avoid = set(equation) or set(set(equation)) or set(list(values)) or set(values)

For example, avoid = {x=0, x=1}, avoid = {{x=0, y=0}, {x=0, y=exp(1)}} or avoid = {[0, 1]}.

Avoid specified values when searching for roots. The left-hand side of each equation must be a variable name, unless equations is a list of procedures in which case only the right-hand side value must be provided. The right-hand side is the corresponding numeric value to avoid. To avoid values for a single variable, specify a set of equations. To avoid values for multiple variables, specify a set of sets of equations. To avoid values for a list of procedures, specify a set of lists of numeric values. To avoid values for a single procedure, specify a set of numeric values.

When using the avoid option, it is recommended that you specify a starting_values option to facilitate converging to a different solution.

complex

Find one root (or all roots, for polynomials) over the complex floating-point numbers.

fulldigits

Prevent fsolve from decreasing Digits for intermediate calculations at high settings of Digits.  With this option, fsolve may escape ill-conditioning problems, but is slower.

useunits

useunits = list or set of variable/unit pairs

useunits = list of units

The fsolve command can automatically deal with equations and expressions that contain units in most situations, and procedures that contain units if the units are given in initial values or ranges, as explained in the Units section. This option can be used in three ways:

When used as just the keyword useunits, this option turns on code for dealing with units if those units had not been detected from the presence in other arguments. This is particularly useful if you specify procedures that return a value with units.

When used to specify a list or set of variable/unit pairs, such as x :: Unit(m), which is possible only if expressions or equations, rather than procedures, are submitted to fsolve, then this associates the given unit with the given variable. This is particularly useful if no initial value or range can be specified for the variable, or if the initial point is 0 (with which no unit can be associated in Maple).

When used to specify a list of units, such as Unit(m), which is possible only if procedures, rather than expressions or equations, are submitted to fsolve, then this associates the given units with the arguments to the procedure, in the order they are given. The number of units given must equal the number of arguments to the procedure.

range, equation, set(equation), or list(range)

For example, 1..2, x = 1..2, {x = 1..2, y = 10..20}, or [1..2, 10..20].

Search for roots only in the specified interval. The ranges are closed intervals. That is, the endpoints are included in the range. The left-hand side of each equation must be a variable name, unless equations is a list of procedures in which case only the right-hand side range values must be provided. The right-hand side is the corresponding interval to search. To specify a range for a single variable, specify an equation. To specify a range for multiple variables, specify a set of equations. To specify a range for a list of procedures, specify a list of ranges.

In conjunction with the complex option, you can specify a complex interval (rectangle in the complex plane) option using the syntax lower_left_point..upper_right_point, for example, x = -1 - I..1 + I.

maxsols = posint

Find only the specified number of least roots. This option is valid only for a single univariate polynomial equation, in which case the fsolve command computes more than one root.

method = NAG, hefroots, ABND, RC, RS, PW, HR, subdivide, or NextZero

Use the specified method for the numeric root finding of an equation of one variable. The methods NAG, ABND, RC, RS, PW, HR, and hefroots pertain to a polynomial in one variable.

The methods subdivide and NextZero pertain to an equation in one variable that is not a polynomial. The method subdivide repeatedly subdivides the specified range amongst roots already found. The method NextZero uses the command RootFinding:-NextZero to repeatedly search past the last root found. The method subdivide may be more suitable for discontinuous expressions or those with non-isolated roots.  When either method = subdivide or method = NextZero is specified, the maxsols option must also be provided.

equation or set(equation)

For example, x = 1.2, {x = 1.2, y = 10}, or [1.2, 10].

Specify starting values for the variables. You cannot specify the variables option if you specify this option.  If the method that fsolve chooses for a particular problem requires more than one starting value, then the additional values are generated by perturbing the given starting values.

The digits keyword option takes a positive integer that specifies the maximal working precision (Digits) that will be set by fsolve. If not specified the value may be increased according to the algorithm.

The absftol keyword option takes a positive float that specifies an acceptance criterion for the norm of the forward error. If not specified the value will be set according to the algorithm.

The epsilon keyword option takes a positive float that specifies an acceptance criterion for the norm of the change in the variables for the current iteration step. If not specified the value will be set according to the algorithm.

Consider the following model of a parabolic suspension cable. We specify the length of a cable and the sag, and we want to know what the span covered by that cable is. According to Karnovsky and Lebed (see References), we have

$\mathrm{cable\_length}=\mathrm{int}\left(\mathrm{sqrt}\left(1+{\left(4\mathrm{ratio}\right)}^2{\left(\frac{2x}{\mathrm{span}}-1\right)}^2\right)\&comma;x=0..\mathrm{span}\right)\mathbf{assuming}0<\mathrm{span},0<\mathrm{ratio}$

$\mathrm{cable\_length}=\frac{\mathrm{span}\left(4\sqrt{16{\mathrm{ratio}}^2+1}\mathrm{ratio}+\mathrm{arcsinh}\left(4\mathrm{ratio}\right)\right)}{8\mathrm{ratio}}$

where $\mathrm{ratio}=\frac{\mathrm{sag}}{\mathrm{span}}$. Approximating for $\mathrm{ratio}$ near 0:

$\mathrm{lhs}\left(\right)=\mathrm{series}\left(\mathrm{rhs}\left(\right)\&comma;\mathrm{ratio}\&comma;8\right)$

$\mathrm{cable\_length}=\mathrm{span}+\frac{8}{3}\mathrm{span}{\mathrm{ratio}}^2-\frac{32}{5}\mathrm{span}{\mathrm{ratio}}^4+\frac{256}{7}\mathrm{span}{\mathrm{ratio}}^6+O\left({\mathrm{ratio}}^8\right)$

$\mathrm{eval}\left(\mathrm{convert}\left(\&comma;'\mathrm{polynom}'\right)\&comma;\mathrm{ratio}=\frac{\mathrm{sag}}{\mathrm{span}}\right)$

$\mathrm{cable\_length}=\mathrm{span}+\frac{8{\mathrm{sag}}^2}{3\mathrm{span}}-\frac{32{\mathrm{sag}}^4}{5{\mathrm{span}}^3}+\frac{256{\mathrm{sag}}^6}{7{\mathrm{span}}^5}$

$\mathrm{cable\_length}≔100\mathrm{Unit}\left(\mathrm{ft}\right)$

$\mathrm{cable\_length}≔100⟦\mathrm{ft}⟧$

$\mathrm{sag}≔2.738\mathrm{Unit}\left(\mathrm{ft}\right)$

$\mathrm{sag}≔2.738⟦\mathrm{ft}⟧$

$\mathrm{fsolve}\left(\&comma;\mathrm{span}=10\mathrm{Unit}\left(\mathrm{ft}\right)..1000\mathrm{Unit}\left(\mathrm{ft}\right)\right)$

$99.80004926⟦\mathrm{ft}⟧$

We reformulate the same problem in terms of a procedure.

$p≔\mathrm{unapply}\left(\mathrm{rhs}\left(\right)-\mathrm{lhs}\left(\right)\&comma;\mathrm{span}\right)$

$p≔\mathrm{span}\mapsto \mathrm{span}+\frac{19.99105067}{\mathrm{span}}\cdot {⟦\mathrm{ft}⟧}^2-\frac{359.6778961}{{\mathrm{span}}^3}\cdot {⟦\mathrm{ft}⟧}^4+\frac{15407.86938}{{\mathrm{span}}^5}\cdot {⟦\mathrm{ft}⟧}^6-100\cdot ⟦\mathrm{ft}⟧$

$\mathrm{fsolve}\left(p\&comma;10\mathrm{Unit}\left(\mathrm{ft}\right)..1000\mathrm{Unit}\left(\mathrm{ft}\right)\right)$

$99.80004926⟦\mathrm{ft}⟧$

A range can be specified with different units, but they have to be consistent - otherwise the fsolve command issues an error message.

$\mathrm{fsolve}\left(\&comma;\mathrm{span}=10\mathrm{Unit}\left(\mathrm{ft}\right)..100\mathrm{Unit}\left(m\right)\right)$

$99.80004926⟦\mathrm{ft}⟧$

$\mathrm{fsolve}\left(\&comma;\mathrm{span}=10\mathrm{Unit}\left(\mathrm{ft}\right)..1000\mathrm{Unit}\left(s\right)\right)$

Error, (in fsolve) units problem, right range limit span = 1000*Units:-Unit(s) is not compatible with Units:-Unit(ft)

If we want the result in fathoms, we can associate that unit with the variable span, even if we do not supply a range.

$\mathrm{fsolve}\left(\&comma;\mathrm{span}\&comma;\mathrm{useunits}=\left\{\mathrm{span}::\mathrm{Unit}\left(\mathrm{fathom}\right)\right\}\right)$

$16.63334154⟦\mathrm{fathom}⟧$