gnuplot: the "sum [<var> = <start>:<end>] <expression>" strategy

Question

In the following script, I would like to plot y(x):

and this function u(x):

EDIT

Plotting y(x) is easy, but I am having problems to plot the function u(x).

u(x) is just the same function as y(x), but summing every step.

Therefore, in order to plot u(x), I have tried the sum [<var> = <start>:<end>] <expression> strategy. I have implemented this notation as:

replot sum[x=1:6] y(x) with line lt -1 lw 1 lc 2 title "u(x)"

in the following script:

 #

 set ylabel "y" font  ", 20"
 set xlabel 'x' font  ", 20"


 set format y "%9.4f"
 set xrange [1:6]

 set yrange [0:20]
 set xtics font ", 15"
 set ytics font ", 15"


 set key font ",17" # Changes the font of the letters of the legend

 y(x) = (2*x)/(x**2)
 plot y(x) with line lt -1 lw 1 lc 1 title "y(x)"
 replot sum[x=1:6] y(x) with line lt -1 lw 1 lc 2 title "u(x)"


 pause -1
 set term post enh color eps
 set output "y_x.eps"
 replot

I am not sure if the sum[x=1:6] y(x) strategy is indeed plotting u(x).

In order to check this, we can do the following:

we know that:

So, what is the value in gnuplot for u(6)? If you run that script, you get:

zooming:

I see that u(6) is reaching the value of 2.0000, and not 3.5835.

This makes me think that the replot sum[x=1:6] u(x) is not plotting u(x_i) (second formula)

How could I plot u(x) ?.

EDIT 2

Running replot sum[i=1:6] y(i) in this script:

set ylabel "y" font  ", 20"
set xlabel 'x' font  ", 20"


 set format y "%9.4f"

 set xrange [1:6]

 set yrange [0:20]
 set xtics font ", 15"
 set ytics font ", 15"


 set key font ",17" # Changes the font of the letters of the legend

 y(x) = (2*x)/(x**2)
 plot y(x) with line lt -1 lw 1 lc 1 title "y(x)"


 replot sum[i=1:6] y(i) with line lt -1 lw 1 lc 2 title "u(x)"

 pause -1
 set term post enh color eps
 set output "y_x.eps"
 replot

produces the following: u(6) = 3.000:

EDIT 3

Using y(x) = (2.*x)/(x**2) or y(x) = (2.*x)/(x**2.), I get u(6) = 4.9:

EDIT 4

Making:

N=100
replot sum[i=0:N-1] y(1. + (i+0.5)*5./N)*5./N with line lt -1 lw 1 lc 5 title "sum(x)"

produces a constant (cyan line) y=3.58. This is the result of the numerical approximation of the summation.

What I really want to achieve is to plot the function u(x) for all the values of x_{i}... in which at every step i, the summation over all the previous steps is performed, and a new value of u is generated. I would like to plot the function u(x)...

Answer 1

In fact, what is actually plotted as u(x) is just 6*y(x), you can check that if you replace in your script the line

plot y(x) with line lt -1 lw 1 lc 1 title "y(x)"

with

plot 6*y(x) with line lt -1 lw 1 lc 1 title "y(x)"

that the line will then coincide with u(x).

Also, note that the function y(x) is not integrable in the interval [0, 6] since it behaves at 0 as 1/x (in your formula, you would obtain expression ln(0)).