Friday, June 3, 2016

The functions e^x, e^-x, e^(-x^2), erf(x) and Taylor Series

The functions e^x, e^-x, e^(-x^2), erf(x) and Taylor Series

Accurate digits are highlighted in green.  Calculations are used with a TI 84 Plus CE.

e^x = 1 + x + x^2/2! + x^3/3! + x^4/4 + … = Σ(x^n/n!, from n = 0 to ∞)

x =
e^x
10 terms
25 terms
50 terms
1
2.718281828
2.718281801
2.718281828
2.718281828
3
20.08553692
20.07966518
20.08553692
20.08553692
5
148.4131591
146.380601
148.4131591
148.4131591
9.9
19930.37044
11869.50538
19930.07221
19930.37044


e^(-x) = 1 – x + x^2/2! – x^3/3! + x^4/4 - … = Σ( (-x)^n/n!, from n = 0 to ∞)

x =
e^(-x)
10 terms
25 terms
50 terms
1
0.3678794412
0.3678794643
0.3678794412
0.3678794412
3
0.0497870684
0.0533258929
0.0497870684
0.0497870684
5
0.006737947
0.8640390763
0.0067379439
0.006737947
9.9
5.017468206E-5
1207.799663
-0.1392914019
5.017463241E-5

I think you know where I’m going.

e^(-x^2) = 1 – x^2 + x^4/2! – x^6/3! + x^8/4! = Σ( (-x)^(2*n)/n!, from n = 0 to ∞)

x =
e^(-x^2)
10 terms
25 terms
50 terms
1
0.3678794412
0.3678794643
0.3678794412
0.3678794412
3
1.234098041E-4
442.2750223
-0.0118646275
1.234194001E-4
5
1.38879439E-11
18613495.8
-2834107793
85689.40174
9.9
2.72143414E-43
2.04347238E13
-3.10254183E24
7.951057508E34

(Something really goes bonkers as x increases and n increases)

Error Function
erf(x) = 2/√π * ∫(e^(-t^2) dt, 0, x)
= 2/√π * (x – x^3/3 + x^5/(5*2!) – x^7/(7*3!) + x^9/(9*4!) - ...)
= 2/√π * Σ( (-x^(2n+1)/((2n+1)*n!) from n = 0 to ∞ )

x =
erf(x)
10 terms
25 terms
50 terms
1
0.8427007929
0.8427007941
0.8427007929
0.8427007929
3
0.9999779095
68.58627744
0.9992050426
0.9999779095
5
1
4853382.901
-3070260210.4
4724.331354
9.9
1
1.076461715E13
-6.7395908E23
*overflows during calculation*
(Result: 8.73442E33 from WolframAlpha)
(erf(x) is practically 1 for x > 3)

Note: 9.9^(2*50+1) ≈ 3.623E100


Thoughts:

*  Taylor series are great when x is near its center point.  In the all the cases above, the center point is x = 0. 

*  The more simple the expression, the better range of accuracy with less terms. 

*  Before you recommend a Taylor Series to approximate f(x), check the accuracy and the range.  A cautionary tale. 

Eddie


This blog is property of Edward Shore, 2016.

1 comment:

  1. Ola Edward,
    Muito legal o blog, Parabéns!
    Notei que informa varias dicas, poderia me auxiliar na programação da Hp prime?
    Quero aplicar a Formula de transformação gasosa: P1*V1/T1=P2*V2/V2
    Qual seria a melhor logica para retornar com o reultado?

    obrigado!!!

    ReplyDelete

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