'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' 'Cephes Math Library Release 2.8: June, 2000 'Copyright by Stephen L. Moshier ' 'Contributors: ' * Sergey Bochkanov (ALGLIB project). Translation from C to ' pseudocode. ' 'See subroutines comments for additional copyrights. ' '>>> SOURCE LICENSE >>> 'This program is free software; you can redistribute it and/or modify 'it under the terms of the GNU General Public License as published by 'the Free Software Foundation (www.fsf.org); either version 2 of the 'License, or (at your option) any later version. ' 'This program is distributed in the hope that it will be useful, 'but WITHOUT ANY WARRANTY; without even the implied warranty of 'MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 'GNU General Public License for more details. ' 'A copy of the GNU General Public License is available at 'http://www.fsf.org/licensing/licenses ' '>>> END OF LICENSE >>> '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' 'Routines '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' 'Airy function ' 'Solution of the differential equation ' 'y"(x) = xy. ' 'The function returns the two independent solutions Ai, Bi 'and their first derivatives Ai'(x), Bi'(x). ' 'Evaluation is by power series summation for small x, 'by rational minimax approximations for large x. ' ' ' 'ACCURACY: 'Error criterion is absolute when function <= 1, relative 'when function > 1, except * denotes relative error criterion. 'For large negative x, the absolute error increases as x^1.5. 'For large positive x, the relative error increases as x^1.5. ' 'Arithmetic domain function # trials peak rms 'IEEE -10, 0 Ai 10000 1.6e-15 2.7e-16 'IEEE 0, 10 Ai 10000 2.3e-14* 1.8e-15* 'IEEE -10, 0 Ai' 10000 4.6e-15 7.6e-16 'IEEE 0, 10 Ai' 10000 1.8e-14* 1.5e-15* 'IEEE -10, 10 Bi 30000 4.2e-15 5.3e-16 'IEEE -10, 10 Bi' 30000 4.9e-15 7.3e-16 ' 'Cephes Math Library Release 2.8: June, 2000 'Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier ' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' Public Sub Airy(ByVal x As Double, _ ByRef Ai As Double, _ ByRef Aip As Double, _ ByRef Bi As Double, _ ByRef Bip As Double) Dim z As Double Dim zz As Double Dim t As Double Dim f As Double Dim g As Double Dim uf As Double Dim ug As Double Dim k As Double Dim zeta As Double Dim theta As Double Dim domflg As Long Dim c1 As Double Dim c2 As Double Dim sqrt3 As Double Dim sqpii As Double Dim AFN As Double Dim AFD As Double Dim AGN As Double Dim AGD As Double Dim APFN As Double Dim APFD As Double Dim APGN As Double Dim APGD As Double Dim AN As Double Dim AD As Double Dim APN As Double Dim APD As Double Dim BN16 As Double Dim BD16 As Double Dim BPPN As Double Dim BPPD As Double sqpii = 0.564189583547756 c1 = 0.355028053887817 c2 = 0.258819403792807 sqrt3 = 1.73205080756888 domflg = 0# If x > 25.77 Then Ai = 0# Aip = 0# Bi = MaxRealNumber Bip = MaxRealNumber Exit Sub End If If x < -2.09 Then domflg = 15# t = Sqr(-x) zeta = -(2# * x * t / 3#) t = Sqr(t) k = sqpii / t z = 1# / zeta zz = z * z AFN = -0.131696323418332 AFN = AFN * zz - 0.626456544431912 AFN = AFN * zz - 0.693158036036933 AFN = AFN * zz - 0.279779981545119 AFN = AFN * zz - 0.04919001326095 AFN = AFN * zz - 4.06265923594885E-03 AFN = AFN * zz - 1.59276496239262E-04 AFN = AFN * zz - 2.77649108155233E-06 AFN = AFN * zz - 1.67787698489115E-08 AFD = 1# AFD = AFD * zz + 13.3560420706553 AFD = AFD * zz + 32.6825032795225 AFD = AFD * zz + 26.73670409415 AFD = AFD * zz + 9.1870740290726 AFD = AFD * zz + 1.47529146771666 AFD = AFD * zz + 0.115687173795188 AFD = AFD * zz + 4.40291641615211E-03 AFD = AFD * zz + 7.54720348287414E-05 AFD = AFD * zz + 4.5185009297058E-07 uf = 1# + zz * AFN / AFD AGN = 1.97339932091686E-02 AGN = AGN * zz + 0.391103029615688 AGN = AGN * zz + 1.06579897599596 AGN = AGN * zz + 0.93916922981665 AGN = AGN * zz + 0.351465656105548 AGN = AGN * zz + 6.33888919628925E-02 AGN = AGN * zz + 5.85804113048388E-03 AGN = AGN * zz + 2.82851600836737E-04 AGN = AGN * zz + 6.98793669997261E-06 AGN = AGN * zz + 8.11789239554389E-08 AGN = AGN * zz + 3.41551784765924E-10 AGD = 1# AGD = AGD * zz + 9.30892908077442 AGD = AGD * zz + 19.8352928718312 AGD = AGD * zz + 15.5646628932865 AGD = AGD * zz + 5.47686069422975 AGD = AGD * zz + 0.954293611618962 AGD = AGD * zz + 8.64580826352392E-02 AGD = AGD * zz + 4.12656523824223E-03 AGD = AGD * zz + 1.01259085116509E-04 AGD = AGD * zz + 1.17166733214414E-06 AGD = AGD * zz + 4.9183457006293E-09 ug = z * AGN / AGD theta = zeta + 0.25 * PI() f = Sin(theta) g = Cos(theta) Ai = k * (f * uf - g * ug) Bi = k * (g * uf + f * ug) APFN = 0.185365624022536 APFN = APFN * zz + 0.886712188052584 APFN = APFN * zz + 0.987391981747399 APFN = APFN * zz + 0.401241082318004 APFN = APFN * zz + 7.10304926289631E-02 APFN = APFN * zz + 5.90618657995662E-03 APFN = APFN * zz + 2.33051409401777E-04 APFN = APFN * zz + 4.08718778289035E-06 APFN = APFN * zz + 2.48379932900442E-08 APFD = 1# APFD = APFD * zz + 14.7345854687503 APFD = APFD * zz + 37.542393343549 APFD = APFD * zz + 31.4657751203046 APFD = APFD * zz + 10.9969125207299 APFD = APFD * zz + 1.78885054766999 APFD = APFD * zz + 0.141733275753663 APFD = APFD * zz + 5.44066067017226E-03 APFD = APFD * zz + 9.39421290654511E-05 APFD = APFD * zz + 5.65978713036027E-07 uf = 1# + zz * APFN / APFD APGN = -3.55615429033082E-02 APGN = APGN * zz - 0.637311518129436 APGN = APGN * zz - 1.70856738884312 APGN = APGN * zz - 1.50221872117317 APGN = APGN * zz - 0.563606665822103 APGN = APGN * zz - 0.102101031120217 APGN = APGN * zz - 9.48396695961445E-03 APGN = APGN * zz - 4.60325307486781E-04 APGN = APGN * zz - 1.14300836484517E-05 APGN = APGN * zz - 1.33415518685547E-07 APGN = APGN * zz - 5.63803833958894E-10 APGD = 1# APGD = APGD * zz + 9.8586580169613 APGD = APGD * zz + 21.6401867356586 APGD = APGD * zz + 17.3130776389749 APGD = APGD * zz + 6.17872175280829 APGD = APGD * zz + 1.08848694396321 APGD = APGD * zz + 9.95005543440888E-02 APGD = APGD * zz + 4.78468199683887E-03 APGD = APGD * zz + 1.18159633322839E-04 APGD = APGD * zz + 1.37480673554219E-06 APGD = APGD * zz + 5.79912514929148E-09 ug = z * APGN / APGD k = sqpii * t Aip = -(k * (g * uf + f * ug)) Bip = k * (f * uf - g * ug) Exit Sub End If If x >= 2.09 Then domflg = 5# t = Sqr(x) zeta = 2# * x * t / 3# g = Exp(zeta) t = Sqr(t) k = 2# * t * g z = 1# / zeta AN = 0.346538101525629 AN = AN * z + 12.0075952739646 AN = AN * z + 76.2796053615235 AN = AN * z + 168.089224934631 AN = AN * z + 159.756391350164 AN = AN * z + 70.5360906840444 AN = AN * z + 14.026469116339 AN = AN * z + 1# AD = 0.56759453263877 AD = AD * z + 14.7562562584847 AD = AD * z + 84.5138970141475 AD = AD * z + 177.3180881454 AD = AD * z + 164.23469287153 AD = AD * z + 71.4778400825576 AD = AD * z + 14.0959135607834 AD = AD * z + 1# f = AN / AD Ai = sqpii * f / k k = -(0.5 * sqpii * t / g) APN = 0.613759184814036 APN = APN * z + 14.7454670787755 APN = APN * z + 82.0584123476061 APN = APN * z + 171.184781360976 APN = APN * z + 159.317847137142 APN = APN * z + 69.9778599330103 APN = APN * z + 13.9470856980482 APN = APN * z + 1# APD = 0.334203677749737 APD = APD * z + 11.1810297306158 APD = APD * z + 71.172735214786 APD = APD * z + 158.778084372838 APD = APD * z + 153.206427475809 APD = APD * z + 68.675230459278 APD = APD * z + 13.8498634758259 APD = APD * z + 1# f = APN / APD Aip = f * k If x > 8.3203353 Then BN16 = -0.253240795869364 BN16 = BN16 * z + 0.575285167332467 BN16 = BN16 * z - 0.329907036873225 BN16 = BN16 * z + 0.06444040689482 BN16 = BN16 * z - 3.82519546641337E-03 BD16 = 1# BD16 = BD16 * z - 7.15685095054035 BD16 = BD16 * z + 10.6039580715665 BD16 = BD16 * z - 5.23246636471251 BD16 = BD16 * z + 0.957395864378384 BD16 = BD16 * z - 0.055082814716355 f = z * BN16 / BD16 k = sqpii * g Bi = k * (1# + f) / t BPPN = 0.465461162774652 BPPN = BPPN * z - 1.08992173800494 BPPN = BPPN * z + 0.638800117371828 BPPN = BPPN * z - 0.126844349553103 BPPN = BPPN * z + 7.6248784434211E-03 BPPD = 1# BPPD = BPPD * z - 8.70622787633159 BPPD = BPPD * z + 13.8993162704553 BPPD = BPPD * z - 7.14116144616431 BPPD = BPPD * z + 1.34008595960681 BPPD = BPPD * z - 7.84273211323342E-02 f = z * BPPN / BPPD Bip = k * t * (1# + f) Exit Sub End If End If f = 1# g = x t = 1# uf = 1# ug = x k = 1# z = x * x * x Do While t > MachineEpsilon uf = uf * z k = k + 1# uf = uf / k ug = ug * z k = k + 1# ug = ug / k uf = uf / k f = f + uf k = k + 1# ug = ug / k g = g + ug t = Abs(uf / f) Loop uf = c1 * f ug = c2 * g If domflg Mod 2# = 0# Then Ai = uf - ug End If If domflg \ 2# Mod 2# = 0# Then Bi = sqrt3 * (uf + ug) End If k = 4# uf = x * x / 2# ug = z / 3# f = uf g = 1# + ug uf = uf / 3# t = 1# Do While t > MachineEpsilon uf = uf * z ug = ug / k k = k + 1# ug = ug * z uf = uf / k f = f + uf k = k + 1# ug = ug / k uf = uf / k g = g + ug k = k + 1# t = Abs(ug / g) Loop uf = c1 * f ug = c2 * g If domflg \ 4# Mod 2# = 0# Then Aip = uf - ug End If If domflg \ 8# Mod 2# = 0# Then Bip = sqrt3 * (uf + ug) End If End Sub

## Monday, July 23, 2018

### Solution of the differential equation

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