6.2198 3198 5865 6 E-16t = 8 to t = 9
1.2791 9044 7828 2 E-12t = 7 to t = 8
9.8530 7832 4938 2 E-10t = 6 to t = 7
3.8566 4984 2341 4 E-7t = 5 to t = 6
3.1384 5902 6124 E-5t = 4 to t = 5
0.0013 1822 6789 7969t = 3 to t = 4
0.0214 0023 3916 549t = 2 to t = 3
0.1359 0512 1983 28t = 1 to t = 2
IntegralIntegration limits
Summing these individual integrals gives a result of 0.1586 5525 3931 46, which has an error of zero,
to 14 digits.
Note that I stopped taking integrals when the individual integral is less than 1E-14, since further,
smaller integrals will not contribute to the sum.
This method can be used with the á operator, or with the nInt() function.
The function nintx(), shown below, automates this process.
nintx(ffx,xv,xx1,xx2,nx)
func
©Numerical integrator with summed subintervals
©dburkett@infinet.com 6 nov 99
©ffx: function to integrate
©xv: variable of integration
©xx1,xx2: lower and upper integration limits
©nx: number of subintervals
local dx
(xx2-xx1)/nx→dx
sum(seq(nint(ffx,xv,xx1+i*dx,xx1+(i+1)*dx),i,0,nx-1))
endfunc
For example, the call nintx(tan(x),x,0,1.5707,5) integrates tan(x) from x=0 to x=1.5707, with 5
subintervals.
The amount of improvement depends on the function. The built-in nint() function has an error of
-29.7E-12 for the tan(x) example above. nintx() with 10 intervals has an error of about 13E-12.
[6.20] Generating random numbers
The rand() function can only be used to generate random integers in the intervals [1,n] or [-n, -1]. Use
this to generate random integers over any range [nl,nh]:
rand(nh-nl+1)+nl-1
If rand() is called with no arguments, it returns a random floating point number between 0 and 1. To
generate uniformly distributed random numbers over the range [fl,fh], use this:
(fh-fl)*rand()+fl
6 - 23
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