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<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><title>Решение уравнений</title><meta name="generator" content="DocBook XSL Stylesheets Vsnapshot"><link rel="home" href="index.html" title="Руководство пользователя Genius"><link rel="up" href="ch11.html" title="Chapter 11. Список функций GEL"><link rel="prev" href="ch11s12.html" title="Functions"><link rel="next" href="ch11s14.html" title="Статистика"></head><body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"><div class="navheader"><table width="100%" summary="Navigation header"><tr><th colspan="3" align="center">Решение уравнений</th></tr><tr><td width="20%" align="left"><a accesskey="p" href="ch11s12.html">Prev</a> </td><th width="60%" align="center">Chapter 11. Список функций GEL</th><td width="20%" align="right"> <a accesskey="n" href="ch11s14.html">Next</a></td></tr></table><hr></div><div class="sect1"><div class="titlepage"><div><div><h2 class="title" style="clear: both"><a name="genius-gel-function-list-equation-solving"></a>Решение уравнений</h2></div></div></div><div class="variablelist"><dl class="variablelist"><dt><span lang="en" class="term"><a name="gel-function-CubicFormula"></a>CubicFormula</span></dt><dd><pre lang="en" class="synopsis">CubicFormula (p)</pre><p lang="en">
Compute roots of a cubic (degree 3) polynomial using the
cubic formula. The polynomial should be given as a
vector of coefficients. That is
<strong class="userinput"><code>4*x^3 + 2*x + 1</code></strong> corresponds to the vector
<strong class="userinput"><code>[1,2,0,4]</code></strong>.
Returns a column vector of the three solutions. The first solution is always
the real one as a cubic always has one real solution.
</p><p lang="en">
See
<a class="ulink" href="http://planetmath.org/CubicFormula" target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/CubicFormula.html" target="_top">Mathworld</a>, or
<a class="ulink" href="https://en.wikipedia.org/wiki/Cubic_equation" target="_top">Wikipedia</a> for more information.
</p></dd><dt><span lang="en" class="term"><a name="gel-function-EulersMethod"></a>EulersMethod</span></dt><dd><pre lang="en" class="synopsis">EulersMethod (f,x0,y0,x1,n)</pre><p lang="en">
Use classical Euler's method to numerically solve y'=f(x,y) for
initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
<code class="varname">x1</code> with <code class="varname">n</code> increments,
returns <code class="varname">y</code> at <code class="varname">x1</code>.
Unless you explicitly want to use Euler's method, you should really
think about using
<a class="link" href="ch11s13.html#gel-function-RungeKutta">RungeKutta</a>
for solving ODE.
</p><p lang="en">
Systems can be solved by just having <code class="varname">y</code> be a
(column) vector everywhere. That is, <code class="varname">y0</code> can
be a vector in which case <code class="varname">f</code> should take a number
<code class="varname">x</code> and a vector of the same size for the second
argument and should return a vector of the same size.
</p><p lang="en">
See
<a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html" target="_top">Mathworld</a> or
<a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method" target="_top">Wikipedia</a> for more information.
</p></dd><dt><span lang="en" class="term"><a name="gel-function-EulersMethodFull"></a>EulersMethodFull</span></dt><dd><pre lang="en" class="synopsis">EulersMethodFull (f,x0,y0,x1,n)</pre><p lang="en">
Use classical Euler's method to numerically solve y'=f(x,y) for
initial <code class="varname">x0</code>, <code class="varname">y0</code> going to
<code class="varname">x1</code> with <code class="varname">n</code> increments,
returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
<code class="varname">x</code> and <code class="varname">y</code> values.
Unless you explicitly want to use Euler's method, you should really
think about using
<a class="link" href="ch11s13.html#gel-function-RungeKuttaFull">RungeKuttaFull</a>
for solving ODE.
Suitable
for plugging into
<a class="link" href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> or
<a class="link" href="ch11s20.html#gel-function-LinePlotDrawPoints">LinePlotDrawPoints</a>.
</p><p lang="en">
Example:
</p><pre lang="en" class="screen"><code class="prompt">genius></code> <strong class="userinput"><code>LinePlotClear();</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>line = EulersMethodFull(`(x,y)=y,0,1.0,3.0,50);</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>LinePlotDrawLine(line,"window","fit","color","blue","legend","Exponential growth");</code></strong>
</pre><p lang="en">
</p><p lang="en">
Systems can be solved by just having <code class="varname">y</code> be a
(column) vector everywhere. That is, <code class="varname">y0</code> can
be a vector in which case <code class="varname">f</code> should take a number
<code class="varname">x</code> and a vector of the same size for the second
argument and should return a vector of the same size.
</p><p lang="en">
The output for a system is still a n by 2 matrix with the second
entry being a vector. If you wish to plot the line, make sure to use row vectors, and then flatten the matrix with
<a class="link" href="ch11s08.html#gel-function-ExpandMatrix">ExpandMatrix</a>,
and pick out the right columns. Example:
</p><pre lang="en" class="screen"><code class="prompt">genius></code> <strong class="userinput"><code>LinePlotClear();</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>lines = EulersMethodFull(`(x,y)=[y@(2),-y@(1)],0,[1.0,1.0],10.0,500);</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>lines = ExpandMatrix(lines);</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>firstline = lines@(,[1,2]);</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>secondline = lines@(,[1,3]);</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>LinePlotWindow = [0,10,-2,2];</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>LinePlotDrawLine(firstline,"color","blue","legend","First");</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>LinePlotDrawPoints(secondline,"color","red","thickness",3,"legend","Second");</code></strong>
</pre><p lang="en">
</p><p lang="en">
See
<a class="ulink" href="http://mathworld.wolfram.com/EulerForwardMethod.html" target="_top">Mathworld</a> or
<a class="ulink" href="https://en.wikipedia.org/wiki/Eulers_method" target="_top">Wikipedia</a> for more information.
</p><p lang="en">Version 1.0.10 onwards.</p></dd><dt><span lang="en" class="term"><a name="gel-function-FindRootBisection"></a>FindRootBisection</span></dt><dd><pre lang="en" class="synopsis">FindRootBisection (f,a,b,TOL,N)</pre><p lang="en">Find root of a function using the bisection method.
<code class="varname">a</code> and <code class="varname">b</code> are the initial guess interval,
<strong class="userinput"><code>f(a)</code></strong> and <strong class="userinput"><code>f(b)</code></strong> should have opposite signs.
<code class="varname">TOL</code> is the desired tolerance and
<code class="varname">N</code> is the limit on the number of iterations to run, 0 means no limit. The function returns a vector <strong class="userinput"><code>[success,value,iteration]</code></strong>, where <code class="varname">success</code> is a boolean indicating success, <code class="varname">value</code> is the last value computed, and <code class="varname">iteration</code> is the number of iterations done.</p></dd><dt><span lang="en" class="term"><a name="gel-function-FindRootFalsePosition"></a>FindRootFalsePosition</span></dt><dd><pre lang="en" class="synopsis">FindRootFalsePosition (f,a,b,TOL,N)</pre><p lang="en">Find root of a function using the method of false position.
<code class="varname">a</code> and <code class="varname">b</code> are the initial guess interval,
<strong class="userinput"><code>f(a)</code></strong> and <strong class="userinput"><code>f(b)</code></strong> should have opposite signs.
<code class="varname">TOL</code> is the desired tolerance and
<code class="varname">N</code> is the limit on the number of iterations to run, 0 means no limit. The function returns a vector <strong class="userinput"><code>[success,value,iteration]</code></strong>, where <code class="varname">success</code> is a boolean indicating success, <code class="varname">value</code> is the last value computed, and <code class="varname">iteration</code> is the number of iterations done.</p></dd><dt><span lang="en" class="term"><a name="gel-function-FindRootMullersMethod"></a>FindRootMullersMethod</span></dt><dd><pre lang="en" class="synopsis">FindRootMullersMethod (f,x0,x1,x2,TOL,N)</pre><p lang="en">Find root of a function using the Muller's method.
<code class="varname">TOL</code> is the desired tolerance and
<code class="varname">N</code> is the limit on the number of iterations to run, 0 means no limit. The function returns a vector <strong class="userinput"><code>[success,value,iteration]</code></strong>, where <code class="varname">success</code> is a boolean indicating success, <code class="varname">value</code> is the last value computed, and <code class="varname">iteration</code> is the number of iterations done.</p></dd><dt><span lang="en" class="term"><a name="gel-function-FindRootSecant"></a>FindRootSecant</span></dt><dd><pre lang="en" class="synopsis">FindRootSecant (f,a,b,TOL,N)</pre><p lang="en">Find root of a function using the secant method.
<code class="varname">a</code> and <code class="varname">b</code> are the initial guess interval,
<strong class="userinput"><code>f(a)</code></strong> and <strong class="userinput"><code>f(b)</code></strong> should have opposite signs.
<code class="varname">TOL</code> is the desired tolerance and
<code class="varname">N</code> is the limit on the number of iterations to run, 0 means no limit. The function returns a vector <strong class="userinput"><code>[success,value,iteration]</code></strong>, where <code class="varname">success</code> is a boolean indicating success, <code class="varname">value</code> is the last value computed, and <code class="varname">iteration</code> is the number of iterations done.</p></dd><dt><span lang="en" class="term"><a name="gel-function-HalleysMethod"></a>HalleysMethod</span></dt><dd><pre lang="en" class="synopsis">HalleysMethod (f,df,ddf,guess,epsilon,maxn)</pre><p lang="en">Find zeros using Halley's method. <code class="varname">f</code> is
the function, <code class="varname">df</code> is the derivative of
<code class="varname">f</code>, and <code class="varname">ddf</code> is the second derivative of
<code class="varname">f</code>. <code class="varname">guess</code> is the initial
guess. The function returns after two successive values are
within <code class="varname">epsilon</code> of each other, or after <code class="varname">maxn</code> tries, in which case the function returns <code class="constant">null</code> indicating failure.
</p><p lang="en">
See also <a class="link" href="ch11s13.html#gel-function-NewtonsMethod"><code class="function">NewtonsMethod</code></a> and <a class="link" href="ch11s19.html#gel-function-SymbolicDerivative"><code class="function">SymbolicDerivative</code></a>.
</p><p lang="en">
Example to find the square root of 10:
</p><pre lang="en" class="screen"><code class="prompt">genius></code> <strong class="userinput"><code>HalleysMethod(`(x)=x^2-10,`(x)=2*x,`(x)=2,3,10^-10,100)</code></strong>
</pre><p lang="en">
</p><p lang="en">
See
<a class="ulink" href="https://en.wikipedia.org/wiki/Halley%27s_method" target="_top">Wikipedia</a> for more information.
</p><p lang="en">Version 1.0.18 onwards.</p></dd><dt><span lang="en" class="term"><a name="gel-function-NewtonsMethod"></a>NewtonsMethod</span></dt><dd><pre lang="en" class="synopsis">NewtonsMethod (f,df,guess,epsilon,maxn)</pre><p lang="en">Find zeros using Newton's method. <code class="varname">f</code> is
the function and <code class="varname">df</code> is the derivative of
<code class="varname">f</code>. <code class="varname">guess</code> is the initial
guess. The function returns after two successive values are
within <code class="varname">epsilon</code> of each other, or after <code class="varname">maxn</code> tries, in which case the function returns <code class="constant">null</code> indicating failure.
</p><p lang="en">
See also <a class="link" href="ch11s15.html#gel-function-NewtonsMethodPoly"><code class="function">NewtonsMethodPoly</code></a> and <a class="link" href="ch11s19.html#gel-function-SymbolicDerivative"><code class="function">SymbolicDerivative</code></a>.
</p><p lang="en">
Example to find the square root of 10:
</p><pre lang="en" class="screen"><code class="prompt">genius></code> <strong class="userinput"><code>NewtonsMethod(`(x)=x^2-10,`(x)=2*x,3,10^-10,100)</code></strong>
</pre><p lang="en">
</p><p lang="en">
See
<a class="ulink" href="https://en.wikipedia.org/wiki/Newtons_method" target="_top">Wikipedia</a> for more information.
</p><p lang="en">Version 1.0.18 onwards.</p></dd><dt><span lang="en" class="term"><a name="gel-function-PolynomialRoots"></a>PolynomialRoots</span></dt><dd><pre lang="en" class="synopsis">PolynomialRoots (p)</pre><p lang="en">
Compute roots of a polynomial (degrees 1 through 4)
using one of the formulas for such polynomials.
The polynomial should be given as a
vector of coefficients. That is
<strong class="userinput"><code>4*x^3 + 2*x + 1</code></strong> corresponds to the vector
<strong class="userinput"><code>[1,2,0,4]</code></strong>.
Returns a column vector of the solutions.
</p><p lang="en">
The function calls
<a class="link" href="ch11s13.html#gel-function-QuadraticFormula">QuadraticFormula</a>,
<a class="link" href="ch11s13.html#gel-function-CubicFormula">CubicFormula</a>, and
<a class="link" href="ch11s13.html#gel-function-QuarticFormula">QuarticFormula</a>.
</p></dd><dt><span lang="en" class="term"><a name="gel-function-QuadraticFormula"></a>QuadraticFormula</span></dt><dd><pre lang="en" class="synopsis">QuadraticFormula (p)</pre><p lang="en">
Compute roots of a quadratic (degree 2) polynomial using the
quadratic formula. The polynomial should be given as a
vector of coefficients. That is
<strong class="userinput"><code>3*x^2 + 2*x + 1</code></strong> corresponds to the vector
<strong class="userinput"><code>[1,2,3]</code></strong>.
Returns a column vector of the two solutions.
</p><p lang="en">
See
<a class="ulink" href="http://planetmath.org/QuadraticFormula" target="_top">Planetmath</a>, or
<a class="ulink" href="http://mathworld.wolfram.com/QuadraticFormula.html" target="_top">Mathworld</a>, or
<a class="ulink" href="https://en.wikipedia.org/wiki/Quadratic_formula" target="_top">Wikipedia</a> for more information.
</p></dd><dt><span lang="en" class="term"><a name="gel-function-QuarticFormula"></a>QuarticFormula</span></dt><dd><pre lang="en" class="synopsis">QuarticFormula (p)</pre><p lang="en">
Compute roots of a quartic (degree 4) polynomial using the
quartic formula. The polynomial should be given as a
vector of coefficients. That is
<strong class="userinput"><code>5*x^4 + 2*x + 1</code></strong> corresponds to the vector
<strong class="userinput"><code>[1,2,0,0,5]</code></strong>.
Returns a column vector of the four solutions.
</p><p lang="en">
See
<a class="ulink" href="http://planetmath.org/QuarticFormula" target="_top">Planetmath</a>,
<a class="ulink" href="http://mathworld.wolfram.com/QuarticEquation.html" target="_top">Mathworld</a>, or
<a class="ulink" href="https://en.wikipedia.org/wiki/Quartic_equation" target="_top">Wikipedia</a> for more information.
</p></dd><dt><span lang="en" class="term"><a name="gel-function-RungeKutta"></a>RungeKutta</span></dt><dd><pre lang="en" class="synopsis">RungeKutta (f,x0,y0,x1,n)</pre><p lang="en">
Use classical non-adaptive fourth order Runge-Kutta method to
numerically solve
y'=f(x,y) for initial <code class="varname">x0</code>, <code class="varname">y0</code>
going to <code class="varname">x1</code> with <code class="varname">n</code>
increments, returns <code class="varname">y</code> at <code class="varname">x1</code>.
</p><p lang="en">
Systems can be solved by just having <code class="varname">y</code> be a
(column) vector everywhere. That is, <code class="varname">y0</code> can
be a vector in which case <code class="varname">f</code> should take a number
<code class="varname">x</code> and a vector of the same size for the second
argument and should return a vector of the same size.
</p><p lang="en">
See
<a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html" target="_top">Mathworld</a> or
<a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods" target="_top">Wikipedia</a> for more information.
</p></dd><dt><span lang="en" class="term"><a name="gel-function-RungeKuttaFull"></a>RungeKuttaFull</span></dt><dd><pre lang="en" class="synopsis">RungeKuttaFull (f,x0,y0,x1,n)</pre><p lang="en">
Use classical non-adaptive fourth order Runge-Kutta method to
numerically solve
y'=f(x,y) for initial <code class="varname">x0</code>, <code class="varname">y0</code>
going to <code class="varname">x1</code> with <code class="varname">n</code>
increments,
returns an <strong class="userinput"><code>n+1</code></strong> by 2 matrix with the
<code class="varname">x</code> and <code class="varname">y</code> values. Suitable
for plugging into
<a class="link" href="ch11s20.html#gel-function-LinePlotDrawLine">LinePlotDrawLine</a> or
<a class="link" href="ch11s20.html#gel-function-LinePlotDrawPoints">LinePlotDrawPoints</a>.
</p><p lang="en">
Example:
</p><pre lang="en" class="screen"><code class="prompt">genius></code> <strong class="userinput"><code>LinePlotClear();</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>line = RungeKuttaFull(`(x,y)=y,0,1.0,3.0,50);</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>LinePlotDrawLine(line,"window","fit","color","blue","legend","Exponential growth");</code></strong>
</pre><p lang="en">
</p><p lang="en">
Systems can be solved by just having <code class="varname">y</code> be a
(column) vector everywhere. That is, <code class="varname">y0</code> can
be a vector in which case <code class="varname">f</code> should take a number
<code class="varname">x</code> and a vector of the same size for the second
argument and should return a vector of the same size.
</p><p lang="en">
The output for a system is still a n by 2 matrix with the second
entry being a vector. If you wish to plot the line, make sure to use row vectors, and then flatten the matrix with
<a class="link" href="ch11s08.html#gel-function-ExpandMatrix">ExpandMatrix</a>,
and pick out the right columns. Example:
</p><pre lang="en" class="screen"><code class="prompt">genius></code> <strong class="userinput"><code>LinePlotClear();</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>lines = RungeKuttaFull(`(x,y)=[y@(2),-y@(1)],0,[1.0,1.0],10.0,100);</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>lines = ExpandMatrix(lines);</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>firstline = lines@(,[1,2]);</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>secondline = lines@(,[1,3]);</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>LinePlotWindow = [0,10,-2,2];</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>LinePlotDrawLine(firstline,"color","blue","legend","First");</code></strong>
<code class="prompt">genius></code> <strong class="userinput"><code>LinePlotDrawPoints(secondline,"color","red","thickness",3,"legend","Second");</code></strong>
</pre><p lang="en">
</p><p lang="en">
See
<a class="ulink" href="http://mathworld.wolfram.com/Runge-KuttaMethod.html" target="_top">Mathworld</a> or
<a class="ulink" href="https://en.wikipedia.org/wiki/Runge-Kutta_methods" target="_top">Wikipedia</a> for more information.
</p><p lang="en">Version 1.0.10 onwards.</p></dd></dl></div></div><div class="navfooter"><hr><table width="100%" summary="Navigation footer"><tr><td width="40%" align="left"><a accesskey="p" href="ch11s12.html">Prev</a> </td><td width="20%" align="center"><a accesskey="u" href="ch11.html">Up</a></td><td width="40%" align="right"> <a accesskey="n" href="ch11s14.html">Next</a></td></tr><tr><td width="40%" align="left" valign="top">Functions </td><td width="20%" align="center"><a accesskey="h" href="index.html">Home</a></td><td width="40%" align="right" valign="top"> Статистика</td></tr></table></div></body></html>
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