Fixed point: Difference between revisions
imported>Dmitrii Kouznetsov |
imported>Dmitrii Kouznetsov |
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<math>b=\sqrt{2}</math> | <math>b=\sqrt{2}</math> | ||
<math>b=\exp(1/\rm e</math> | <math>b=\exp(1/\rm e</math> | ||
<math>b=2</math>. In the last case, there are no real solutions, but the complex fixed points are complex numbers <math>L_2</math> and <math>L_2^*</math>. Within few hundred iterations of equation (14), they can be approcimated with many | <math>b=2</math>. In the last case, there are no real solutions, but the complex fixed points are complex numbers <math>L_2</math> and <math>L_2^*</math>. Within few hundred iterations of equation (14), they can be approcimated with many decimal digits; | ||
:(15)<math> L_2 \approx 0.824678546142074222314065+1.56743212384964786105857 \!~\rm i | :(15)<math> L_2 \approx 0.824678546142074222314065+1.56743212384964786105857 \!~\rm i | ||
</math>. | </math>. | ||
Revision as of 05:18, 31 May 2008
Fixed point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L } of a functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F } is solution of equation
- (1) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F[L]=L}
Simple examples
Elementary functions
In particular, functor can be elementaty function. For example, 0 and 1 are fixed points of function sqrt, because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{0}=0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{1}=1} .
In similar way, 0 is fixed point of sine function, because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(0)=0} .
Operators
Functor in the equation (1) can be a linear operator. In this case, the fixed point of functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is its eigenfunction with eigenvalue equal to unity.
Exponential if fixed point or operator of differentiation D, because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm D}~ \exp(x) =\exp'(x)=\exp(x)}
- (2) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(x)=\exp(-x^2/2)} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~x \in} reals
is fixed point of the Fourier operator, defined with its action on a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} :
- (3) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(g)(p)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} g(x)\exp(-{\rm i}px) {\rm d}x }
in general, functors have no need to be linear, so, there is no associativity at application of several functors in row, and parenthesis are necessary in the left hand side of eapression (3). [1]
Fixed points of exponential and fixed points of logarithm
Fixed points can be searched graphically. Fig.1 shows the graphical search of fixed points of logarithm, i.e., soluitons of the equaiton
- (10) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\ln(L) }
There are no real solutions fot this equation, but there are two complex-congjugated solutions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{\rm e}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{\rm e}^*} . However, the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{\rm e}} cannot be estimated well from the figure (1), but the straigtgorward iteration allows the precise estimate. Few hundreds of iterations are sufficient to get error of order of last significant figure in the approximation
- (11) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{\rm e} \approx 0.318131505204764135312654+1.33723570143068940890116 \!~\rm i}
Fixed points of logarithm should not be confised with fixed points of exponential, shown in FIgure 2. Therse fixed points are solutions of equaitons
- (12) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\exp(L)}
They can be expressed also as solution of equation
- (13) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\log(L)+2\pi \!~\rm i ~m~} for some integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m}
For example, at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=1}
- (13)Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{\rm e, 1}\approx 2.062277729598284 + 7.5886311784725127 \!~\rm i }
is fixed point of the exponential, but is not the fixed point of natural logarithm.
In the case of exponential and natural logarithm, all fixed points are complex. However, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_b} the real fixed points exist at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1< b \le \exp(1/\rm e)} . For example, at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(1/ \rm e)>} , number e is fixed point of both, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_b} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp_b} ; and at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\sqrt{2}} , numbers 2 and 4 are their fixed points.
Finding of real fixed points is presented graphically in Figure 3. The black curve represents the identical function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=f(x)=x} in the left hand side of equation
- (14) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=\log_b(z)}
the colored curves represent the function in the right hand side for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\sqrt{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(1/\rm e} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=2} . In the last case, there are no real solutions, but the complex fixed points are complex numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_2} and . Within few hundred iterations of equation (14), they can be approcimated with many decimal digits;
- (15).
The fixed points of logarithm determine the asymptotic properties of analytic extension of tetration.
Notes
- ↑ Note that that there is certain ambiguity in commonly used writing of mathematical formulas, omiting sign * of multiplication; in equation (3), expression does not mean that is multiplied to value of ; it means that result of action of operator on function , whith is also function, is evaluated at argument .
