calc: Multiple Solutions
11.6.1 Multiple Solutions
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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; 
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).
Info Catalog
calc: Solving Equations
calc: Solving Equations
calc: Solving Systems of Equations