 RootFinding - Maple Programming Help

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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 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 $f\left(z\right)$ within the rectangular region $a<=\mathrm{Re}\left(z\right)<=b$, $c<=\mathrm{Im}\left(z\right)<=d$ 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 are 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.
 • $\mathrm{modulo}=a+Ib$ and $\mathrm{modulo_s}=a+Ib$
 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 $0<=\mathrm{Re}\left(z\right)<=a$ and $0<=\mathrm{Im}\left(z\right)<=b$; With 'modulo_s' they will be reduced to the region $-\frac{a}{2}<=\mathrm{Re}\left(z\right)<=\frac{a}{2}$, $-\frac{b}{2}<=\mathrm{Im}\left(z\right)<=\frac{b}{2}$.
 • 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

 > $\mathrm{with}\left(\mathrm{RootFinding}\right)$
 $\left[{\mathrm{Analytic}}{,}{\mathrm{AnalyticZerosFound}}{,}{\mathrm{BivariatePolynomial}}{,}{\mathrm{EnclosingBox}}{,}{\mathrm{HasRealRoots}}{,}{\mathrm{Homotopy}}{,}{\mathrm{Isolate}}{,}{\mathrm{NextZero}}{,}{\mathrm{Parametric}}{,}{\mathrm{WitnessPoints}}\right]$ (1)
 > $f≔\mathrm{tan}\left(\mathrm{sin}\left(x\right)\right)-1$
 ${f}{≔}{\mathrm{tan}}{}\left({\mathrm{sin}}{}\left({x}\right)\right){-}{1}$ (2)
 > $\mathrm{Analytic}\left(f,x,-\frac{I}{2}..1+I\right)$
 ${0.903339110766515}$ (3)
 > $g≔23{x}^{5}+105{x}^{4}-10{x}^{2}+17x$
 ${g}{≔}{23}{}{{x}}^{{5}}{+}{105}{}{{x}}^{{4}}{-}{10}{}{{x}}^{{2}}{+}{17}{}{x}$ (4)
 > $\mathrm{Analytic}\left(g,x,\mathrm{re}=-5..1,\mathrm{im}=-1..1\right)$
 ${0.}{}{I}{,}{-0.637181318531050}{,}{0.304066454284907}{-}{0.404061905751759}{}{I}{,}{0.304066454284907}{+}{0.404061905751759}{}{I}{,}{-4.53616898134312}$ (5)
 > $h≔10-\left(\mathrm{ln}\left(v+{\left({v}^{2}-1\right)}^{\frac{1}{2}}\right)-\mathrm{ln}\left(3+{\left({3}^{2}-1\right)}^{\frac{1}{2}}\right)\right)$
 ${h}{≔}{10}{-}{\mathrm{ln}}{}\left({v}{+}\sqrt{{{v}}^{{2}}{-}{1}}\right){+}{\mathrm{ln}}{}\left({3}{+}\sqrt{{8}}\right)$ (6)
 > $\mathrm{Analytic}\left(h,v,\mathrm{re}=2000..100000,\mathrm{im}=-100000..100000,\mathrm{digits}=32\right)$
 ${64189.825354267506015885528786917}$ (7)
 > $\mathrm{Analytic}\left(\mathrm{sin}\left(x\right),x,\mathrm{re}=-10..10,\mathrm{im}=-10..10,\mathrm{digits}=10\right)$
 ${0.}{}{I}{,}{-6.283185305}{,}{-3.141592654}{,}{-9.424777960}{,}{6.283185305}{,}{3.141592654}{,}{9.424777960}{,}{0.}{}{I}{,}{0.}{}{I}$ (8)
 > $\mathrm{f0}≔\mathrm{sin}\left(x\right)-{2}^{x}$
 ${\mathrm{f0}}{≔}{\mathrm{sin}}{}\left({x}\right){-}{{2}}^{{x}}$ (9)
 > $\mathrm{Analytic}\left(\mathrm{f0},x,\mathrm{re}=-10..10,\mathrm{im}=-10..10,\mathrm{digits}=10\right)$
 ${-9.426231485}{,}{-3.247111097}{,}{-6.270228880}{,}{0.6634376855}{-}{1.171931328}{}{I}{,}{4.980776967}{-}{4.145343143}{}{I}{,}{9.224685705}{-}{7.087212710}{}{I}{,}{0.6634376855}{+}{1.171931328}{}{I}{,}{4.980776967}{+}{4.145343143}{}{I}{,}{9.224685705}{+}{7.087212710}{}{I}$ (10)
 > $\mathrm{numlist}≔\left[\frac{2}{3},\frac{1+I}{5},-\frac{1}{2}-\frac{I}{2},7+I\right]:$
 > $\mathrm{f1}≔\mathrm{mul}\left(z-\mathrm{z0},\mathrm{z0}=\mathrm{numlist}\right)$
 ${\mathrm{f1}}{≔}\left({z}{-}\frac{{2}}{{3}}\right){}\left({z}{-}\frac{{1}}{{5}}{-}\frac{{I}}{{5}}\right){}\left({z}{+}\frac{{1}}{{2}}{+}\frac{{I}}{{2}}\right){}\left({z}{-}{7}{-}{I}\right)$ (11)
 > $\mathrm{Analytic}\left(\mathrm{f1},z,\mathrm{re}=-2..10,\mathrm{im}=-2..2\right)$
 ${0.666666666666665}{,}{0.200000000000000}{+}{0.200000000000000}{}{I}{,}{7.00000000000000}{+}{I}{,}{-0.50000000000000}{-}{0.50000000000000}{}{I}$ (12)

The following examples may be executed in Maple.

 > $\mathrm{numlist}≔\left[\frac{2}{3},\frac{1+I}{5},-\frac{1}{2}-\frac{I}{2},7+I\right]:$
 > $\mathrm{f2}≔\mathrm{add}\left({\mathrm{z0}}^{z},\mathrm{z0}=\mathrm{numlist}\right)$
 > $\mathrm{RootFinding}:-\mathrm{Analytic}\left(\mathrm{f2},z,\mathrm{re}=-10..10,\mathrm{im}=-10..10,\mathrm{plot}\right)$
 > $\mathrm{RootFinding}:-\mathrm{AnalyticZerosFound}\left(\right)$
 > $\mathrm{f3}≔\mathrm{add}\left(\frac{1}{{n}^{z}},n=1..10\right)$
 > $\mathrm{RootFinding}:-\mathrm{Analytic}\left(\mathrm{f3},z,\mathrm{re}=-5..2,\mathrm{im}=-1..50,\mathrm{plot},\mathrm{extra}\right)$
 > $\mathrm{f4}≔{\left(-100I{x}^{2}+78{x}^{4}+39I{x}^{4}-94{x}^{3}-17x\right)}^{4}+19x-73Ix-25-4I-59{x}^{5}$
 > $\mathrm{zeros}≔\mathrm{RootFinding}:-\mathrm{Analytic}\left(\mathrm{f4},x,\mathrm{re}=-10..10,\mathrm{im}=-10..10\right)$
 > $\mathrm{plots}\left[\mathrm{complexplot}\right]\left(\left\{\mathrm{zeros}\right\},\mathrm{style}=\mathrm{point},\mathrm{axes}=\mathrm{boxed}\right)$