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RootFinding

  

Analytic

  

zeros of an analytic function in one variable

  

AnalyticZerosFound

  

zeros of an analytic function in one variable

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Analytic(f, z, a+c*I..b+d*I, ...)

Analytic(f, z, re=a..b, im=c..d, ...)

AnalyticZerosFound()

Parameters

f

-

an analytic function of z (or an equation defining such)

z

-

(optional) an unknown

a, b, c, d

-

real constants

Description

• 

For an analytic function f of z, Analytic attempts to find all complex zeros of fz within the rectangular region a<=Rez<=b, c<=Imz<=d in the complex plane. It may also find some zeros outside but close to the boundary of the region.

• 

Setting of infolevel[RootFinding:-Analytic] to be between 1 and 7 will result in detailed information concerning the solving process being displayed.

• 

The AnalyticZerosFound() calling sequence returns a sequence of the zeros which have been located. These may be accessed after Analytic returns, or if its computation is interrupted.

Options

  

The remaining arguments are interpreted as options.

• 

digits=n

  

Set Digits := n; for the purpose of the calculation. Thus the accuracy of the zeros computed is usually less than n digits since only n digits is used for the calculation. The default is the value of Digits already set by the user; the minimum is 5. If hardware floats are in use, this default may be increased to a maximum of trunc(evalhf(Digits)).

• 

iterations=n

  

The number of iterations of Newton's method to be applied for each starting point. The default is 50.

• 

continue

  

Continue with an interrupted calculation, remembering already found zeros. This option may also be used to change the boundaries of the region without losing already found zeros. This option may only be used with the same function and variable as the original call.

• 

plot

  

Return a plot of the zeros instead of the zeros. Note, the zeros are still available to the user through the command RootFinding:-AnalyticZerosFound(), and are available until the routine RootFinding:-Analytic is once again used.

• 

modulo&equals;a&plus;Ib and modulo_s&equals;a&plus;Ib

  

When plotting, the zeros will be reduced modulo a in the real direction and modulo b in the imaginary direction. With the option 'modulo' they will be reduced to the region 0<=Rez<=a and 0<=Imz<=b; With 'modulo_s' they will be reduced to the region a2<=Rez<=a2, b2<=Imz<=b2.

• 

extra

  

For plotting purposes, remember the rectangular subregions in which the zeros were isolated, as well as the traces of the Newton iterations. These will appear when the zeros are plotted.

• 

viewall

  

For plotting purposes, view the entire specified region. Do not restrict the viewing region to the area where zeros are found.

Examples

withRootFinding

Analytic&comma;AnalyticZerosFound&comma;BivariatePolynomial&comma;EnclosingBox&comma;HasRealRoots&comma;Homotopy&comma;Isolate&comma;NextZero&comma;Parametric&comma;WitnessPoints

(1)

ftansinx1

f:=tansinx1

(2)

Analyticf&comma;x&comma;I2..1&plus;I

0.903339110766515

(3)

g23x5&plus;105x410x2&plus;17x

g:=23x5&plus;105x410x2&plus;17x

(4)

Analyticg&comma;x&comma;re&equals;5..1&comma;im&equals;1..1

0.&plus;0.I&comma;0.637181318531050&comma;0.3040664542849070.404061905751759I&comma;0.304066454284907&plus;0.404061905751759I&comma;4.53616898134312

(5)

h10lnv&plus;v2112ln3&plus;32112

h:=10lnv&plus;v21&plus;ln3&plus;8

(6)

Analytich&comma;v&comma;re&equals;2000..100000&comma;im&equals;100000..100000&comma;digits&equals;32

64189.825354267506015885528786917

(7)

Analyticsinx&comma;x&comma;re&equals;10..10&comma;im&equals;10..10&comma;digits&equals;10

0.&plus;0.I&comma;6.283185305&comma;3.141592654&comma;9.424777960&comma;6.283185305&comma;3.141592654&comma;9.424777960&comma;0.&plus;0.I&comma;0.&plus;0.I

(8)

f0sinx2x

f0:=sinx2x

(9)

Analyticf0&comma;x&comma;re&equals;10..10&comma;im&equals;10..10&comma;digits&equals;10

9.426231485&comma;3.247111097&comma;6.270228880&comma;0.66343768551.171931328I&comma;4.9807769674.145343143I&comma;9.2246857057.087212710I&comma;0.6634376855&plus;1.171931328I&comma;4.980776967&plus;4.145343143I&comma;9.224685705&plus;7.087212710I

(10)

numlist23&comma;1&plus;I5&comma;12I2&comma;7&plus;I&colon;

f1mulzz0&comma;z0&equals;numlist

f1:=z23z1515Iz&plus;12&plus;12Iz7I

(11)

Analyticf1&comma;z&comma;re&equals;2..10&comma;im&equals;2..2

0.666666666666665&comma;0.200000000000000&plus;0.200000000000000I&comma;7.00000000000000&plus;1.00000000000000I&comma;0.5000000000000000.499999999999999I

(12)

The following examples may be executed in Maple.

numlist23&comma;1&plus;I5&comma;12I2&comma;7&plus;I&colon;

f2addz0z&comma;z0&equals;numlist

RootFinding:-Analyticf2&comma;z&comma;re&equals;10..10&comma;im&equals;10..10&comma;plot

RootFinding:-AnalyticZerosFound

f3add1nz&comma;n&equals;1..10

RootFinding:-Analyticf3&comma;z&comma;re&equals;5..2&comma;im&equals;1..50&comma;plot&comma;extra

f4100Ix2&plus;78x4&plus;39Ix494x317x4&plus;19x73Ix254I59x5

zerosRootFinding:-Analyticf4&comma;x&comma;re&equals;10..10&comma;im&equals;10..10

plots&lsqb;complexplot&rsqb;zeros&comma;style&equals;point&comma;axes&equals;boxed

See Also

fsolve

plot

RootFinding

 


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