x)} unless {x} is a number:

In> NFunction("t1", "t", {x})
Out> True;
In> t1(x);
Out> t1(x);
In> t1(0.2);
Out> 0.4;

Now we can plot the function.

In> Plot2D(t1(x), -0.1: 1.1) Out> True;

MacroRule

infix Where(expr, x==v)

substitute result into expression

param expr

expression to evaluate

param x

variable to set

param v

value to substitute for variable

The operator {Where} fills in values for variables, in its simplest form. It accepts sets of variable/value pairs defined as var1==val1 And var2==val2 And ... and fills in the corresponding values. Lists of value pairs are also possible, as: {var1==val1 And var2==val2, var1==val3 And var2==val4} These values might be obtained through {Solve}.

Example
In> x^2+y^2 Where x==2
Out> y^2+4;
In> x^2+y^2 Where x==2 And y==3
Out> 13;
In> x^2+y^2 Where {x==2 And y==3}
Out> {13};
In> x^2+y^2 Where {x==2 And y==3,x==4 And y==5}
Out> {13,41};

Solve, AddTo

infix AddTo(eq1,eq2)

add an equation to a set of equations or set of set of equations

param eq

(set of) set of equations

Given two (sets of) sets of equations, the command AddTo combines multiple sets of equations into one. A list {a,b} means that a is a solution, OR b is a solution. AddTo then acts as a AND operation: (a or b) and (c or d) => (a or b) Addto (c or d) => (a and c) or (a and d) or (b and c) or (b and d) This function is useful for adding an identity to an already existing set of equations. Suppose a solve command returned {a>=0 And x==a,a<0 And x== -a} from an expression x==Abs(a), then a new identity a==2 could be added as follows: In> a==2 AddTo {a>=0 And x==a,a<0 And x== -a} Out> {a==2 And a>=0 And x==a,a==2 And a<0 And x== -a}; Passing this set of set of identities back to solve, solve should recognize that the second one is not a possibility any more, since a==2 And a<0 can never be true at the same time.

Example
In> {A==2,c==d} AddTo {b==3 And d==2}
Out> {A==2 And b==3 And d==2,c==d
And b==3 And d==2};
In> {A==2,c==d} AddTo {b==3, d==2}
Out> {A==2 And b==3,A==2 And d==2,c==d
And b==3,c==d And d==2};

Where, Solve