File:TrivFctRootExm.pdf
Summary
| Description |
English: The function g: R→R is a trivial 5th functional root of f: R+→R+ defined by f(x) = sin(x). The computation of f(π/6) = 1/2 = g5(π/6) is shown. See File:TrivFctRootDef.pdf for the definition of g, given an arbitrary f: R+→R+ and n∈N, and the computation of gn(x) = f(x) for all x∈R+. |
| Date | |
| Source | Own work |
| Author | Jochen Burghardt |
| Other versions | File:TrivFctRootExm.pdf * File:TrivFctRootExm svg.svg * File:TrivFctRootDef.pdf |
C source code for example coordinates |
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#include <stdio.h>
#include <math.h>
static int const n = 5;
static double f(double x) { return sin(x); }
static double g(double x) {
if (0.0 <= x ) return atan(x) - M_PI;
if (-(n-1)*M_PI/2 <= x && x < 0 ) return x - M_PI/2;
if (-n*M_PI/2 <= x && x < -(n-1)*M_PI/2) return f(tan(x+n*M_PI/2));
return 0.0;
}
int main(void) {
int i;
double gi;
while (1) {
printf("enter:\n");
scanf(" %lf",&gi);
for (i=0; i<=n; ++i) {
printf("g^%d(x) = %f\n",i,gi);
gi = g(gi);
}
}
return 0;
}
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Gnuplot source code |
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# given:
n = 5
f(x) = sin(x)
# trivial function root:
g(x) = ( 0 <= x ? atan(x) - pi \
: -(n-1)*pi/2 <= x && x < 0 ? x - pi/2 \
: -n*pi/2 <= x && x < -(n-1)*pi/2 ? f(tan(x+n*pi/2)) \
: 0 \
)
xmin = -8
xmax = 7
ymin = -8
ymax = 2
set xrange[xmin:xmax]
set yrange[ymin:ymax]
set grid
set xtics pi/2
set ytics pi/2
set format x "%5.2f"
set format y "%5.2f"
set key right bottom
# coordinate cross:
set arrow from xmin , 0 to xmax , 0 nohead lt 8
set arrow from 0 , ymin to 0 , ymax nohead lt 8
# shift ranges:
set arrow from -1*pi/2 , ymin to -1*pi/2 , ymax nohead lt 5
set arrow from -2*pi/2 , ymin to -2*pi/2 , ymax nohead lt 5
set arrow from -3*pi/2 , ymin to -3*pi/2 , ymax nohead lt 5
set arrow from -4*pi/2 , ymin to -4*pi/2 , ymax nohead lt 5
set arrow from -5*pi/2 , ymin to -5*pi/2 , ymax nohead lc 5
# 1 input/output example f:
set arrow from 0.523599 , 0 to 0.523599 , 0.500000 lt 1
# 5 input/output examples g:
set arrow from 0.523599 , 0 to 0.523599 , -2.659245 lt 2
set arrow from 0.523599 , -2.659245 to -2.659245 , -2.659245 lt 2
set arrow from -2.659245 , -2.659245 to -2.659245 , -4.230041 lt 2
set arrow from -2.659245 , -4.230041 to -4.230041 , -4.230041 lt 2
set arrow from -4.230041 , -4.230041 to -4.230041 , -5.800837 lt 2
set arrow from -4.230041 , -5.800837 to -5.800837 , -5.800837 lt 2
set arrow from -5.800837 , -5.800837 to -5.800837 , -7.371634 lt 2
set arrow from -5.800837 , -7.371634 to -7.371634 , -7.371634 lt 2
set arrow from -7.371634 , -7.371634 to -7.371634 , 0.500000 lt 2
plot x with lines lt 7
replot [x=0:xmax] f(x) with lines lw 3 lt 1 title"f(x)"
replot g(x) with lines lw 3 lt 2
pause -1
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