Some equations have more than one solution. The Hyperbolic flag
(H a S) [fsolve] tells the solver to report the fully
general family of solutions. It will invent variables n1,
n2, …, which represent independent arbitrary integers, and
s1, s2, …, which represent independent arbitrary
signs (either +1 or -1). If you don’t use the Hyperbolic
flag, Calc will use zero in place of all arbitrary integers, and plus
one in place of all arbitrary signs. Note that variables like n1
and s1 are not given any special interpretation in Calc except by
the equation solver itself. As usual, you can use the s l
(calc-let) command to obtain solutions for various actual values
of these variables.
For example, ' x^2 = y RET H a S x RET solves to
get ‘x = s1 sqrt(y)’, indicating that the two solutions to the
equation are ‘sqrt(y)’ and ‘-sqrt(y)’. Another way to
think about it is that the square-root operation is really a
two-valued function; since every Calc function must return a
single result, sqrt chooses to return the positive result.
Then H a S doctors this result using s1 to indicate
the full set of possible values of the mathematical square-root.
There is a similar phenomenon going the other direction: Suppose
we solve ‘sqrt(y) = x’ for y. Calc squares both sides
to get ‘y = x^2’. This is correct, except that it introduces
some dubious solutions. Consider solving ‘sqrt(y) = -3’:
Calc will report ‘y = 9’ as a valid solution, which is true
in the mathematical sense of square-root, but false (there is no
solution) for the actual Calc positive-valued sqrt. This
happens for both a S and H a S.
If you store a positive integer in the Calc variable GenCount,
then Calc will generate formulas of the form ‘as(n)’ for
arbitrary signs, and ‘an(n)’ for arbitrary integers,
where n represents successive values taken by incrementing
GenCount by one. While the normal arbitrary sign and
integer symbols start over at s1 and n1 with each
new Calc command, the GenCount approach will give each
arbitrary value a name that is unique throughout the entire Calc
session. Also, the arbitrary values are function calls instead
of variables, which is advantageous in some cases. For example,
you can make a rewrite rule that recognizes all arbitrary signs
using a pattern like ‘as(n)’. The s l command only works
on variables, but you can use the a b (calc-substitute)
command to substitute actual values for function calls like ‘as(3)’.
The s G (calc-edit-GenCount) command is a convenient
way to create or edit this variable. Press C-c C-c to finish.
If you have not stored a value in GenCount, or if the value
in that variable is not a positive integer, the regular
s1/n1 notation is used.
With the Inverse flag, I a S [finv] treats the expression
on top of the stack as a function of the specified variable and solves
to find the inverse function, written in terms of the same variable.
For example, I a S x inverts ‘2x + 6’ to ‘x/2 - 3’.
You can use both Inverse and Hyperbolic [ffinv] to obtain a
fully general inverse, as described above.
Some equations, specifically polynomials, have a known, finite number
of solutions. The a P (calc-poly-roots) [roots]
command uses H a S to solve an equation in general form, then, for
all arbitrary-sign variables like s1, and all arbitrary-integer
variables like n1 for which n1 only usefully varies over
a finite range, it expands these variables out to all their possible
values. The results are collected into a vector, which is returned.
For example, ‘roots(x^4 = 1, x)’ returns the four solutions
‘[1, -1, (0, 1), (0, -1)]’. Generally an nth degree
polynomial will always have n roots on the complex plane.
(If you have given a real declaration for the solution
variable, then only the real-valued solutions, if any, will be
reported; see Declarations.)
Note that because a P uses H a S, it is able to deliver
symbolic solutions if the polynomial has symbolic coefficients. Also
note that Calc’s solver is not able to get exact symbolic solutions
to all polynomials. Polynomials containing powers up to ‘x^4’
can always be solved exactly; polynomials of higher degree sometimes
can be: ‘x^6 + x^3 + 1’ is converted to ‘(x^3)^2 + (x^3) + 1’,
which can be solved for ‘x^3’ using the quadratic equation, and then
for ‘x’ by taking cube roots. But in many cases, like
‘x^6 + x + 1’, Calc does not know how to rewrite the polynomial
into a form it can solve. The a P command can still deliver a
list of numerical roots, however, provided that Symbolic mode (m s)
is not turned on. (If you work with Symbolic mode on, recall that the
N (calc-eval-num) key is a handy way to reevaluate the
formula on the stack with Symbolic mode temporarily off.) Naturally,
a P can only provide numerical roots if the polynomial coefficients
are all numbers (real or complex).