Texas Instruments TI-92+, Tip List

Page: 272 / 507

(Credit to Ralf Fritzsch)
[6.60] Faster, more accurate exponential integrals
A post on the comp.sys.hp48 newsgroup announced that the HP-49G did not accurately find a
numerical value for this integral
¶
−
3.724
∞
e
−
x
x dx
Except for a sign change, this is identical to the special function called the exponential integral:
Ei(x)
= −
¶
−
x
∞
e
−
x
x dx
The TI-89 / TI-92 Plus also cannot find accurate values for this integral, and it is no wonder, since at
the integrand singularity at x = 0, the limit is +infinity from the right and -infinity from the left. One
expedient solution is a routine specifically designed to find the exponential integral:
ei(x)
Func
©(x) exponential integral Ei
©
1 3jul02/dburkett@infinet.com
local eps,euler,fpmin,maxit,k,fact,prev,sum 1 ,term
6 ⁻1 2 → eps © Desired relative accuracy
.5772 1 566490 1 53 → euler © Euler's constant
1 00 → maxit © Maximum iterations allowed for convergence
1⁻ 900 → fpmin © Number near floating-point minimum
if x ≤ 0:return "ei arg error" © Argument must be > 0
if x<fpmin then © Handle very small arguments
return euler+ln(x)
elseif x
≤⁻ ln(eps) then © Use power series for x < 25.840
0 → sum 1
1→ fact
for k, 1 ,maxit
fact*x/k → fact
fact/k → term
sum 1 +term → sum 1
if term<eps*sum 1
return sum 1 +ln(x)+euler
endfor
return "ei failed series"
© Return error string if convergence fails
else © Use asymptotic expansion for large arguments
0 → sum 1
1→ term
for k, 1 ,maxit
term → prev
term*k/x → term
if term<eps:goto l 1
if term<prev then
sum
1 +term → sum 1
else
sum
1 -prev → sum 1
goto l 1
endif
endfor
lbl l
1
return ℯ ^(x)*( 1 +sum 1 )/x
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