Revert polynomials, lists, character strings and tables, invert series expansions
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revert(object
)
revert
reverses the ordering of the elements
in a list and the ordering of characters in a string, as well as the
ordering of the coefficients in a polynomial. For tables, it swaps
indices and entries. For a series expansion, it returns the functional
inverse.
revert
is a general function to compute inverses
with respect to functional composition, or to reverse the order of
operands. This type of functionality may be extended to further types
of objects via overloading.
Currently, the MuPAD^{®} library provides functionality for strings,
polynomials, lists, and tables, where revert
reverses
the order of the elements, coefficients, or characters, respectively.
In tables, entries are turned into indices and vice versa. E.g., revert(table(x
= y, 2 = 4))
yields the table table(y = x, 4 =
2)
. For series expansions, the functional inverse is returned.
For all other types of MuPAD objects that do not overload revert
,
the symbolic expression revert(object)
is returned.
revert
operates on lists and character strings:
revert([1, 2, 3, 4, 5])
revert("nuf si DAPuM ni gnimmargorP")
revert
operates on series:
revert(series(sin(x), x)) = series(arcsin(x), x)
revert
operates on tables:
t := table(): t[x] := 1: t[y] := 2: t[z] := 3: T := revert(t): T[1], T[2], T[3]
Beware: if an entry is stored under several distinct indices, revert
reduces
the number of table operands:
revert(table(x = 1, y = 1, z = 3))
The functional inverse of the expansion of exp
around x = 0
is
the expansion of the inverse function ln
around x = exp(0) = 1
:
revert(series(exp(x), x, 3)) = series(ln(x), x = 1, 2)
delete t, T:
revert
computes the reverse of a polynomial:
revert(poly(x^3 + 2*x + 5))
The same works for multivariate polynomials, too:
revert(poly(x^3 + 2*x*y + 5*x + 6*y + 7))
We could have achieved the same by substituting all indeterminates
by their inverses; however, revert
works faster.
numer(evalp(poly(x^3 + 2*x*y + 5*x + 6*y + 7), x = 1/x, y = 1/y))
For all other types of objects, a symbolic function call is returned:
revert(x + y)
The following series expansion is not of type Series::Puiseux
.
Instead, a generalized expansion of type Series::gseries
is produced.
Consequently, revert
does not compute an inverse:
revert(series(exp(x)/(1 + x), x = infinity, 3))

A polynomial, a list, a character
string, a table, or a series
expansion of type 
Object of the same type as the input object, or a symbolic call
of type "revert"
.
object