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allvalues

compute all possible values of expressions involving RootOfs

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

allvalues(expr)

allvalues(expr, opt1, opt2, ...)

Parameters

expr

-

algebraic expression, Matrix, Vector, list, or set of expressions

opt1, opt2, ...

-

(optional) sequence of options

Description

• 

Typically, a RootOf represents more than one value. Thus, expressions involving RootOfs generally evaluate to more than one value or expression. The allvalues command returns, in an expression sequence, all such values (or expressions) generated by the combinations of different values of the RootOfs.

• 

The allvalues command attempts to find symbolic representations of the roots using solve.  If in calling solve it returns only a partial set of roots and sets _SolutionsMayBeLost to true, then allvalues sets the global variable _ValuesMayBeLost to true and continues to work with set of partial solutions.

  

- For roots of polynomials, allvalues attempts to return representations in terms of radicals, unless option 'implicit' is specified (see below). If no such expression is found, then indexed RootOfs are used.

  

- For roots of expressions that are not polynomials, symbolic representations may or may not be available:

  

When symbolic solutions are found, note that some solutions may be missing. In some cases, the solution set is parameterized by names with the following prefix: _Z is for integer values, _NN for non-negative integer values, _B for boolean values (0 or 1). See also solve.

  

If solve cannot find symbolic representations of the roots, then fsolve is used to search for numerical approximations. If numerical approximations are found, RootOfs with a numerical selector are returned.  If it cannot be determined that fsolve has found all the roots, then the global variable _ValuesMayBeLost will be set to true.

  

If no symbolic representation nor numerical approximations are found, the RootOf is not expanded. However, if Maple can prove that there is no complex solutions to the equation corresponding to the RootOf, NULL is returned.

• 

The following options are supported:

  

- The name 'independent' or 'dependent': If the option 'dependent' is given, then a particular RootOfp represents the same root at each occurrence within expr. This is the default. In contrast, if the option is 'independent', then each occurrence of RootOfp is to be treated independently. This option has no effect on RootOfs containing a selector that makes them single-valued.

  

- The name 'explicit' or 'implicit': If the option 'explicit' is given, then allvalues tries to find radical expressions for roots of polynomials. This is the default unless _EnvExplicit has a numerical value. In the latter case only the input of length smaller than that value will be made explicit. If the option 'implicit' is given, then indexed RootOfs are used.

  

- A RootOf or a set of RootOfs: The RootOfs specified is not expanded by allvalues and remains inert.

• 

The allvalues command takes selectors of RootOf into account:

  

- If the selector is a numerical approximation, then the RootOf is considered as a single-valued object, namely the closest root to the selector.

  

- If the selector is a range, then the roots are sought in this range.

  

- If the selector is an index of the form index=expr1, then the RootOf is considered as a single-valued object, except when expr1 is a set or a range.

  

- If the selector is a label of the form label=expr2, then the RootOf is viewed as a multi-valued object.

• 

Nested RootOfs are supported by allvalues.

Examples

e1RootOf_Z21+1RootOf_Z412

e1:=RootOf_Z21+1RootOf_Z412

(1)

allvaluese1

2,0,0,2,2,0,0,2

(2)

e2RootOf_Z31

e2:=RootOf_Z31

(3)

allvaluese2

1,12+12I3,1212I3

(4)

allvaluese2,'implicit'

1,RootOf_Z2+_Z+1,index=1,RootOf_Z2+_Z+1,index=2

(5)

f1RootOfx1cosxx

f1:=RootOf_Z1cos_Z_Z

(6)

allvaluesf1

1,RootOf_Zcos_Z,0.7390851332

(7)

f2RootOfx+1cosxx,1

f2:=RootOf_Z+1cos_Z_Z,1

(8)

allvaluesf2

RootOf_Zcos_Z,1

(9)

allvaluessinRootOf_Z2a2RootOf_Z21

sina2,sina2,sina2,sina2

(10)

e3RootOf_Z21+RootOf_Z22

e3:=RootOf_Z21+RootOf_Z22

(11)

allvaluese3,RootOf_Z22

1RootOf_Z22,1RootOf_Z22

(12)

allvaluese3,RootOf_Z22,implicit

RootOf_Z21+RootOf_Z22,index=1,RootOf_Z21+RootOf_Z22,index=2

(13)

allvaluese3,e3

RootOf_Z2+21,RootOf_Z221

(14)

allvaluese3,e3,RootOf_Z22

RootOf_Z21+RootOf_Z22

(15)

e4RootOf_Z211RootOf_Z21

e4:=RootOf_Z211RootOf_Z21

(16)

allvaluese4

0,0

(17)

allvaluese4,'independent'

0,2,2,0

(18)

e5allvaluesRootOf_Z5_Z+1

e5:=RootOf_Z5_Z+1,index=1,RootOf_Z5_Z+1,index=2,RootOf_Z5_Z+1,index=3,RootOf_Z5_Z+1,index=4,RootOf_Z5_Z+1,index=5

(19)

evalfe5

1.167303978,0.18123244451.083954101I,0.1812324445+1.083954101I,0.76488443360.3524715460I,0.7648844336+0.3524715460I

(20)

e6RootOfx22+RootOfx22,index=1

e6:=RootOf_Z22+RootOf_Z22,index=1

(21)

allvaluese6

22,0

(22)

allvaluese6,implicit

2RootOf_Z22,index=1,RootOf_Z22,index=2+RootOf_Z22,index=1

(23)

e7RootOf_Z22,label=1+RootOf_Z22,label=2

e7:=RootOf_Z22,label=1+RootOf_Z22,label=2

(24)

allvaluese7

22,0,0,22

(25)

e8RootOfsin_Z

e8:=RootOfsin_Z

(26)

allvaluese8

π_Z3~

(27)

e9RootOfⅇ_Z

e9:=RootOfⅇ_Z

(28)

allvaluese9

e10solvex2+y23,y+12x1

e10:=x=RootOf_Z2+3_Z+33,y=RootOf_Z2+3_Z+3,x=RootOf_Z2+_Z1+1,y=RootOf_Z2+_Z1

(29)

mapevalf@allvalues,e10

x=1.5000000000.8660254040I,y=1.500000000+0.8660254040I,x=1.500000000+0.8660254040I,y=1.5000000000.8660254040I,x=1.618033988,y=0.6180339880,x=0.6180339880,y=1.618033988

(30)

See Also

convert/radical

evala

fsolve

RootOf

solve

 


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