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amos.go
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// Copyright ©2016 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package amos
import (
"math"
"math/cmplx"
)
/*
The AMOS functions are included in SLATEC, and the SLATEC guide (http://www.netlib.org/slatec/guide) explicitly states:
"The Library is in the public domain and distributed by the Energy
Science and Technology Software Center."
Mention of AMOS's inclusion in SLATEC goes back at least to this 1985 technical report from Sandia National Labs: http://infoserve.sandia.gov/sand_doc/1985/851018.pdf
*/
// math.NaN() are for padding to keep indexing easy.
var imach = []int{-0, 5, 6, 0, 0, 32, 4, 2, 31, 2147483647, 2, 24, -125, 127, 53, -1021, 1023}
var dmach = []float64{math.NaN(), 2.23e-308, 1.79e-308, 1.11e-16, 2.22e-16, 0.30103000998497009}
func abs(a int) int {
if a >= 0 {
return a
}
return -a
}
func min(a, b int) int {
if a < b {
return a
}
return b
}
func max(a, b int) int {
if a > b {
return a
}
return b
}
func Zairy(ZR, ZI float64, ID, KODE int) (AIR, AII float64, NZ int) {
// zairy is adapted from the original Netlib code by Donald Amos.
// http://www.netlib.no/netlib/amos/zairy.f
// Original comment:
/*
C***BEGIN PROLOGUE ZAIRY
C***DATE WRITTEN 830501 (YYMMDD)
C***REVISION DATE 890801 (YYMMDD)
C***CATEGORY NO. B5K
C***KEYWORDS AIRY FUNCTION,BESSEL FUNCTIONS OF ORDER ONE THIRD
C***AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
C***PURPOSE TO COMPUTE AIRY FUNCTIONS AI(Z) AND DAI(Z) FOR COMPLEX Z
C***DESCRIPTION
C
C ***A DOUBLE PRECISION ROUTINE***
C ON KODE=1, ZAIRY COMPUTES THE COMPLEX AIRY FUNCTION AI(Z) OR
C ITS DERIVATIVE DAI(Z)/DZ ON ID=0 OR ID=1 RESPECTIVELY. ON
C KODE=2, A SCALING OPTION CEXP(ZTA)*AI(Z) OR CEXP(ZTA)*
C DAI(Z)/DZ IS PROVIDED TO REMOVE THE EXPONENTIAL DECAY IN
C -PI/3<ARG(Z)<PI/3 AND THE EXPONENTIAL GROWTH IN
C PI/3<ABS(ARG(Z))<PI WHERE ZTA=(2/3)*Z*CSQRT(Z).
C
C WHILE THE AIRY FUNCTIONS AI(Z) AND DAI(Z)/DZ ARE ANALYTIC IN
C THE WHOLE Z PLANE, THE CORRESPONDING SCALED FUNCTIONS DEFINED
C FOR KODE=2 HAVE A CUT ALONG THE NEGATIVE REAL AXIS.
C DEFINTIONS AND NOTATION ARE FOUND IN THE NBS HANDBOOK OF
C MATHEMATICAL FUNCTIONS (REF. 1).
C
C INPUT ZR,ZI ARE DOUBLE PRECISION
C ZR,ZI - Z=CMPLX(ZR,ZI)
C ID - ORDER OF DERIVATIVE, ID=0 OR ID=1
C KODE - A PARAMETER TO INDICATE THE SCALING OPTION
C KODE= 1 returnS
C AI=AI(Z) ON ID=0 OR
C AI=DAI(Z)/DZ ON ID=1
C = 2 returnS
C AI=CEXP(ZTA)*AI(Z) ON ID=0 OR
C AI=CEXP(ZTA)*DAI(Z)/DZ ON ID=1 WHERE
C ZTA=(2/3)*Z*CSQRT(Z)
C
C OUTPUT AIR,AII ARE DOUBLE PRECISION
C AIR,AII- COMPLEX ANSWER DEPENDING ON THE CHOICES FOR ID AND
C KODE
C NZ - UNDERFLOW INDICATOR
C NZ= 0 , NORMAL return
C NZ= 1 , AI=CMPLX(0.0E0,0.0E0) DUE TO UNDERFLOW IN
C -PI/3<ARG(Z)<PI/3 ON KODE=1
C IERR - ERROR FLAG
C IERR=0, NORMAL return - COMPUTATION COMPLETED
C IERR=1, INPUT ERROR - NO COMPUTATION
C IERR=2, OVERFLOW - NO COMPUTATION, REAL(ZTA)
C TOO LARGE ON KODE=1
C IERR=3, CABS(Z) LARGE - COMPUTATION COMPLETED
C LOSSES OF SIGNIFCANCE BY ARGUMENT REDUCTION
C PRODUCE LESS THAN HALF OF MACHINE ACCURACY
C IERR=4, CABS(Z) TOO LARGE - NO COMPUTATION
C COMPLETE LOSS OF ACCURACY BY ARGUMENT
C REDUCTION
C IERR=5, ERROR - NO COMPUTATION,
C ALGORITHM TERMINATION CONDITION NOT MET
C
C***LONG DESCRIPTION
C
C AI AND DAI ARE COMPUTED FOR CABS(Z)>1.0 FROM THE K BESSEL
C FUNCTIONS BY
C
C AI(Z)=C*SQRT(Z)*K(1/3,ZTA) , DAI(Z)=-C*Z*K(2/3,ZTA)
C C=1.0/(PI*SQRT(3.0))
C ZTA=(2/3)*Z**(3/2)
C
C WITH THE POWER SERIES FOR CABS(Z)<=1.0.
C
C IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE-
C MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z IS LARGE, LOSSES
C OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR. CONSEQUENTLY, IF
C THE MAGNITUDE OF ZETA=(2/3)*Z**1.5 EXCEEDS U1=SQRT(0.5/UR),
C THEN LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR
C FLAG IERR=3 IS TRIGGERED WHERE UR=math.Max(dmach[4),1.0D-18) IS
C DOUBLE PRECISION UNIT ROUNDOFF LIMITED TO 18 DIGITS PRECISION.
C ALSO, if THE MAGNITUDE OF ZETA IS LARGER THAN U2=0.5/UR, THEN
C ALL SIGNIFICANCE IS LOST AND IERR=4. IN ORDER TO USE THE INT
C FUNCTION, ZETA MUST BE FURTHER RESTRICTED NOT TO EXCEED THE
C LARGEST INTEGER, U3=I1MACH(9). THUS, THE MAGNITUDE OF ZETA
C MUST BE RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2,
C AND U3 ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE
C PRECISION ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE
C PRECISION ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMIT-
C ING IN THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT THE MAG-
C NITUDE OF Z CANNOT EXCEED 3.1E+4 IN SINGLE AND 2.1E+6 IN
C DOUBLE PRECISION ARITHMETIC. THIS ALSO MEANS THAT ONE CAN
C EXPECT TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES,
C NO DIGITS IN SINGLE PRECISION AND ONLY 7 DIGITS IN DOUBLE
C PRECISION ARITHMETIC. SIMILAR CONSIDERATIONS HOLD FOR OTHER
C MACHINES.
C
C THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX
C BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT
C ROUNDOFF,1.0E-18) IS THE NOMINAL PRECISION AND 10**S REPRE-
C SENTS THE INCREASE IN ERROR DUE TO ARGUMENT REDUCTION IN THE
C ELEMENTARY FUNCTIONS. HERE, S=MAX(1,ABS(LOG10(CABS(Z))),
C ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF
C CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY
C HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN
C ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE OTHER BY
C SEVERAL ORDERS OF MAGNITUDE. if ONE COMPONENT IS 10**K LARGER
C THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K,
C 0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS
C THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER
C COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY
C BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER
C COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE
C MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES,
C THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P,
C OR -PI/2+P.
C
C***REFERENCES HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ
C AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF
C COMMERCE, 1955.
C
C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983
C
C A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85-
C 1018, MAY, 1985
C
C A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS.
C MATH. SOFTWARE, 1986
*/
var AI, CONE, CSQ, CY, S1, S2, TRM1, TRM2, Z, ZTA, Z3 complex128
var AA, AD, AK, ALIM, ATRM, AZ, AZ3, BK,
CC, CK, COEF, CONEI, CONER, CSQI, CSQR, C1, C2, DIG,
DK, D1, D2, ELIM, FID, FNU, PTR, RL, R1M5, SFAC, STI, STR,
S1I, S1R, S2I, S2R, TOL, TRM1I, TRM1R, TRM2I, TRM2R, TTH, ZEROI,
ZEROR, ZTAI, ZTAR, Z3I, Z3R, ALAZ, BB float64
var IERR, IFLAG, K, K1, K2, MR, NN int
var tmp complex128
// Extra element for padding.
CYR := []float64{math.NaN(), 0}
CYI := []float64{math.NaN(), 0}
_ = AI
_ = CONE
_ = CSQ
_ = CY
_ = S1
_ = S2
_ = TRM1
_ = TRM2
_ = Z
_ = ZTA
_ = Z3
TTH = 6.66666666666666667e-01
C1 = 3.55028053887817240e-01
C2 = 2.58819403792806799e-01
COEF = 1.83776298473930683e-01
ZEROR = 0
ZEROI = 0
CONER = 1
CONEI = 0
NZ = 0
if ID < 0 || ID > 1 {
IERR = 1
}
if KODE < 1 || KODE > 2 {
IERR = 1
}
if IERR != 0 {
return
}
AZ = cmplx.Abs(complex(ZR, ZI))
TOL = math.Max(dmach[4], 1.0e-18)
FID = float64(ID)
if AZ > 1.0e0 {
goto Seventy
}
// POWER SERIES FOR CABS(Z)<=1.
S1R = CONER
S1I = CONEI
S2R = CONER
S2I = CONEI
if AZ < TOL {
goto OneSeventy
}
AA = AZ * AZ
if AA < TOL/AZ {
goto Forty
}
TRM1R = CONER
TRM1I = CONEI
TRM2R = CONER
TRM2I = CONEI
ATRM = 1.0e0
STR = ZR*ZR - ZI*ZI
STI = ZR*ZI + ZI*ZR
Z3R = STR*ZR - STI*ZI
Z3I = STR*ZI + STI*ZR
AZ3 = AZ * AA
AK = 2.0e0 + FID
BK = 3.0e0 - FID - FID
CK = 4.0e0 - FID
DK = 3.0e0 + FID + FID
D1 = AK * DK
D2 = BK * CK
AD = math.Min(D1, D2)
AK = 24.0e0 + 9.0e0*FID
BK = 30.0e0 - 9.0e0*FID
for K = 1; K <= 25; K++ {
STR = (TRM1R*Z3R - TRM1I*Z3I) / D1
TRM1I = (TRM1R*Z3I + TRM1I*Z3R) / D1
TRM1R = STR
S1R = S1R + TRM1R
S1I = S1I + TRM1I
STR = (TRM2R*Z3R - TRM2I*Z3I) / D2
TRM2I = (TRM2R*Z3I + TRM2I*Z3R) / D2
TRM2R = STR
S2R = S2R + TRM2R
S2I = S2I + TRM2I
ATRM = ATRM * AZ3 / AD
D1 = D1 + AK
D2 = D2 + BK
AD = math.Min(D1, D2)
if ATRM < TOL*AD {
goto Forty
}
AK = AK + 18.0e0
BK = BK + 18.0e0
}
Forty:
if ID == 1 {
goto Fifty
}
AIR = S1R*C1 - C2*(ZR*S2R-ZI*S2I)
AII = S1I*C1 - C2*(ZR*S2I+ZI*S2R)
if KODE == 1 {
return
}
tmp = cmplx.Sqrt(complex(ZR, ZI))
STR = real(tmp)
STI = imag(tmp)
ZTAR = TTH * (ZR*STR - ZI*STI)
ZTAI = TTH * (ZR*STI + ZI*STR)
tmp = cmplx.Exp(complex(ZTAR, ZTAI))
STR = real(tmp)
STI = imag(tmp)
PTR = AIR*STR - AII*STI
AII = AIR*STI + AII*STR
AIR = PTR
return
Fifty:
AIR = -S2R * C2
AII = -S2I * C2
if AZ <= TOL {
goto Sixty
}
STR = ZR*S1R - ZI*S1I
STI = ZR*S1I + ZI*S1R
CC = C1 / (1.0e0 + FID)
AIR = AIR + CC*(STR*ZR-STI*ZI)
AII = AII + CC*(STR*ZI+STI*ZR)
Sixty:
if KODE == 1 {
return
}
tmp = cmplx.Sqrt(complex(ZR, ZI))
STR = real(tmp)
STI = imag(tmp)
ZTAR = TTH * (ZR*STR - ZI*STI)
ZTAI = TTH * (ZR*STI + ZI*STR)
tmp = cmplx.Exp(complex(ZTAR, ZTAI))
STR = real(tmp)
STI = imag(tmp)
PTR = STR*AIR - STI*AII
AII = STR*AII + STI*AIR
AIR = PTR
return
// CASE FOR CABS(Z)>1.0.
Seventy:
FNU = (1.0e0 + FID) / 3.0e0
/*
SET PARAMETERS RELATED TO MACHINE CONSTANTS.
TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0D-18.
ELIM IS THE APPROXIMATE EXPONENTIAL OVER-&&UNDERFLOW LIMIT.
EXP(-ELIM)<EXP(-ALIM)=EXP(-ELIM)/TOL AND
EXP(ELIM)>EXP(ALIM)=EXP(ELIM)*TOL ARE INTERVALS NEAR
UNDERFLOW&&OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE.
RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LA>=Z.
DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10**(-DIG).
*/
K1 = imach[15]
K2 = imach[16]
R1M5 = dmach[5]
K = min(abs(K1), abs(K2))
ELIM = 2.303e0 * (float64(K)*R1M5 - 3.0e0)
K1 = imach[14] - 1
AA = R1M5 * float64(K1)
DIG = math.Min(AA, 18.0e0)
AA = AA * 2.303e0
ALIM = ELIM + math.Max(-AA, -41.45e0)
RL = 1.2e0*DIG + 3.0e0
ALAZ = math.Log(AZ)
// TEST FOR PROPER RANGE.
AA = 0.5e0 / TOL
BB = float64(float32(imach[9])) * 0.5e0
AA = math.Min(AA, BB)
AA = math.Pow(AA, TTH)
if AZ > AA {
goto TwoSixty
}
AA = math.Sqrt(AA)
if AZ > AA {
IERR = 3
}
tmp = cmplx.Sqrt(complex(ZR, ZI))
CSQR = real(tmp)
CSQI = imag(tmp)
ZTAR = TTH * (ZR*CSQR - ZI*CSQI)
ZTAI = TTH * (ZR*CSQI + ZI*CSQR)
// RE(ZTA)<=0 WHEN RE(Z)<0, ESPECIALLY WHEN IM(Z) IS SMALL.
IFLAG = 0
SFAC = 1.0e0
AK = ZTAI
if ZR >= 0.0e0 {
goto Eighty
}
BK = ZTAR
CK = -math.Abs(BK)
ZTAR = CK
ZTAI = AK
Eighty:
if ZI != 0.0e0 {
goto Ninety
}
if ZR > 0.0e0 {
goto Ninety
}
ZTAR = 0.0e0
ZTAI = AK
Ninety:
AA = ZTAR
if AA >= 0.0e0 && ZR > 0.0e0 {
goto OneTen
}
if KODE == 2 {
goto OneHundred
}
// OVERFLOW TEST.
if AA > (-ALIM) {
goto OneHundred
}
AA = -AA + 0.25e0*ALAZ
IFLAG = 1
SFAC = TOL
if AA > ELIM {
goto TwoSeventy
}
OneHundred:
// CBKNU AND CACON return EXP(ZTA)*K(FNU,ZTA) ON KODE=2.
MR = 1
if ZI < 0.0e0 {
MR = -1
}
ZTAR, ZTAI, FNU, KODE, MR, _, CYR, CYI, NN, RL, TOL, ELIM, ALIM = Zacai(ZTAR, ZTAI, FNU, KODE, MR, 1, CYR, CYI, NN, RL, TOL, ELIM, ALIM)
if NN < 0 {
goto TwoEighty
}
NZ = NZ + NN
goto OneThirty
OneTen:
if KODE == 2 {
goto OneTwenty
}
// UNDERFLOW TEST.
if AA < ALIM {
goto OneTwenty
}
AA = -AA - 0.25e0*ALAZ
IFLAG = 2
SFAC = 1.0e0 / TOL
if AA < (-ELIM) {
goto TwoTen
}
OneTwenty:
ZTAR, ZTAI, FNU, KODE, _, CYR, CYI, NZ, TOL, ELIM, ALIM = Zbknu(ZTAR, ZTAI, FNU, KODE, 1, CYR, CYI, NZ, TOL, ELIM, ALIM)
OneThirty:
S1R = CYR[1] * COEF
S1I = CYI[1] * COEF
if IFLAG != 0 {
goto OneFifty
}
if ID == 1 {
goto OneFourty
}
AIR = CSQR*S1R - CSQI*S1I
AII = CSQR*S1I + CSQI*S1R
return
OneFourty:
AIR = -(ZR*S1R - ZI*S1I)
AII = -(ZR*S1I + ZI*S1R)
return
OneFifty:
S1R = S1R * SFAC
S1I = S1I * SFAC
if ID == 1 {
goto OneSixty
}
STR = S1R*CSQR - S1I*CSQI
S1I = S1R*CSQI + S1I*CSQR
S1R = STR
AIR = S1R / SFAC
AII = S1I / SFAC
return
OneSixty:
STR = -(S1R*ZR - S1I*ZI)
S1I = -(S1R*ZI + S1I*ZR)
S1R = STR
AIR = S1R / SFAC
AII = S1I / SFAC
return
OneSeventy:
AA = 1.0e+3 * dmach[1]
S1R = ZEROR
S1I = ZEROI
if ID == 1 {
goto OneNinety
}
if AZ <= AA {
goto OneEighty
}
S1R = C2 * ZR
S1I = C2 * ZI
OneEighty:
AIR = C1 - S1R
AII = -S1I
return
OneNinety:
AIR = -C2
AII = 0.0e0
AA = math.Sqrt(AA)
if AZ <= AA {
goto TwoHundred
}
S1R = 0.5e0 * (ZR*ZR - ZI*ZI)
S1I = ZR * ZI
TwoHundred:
AIR = AIR + C1*S1R
AII = AII + C1*S1I
return
TwoTen:
NZ = 1
AIR = ZEROR
AII = ZEROI
return
TwoSeventy:
NZ = 0
IERR = 2
return
TwoEighty:
if NN == (-1) {
goto TwoSeventy
}
NZ = 0
IERR = 5
return
TwoSixty:
IERR = 4
NZ = 0
return
}
// sbknu computes the k bessel function in the right half z plane.
func Zbknu(ZR, ZI, FNU float64, KODE, N int, YR, YI []float64, NZ int, TOL, ELIM, ALIM float64) (ZRout, ZIout, FNUout float64, KODEout, Nout int, YRout, YIout []float64, NZout int, TOLout, ELIMout, ALIMout float64) {
/* Old dimension comment.
DIMENSION YR(N), YI(N), CC(8), CSSR(3), CSRR(3), BRY(3), CYR(2),
* CYI(2)
*/
// TODO(btracey): Find which of these are inputs/outputs/both and clean up
// the function call.
// YR and YI have length n (but n+1 with better indexing)
var AA, AK, ASCLE, A1, A2, BB, BK, CAZ,
CBI, CBR, CCHI, CCHR, CKI, CKR, COEFI, COEFR, CONEI, CONER,
CRSCR, CSCLR, CSHI, CSHR, CSI, CSR, CTWOR,
CZEROI, CZEROR, CZI, CZR, DNU, DNU2, DPI, ETEST, FC, FHS,
FI, FK, FKS, FMUI, FMUR, FPI, FR, G1, G2, HPI, PI, PR, PTI,
PTR, P1I, P1R, P2I, P2M, P2R, QI, QR, RAK, RCAZ, RTHPI, RZI,
RZR, R1, S, SMUI, SMUR, SPI, STI, STR, S1I, S1R, S2I, S2R, TM,
TTH, T1, T2, ELM, CELMR, ZDR, ZDI, AS, ALAS, HELIM float64
var I, IFLAG, INU, K, KFLAG, KK, KMAX, KODED, IDUM, J, IC, INUB, NW int
var sinh, cosh complex128
//var sin, cos float64
var tmp, p complex128
var CSSR, CSRR, BRY [4]float64
var CYR, CYI [3]float64
KMAX = 30
CZEROR = 0
CZEROI = 0
CONER = 1
CONEI = 0
CTWOR = 2
R1 = 2
DPI = 3.14159265358979324e0
RTHPI = 1.25331413731550025e0
SPI = 1.90985931710274403e0
HPI = 1.57079632679489662e0
FPI = 1.89769999331517738e0
TTH = 6.66666666666666666e-01
CC := [9]float64{math.NaN(), 5.77215664901532861e-01, -4.20026350340952355e-02,
-4.21977345555443367e-02, 7.21894324666309954e-03,
-2.15241674114950973e-04, -2.01348547807882387e-05,
1.13302723198169588e-06, 6.11609510448141582e-09}
CAZ = cmplx.Abs(complex(ZR, ZI))
CSCLR = 1.0e0 / TOL
CRSCR = TOL
CSSR[1] = CSCLR
CSSR[2] = 1.0e0
CSSR[3] = CRSCR
CSRR[1] = CRSCR
CSRR[2] = 1.0e0
CSRR[3] = CSCLR
BRY[1] = 1.0e+3 * dmach[1] / TOL
BRY[2] = 1.0e0 / BRY[1]
BRY[3] = dmach[2]
NZ = 0
IFLAG = 0
KODED = KODE
RCAZ = 1.0e0 / CAZ
STR = ZR * RCAZ
STI = -ZI * RCAZ
RZR = (STR + STR) * RCAZ
RZI = (STI + STI) * RCAZ
INU = int(float32(FNU + 0.5))
DNU = FNU - float64(INU)
if math.Abs(DNU) == 0.5e0 {
goto OneTen
}
DNU2 = 0.0e0
if math.Abs(DNU) > TOL {
DNU2 = DNU * DNU
}
if CAZ > R1 {
goto OneTen
}
// SERIES FOR CABS(Z)<=R1.
FC = 1.0e0
tmp = cmplx.Log(complex(RZR, RZI))
SMUR = real(tmp)
SMUI = imag(tmp)
FMUR = SMUR * DNU
FMUI = SMUI * DNU
tmp = complex(FMUR, FMUI)
sinh = cmplx.Sinh(tmp)
cosh = cmplx.Cosh(tmp)
CSHR = real(sinh)
CSHI = imag(sinh)
CCHR = real(cosh)
CCHI = imag(cosh)
if DNU == 0.0e0 {
goto Ten
}
FC = DNU * DPI
FC = FC / math.Sin(FC)
SMUR = CSHR / DNU
SMUI = CSHI / DNU
Ten:
A2 = 1.0e0 + DNU
// GAM(1-Z)*GAM(1+Z)=PI*Z/SIN(PI*Z), T1=1/GAM(1-DNU), T2=1/GAM(1+DNU).
T2 = math.Exp(-dgamln(A2, IDUM))
T1 = 1.0e0 / (T2 * FC)
if math.Abs(DNU) > 0.1e0 {
goto Forty
}
// SERIES FOR F0 TO RESOLVE INDETERMINACY FOR SMALL ABS(DNU).
AK = 1.0e0
S = CC[1]
for K = 2; K <= 8; K++ {
AK = AK * DNU2
TM = CC[K] * AK
S = S + TM
if math.Abs(TM) < TOL {
goto Thirty
}
}
Thirty:
G1 = -S
goto Fifty
Forty:
G1 = (T1 - T2) / (DNU + DNU)
Fifty:
G2 = (T1 + T2) * 0.5e0
FR = FC * (CCHR*G1 + SMUR*G2)
FI = FC * (CCHI*G1 + SMUI*G2)
tmp = cmplx.Exp(complex(FMUR, FMUI))
STR = real(tmp)
STI = imag(tmp)
PR = 0.5e0 * STR / T2
PI = 0.5e0 * STI / T2
tmp = complex(0.5, 0) / complex(STR, STI)
PTR = real(tmp)
PTI = imag(tmp)
QR = PTR / T1
QI = PTI / T1
S1R = FR
S1I = FI
S2R = PR
S2I = PI
AK = 1.0e0
A1 = 1.0e0
CKR = CONER
CKI = CONEI
BK = 1.0e0 - DNU2
if INU > 0 || N > 1 {
goto Eighty
}
// GENERATE K(FNU,Z), 0.0E0 <= FNU < 0.5E0 AND N=1.
if CAZ < TOL {
goto Seventy
}
tmp = complex(ZR, ZI) * complex(ZR, ZI)
CZR = real(tmp)
CZI = imag(tmp)
CZR = 0.25e0 * CZR
CZI = 0.25e0 * CZI
T1 = 0.25e0 * CAZ * CAZ
Sixty:
FR = (FR*AK + PR + QR) / BK
FI = (FI*AK + PI + QI) / BK
STR = 1.0e0 / (AK - DNU)
PR = PR * STR
PI = PI * STR
STR = 1.0e0 / (AK + DNU)
QR = QR * STR
QI = QI * STR
STR = CKR*CZR - CKI*CZI
RAK = 1.0e0 / AK
CKI = (CKR*CZI + CKI*CZR) * RAK
CKR = STR * RAK
S1R = CKR*FR - CKI*FI + S1R
S1I = CKR*FI + CKI*FR + S1I
A1 = A1 * T1 * RAK
BK = BK + AK + AK + 1.0e0
AK = AK + 1.0e0
if A1 > TOL {
goto Sixty
}
Seventy:
YR[1] = S1R
YI[1] = S1I
if KODED == 1 {
return ZR, ZI, FNU, KODE, N, YR, YI, NZ, TOL, ELIM, ALIM
}
tmp = cmplx.Exp(complex(ZR, ZI))
STR = real(tmp)
STI = imag(tmp)
tmp = complex(S1R, S1I) * complex(STR, STI)
YR[1] = real(tmp)
YI[1] = imag(tmp)
return ZR, ZI, FNU, KODE, N, YR, YI, NZ, TOL, ELIM, ALIM
// GENERATE K(DNU,Z) AND K(DNU+1,Z) FOR FORWARD RECURRENCE.
Eighty:
if CAZ < TOL {
goto OneHundred
}
tmp = complex(ZR, ZI) * complex(ZR, ZI)
CZR = real(tmp)
CZI = imag(tmp)
CZR = 0.25e0 * CZR
CZI = 0.25e0 * CZI
T1 = 0.25e0 * CAZ * CAZ
Ninety:
FR = (FR*AK + PR + QR) / BK
FI = (FI*AK + PI + QI) / BK
STR = 1.0e0 / (AK - DNU)
PR = PR * STR
PI = PI * STR
STR = 1.0e0 / (AK + DNU)
QR = QR * STR
QI = QI * STR
STR = CKR*CZR - CKI*CZI
RAK = 1.0e0 / AK
CKI = (CKR*CZI + CKI*CZR) * RAK
CKR = STR * RAK
S1R = CKR*FR - CKI*FI + S1R
S1I = CKR*FI + CKI*FR + S1I
STR = PR - FR*AK
STI = PI - FI*AK
S2R = CKR*STR - CKI*STI + S2R
S2I = CKR*STI + CKI*STR + S2I
A1 = A1 * T1 * RAK
BK = BK + AK + AK + 1.0e0
AK = AK + 1.0e0
if A1 > TOL {
goto Ninety
}
OneHundred:
KFLAG = 2
A1 = FNU + 1.0e0
AK = A1 * math.Abs(SMUR)
if AK > ALIM {
KFLAG = 3
}
STR = CSSR[KFLAG]
P2R = S2R * STR
P2I = S2I * STR
tmp = complex(P2R, P2I) * complex(RZR, RZI)
S2R = real(tmp)
S2I = imag(tmp)
S1R = S1R * STR
S1I = S1I * STR
if KODED == 1 {
goto TwoTen
}
tmp = cmplx.Exp(complex(ZR, ZI))
FR = real(tmp)
FI = imag(tmp)
tmp = complex(S1R, S1I) * complex(FR, FI)
S1R = real(tmp)
S1I = imag(tmp)
tmp = complex(S2R, S2I) * complex(FR, FI)
S2R = real(tmp)
S2I = imag(tmp)
goto TwoTen
// IFLAG=0 MEANS NO UNDERFLOW OCCURRED
// IFLAG=1 MEANS AN UNDERFLOW OCCURRED- COMPUTATION PROCEEDS WITH
// KODED=2 AND A TEST FOR ON SCALE VALUES IS MADE DURING FORWARD RECURSION
OneTen:
tmp = cmplx.Sqrt(complex(ZR, ZI))
STR = real(tmp)
STI = imag(tmp)
tmp = complex(RTHPI, CZEROI) / complex(STR, STI)
COEFR = real(tmp)
COEFI = imag(tmp)
KFLAG = 2
if KODED == 2 {
goto OneTwenty
}
if ZR > ALIM {
goto TwoNinety
}
STR = math.Exp(-ZR) * CSSR[KFLAG]
//sin, cos = math.Sincos(ZI)
STI = -STR * math.Sin(ZI)
STR = STR * math.Cos(ZI)
tmp = complex(COEFR, COEFI) * complex(STR, STI)
COEFR = real(tmp)
COEFI = imag(tmp)
OneTwenty:
if math.Abs(DNU) == 0.5e0 {
goto ThreeHundred
}
// MILLER ALGORITHM FOR CABS(Z)>R1.
AK = math.Cos(DPI * DNU)
AK = math.Abs(AK)
if AK == CZEROR {
goto ThreeHundred
}
FHS = math.Abs(0.25e0 - DNU2)
if FHS == CZEROR {
goto ThreeHundred
}
// COMPUTE R2=F(E). if CABS(Z)>=R2, USE FORWARD RECURRENCE TO
// DETERMINE THE BACKWARD INDEX K. R2=F(E) IS A STRAIGHT LINE ON
// 12<=E<=60. E IS COMPUTED FROM 2**(-E)=B**(1-I1MACH(14))=
// TOL WHERE B IS THE BASE OF THE ARITHMETIC.
T1 = float64(imach[14] - 1)
T1 = T1 * dmach[5] * 3.321928094e0
T1 = math.Max(T1, 12.0e0)
T1 = math.Min(T1, 60.0e0)
T2 = TTH*T1 - 6.0e0
if ZR != 0.0e0 {
goto OneThirty
}
T1 = HPI
goto OneFourty
OneThirty:
T1 = math.Atan(ZI / ZR)
T1 = math.Abs(T1)
OneFourty:
if T2 > CAZ {
goto OneSeventy
}
// FORWARD RECURRENCE LOOP WHEN CABS(Z)>=R2.
ETEST = AK / (DPI * CAZ * TOL)
FK = CONER
if ETEST < CONER {
goto OneEighty
}
FKS = CTWOR
CKR = CAZ + CAZ + CTWOR
P1R = CZEROR
P2R = CONER
for I = 1; I <= KMAX; I++ {
AK = FHS / FKS
CBR = CKR / (FK + CONER)
PTR = P2R
P2R = CBR*P2R - P1R*AK
P1R = PTR
CKR = CKR + CTWOR
FKS = FKS + FK + FK + CTWOR
FHS = FHS + FK + FK
FK = FK + CONER
STR = math.Abs(P2R) * FK
if ETEST < STR {
goto OneSixty
}
}
goto ThreeTen
OneSixty:
FK = FK + SPI*T1*math.Sqrt(T2/CAZ)
FHS = math.Abs(0.25 - DNU2)
goto OneEighty
OneSeventy:
// COMPUTE BACKWARD INDEX K FOR CABS(Z)<R2.
A2 = math.Sqrt(CAZ)
AK = FPI * AK / (TOL * math.Sqrt(A2))
AA = 3.0e0 * T1 / (1.0e0 + CAZ)
BB = 14.7e0 * T1 / (28.0e0 + CAZ)
AK = (math.Log(AK) + CAZ*math.Cos(AA)/(1.0e0+0.008e0*CAZ)) / math.Cos(BB)
FK = 0.12125e0*AK*AK/CAZ + 1.5e0
OneEighty:
// BACKWARD RECURRENCE LOOP FOR MILLER ALGORITHM.
K = int(float32(FK))
FK = float64(K)
FKS = FK * FK
P1R = CZEROR
P1I = CZEROI
P2R = TOL
P2I = CZEROI
CSR = P2R
CSI = P2I
for I = 1; I <= K; I++ {
A1 = FKS - FK
AK = (FKS + FK) / (A1 + FHS)
RAK = 2.0e0 / (FK + CONER)
CBR = (FK + ZR) * RAK
CBI = ZI * RAK
PTR = P2R
PTI = P2I
P2R = (PTR*CBR - PTI*CBI - P1R) * AK
P2I = (PTI*CBR + PTR*CBI - P1I) * AK
P1R = PTR
P1I = PTI
CSR = CSR + P2R
CSI = CSI + P2I
FKS = A1 - FK + CONER
FK = FK - CONER
}
// COMPUTE (P2/CS)=(P2/CABS(CS))*(CONJG(CS)/CABS(CS)) FOR BETTER SCALING.
TM = cmplx.Abs(complex(CSR, CSI))
PTR = 1.0e0 / TM
S1R = P2R * PTR
S1I = P2I * PTR
CSR = CSR * PTR
CSI = -CSI * PTR
tmp = complex(COEFR, COEFI) * complex(S1R, S1I)
STR = real(tmp)
STI = imag(tmp)
tmp = complex(STR, STI) * complex(CSR, CSI)
S1R = real(tmp)
S1I = imag(tmp)
if INU > 0 || N > 1 {
goto TwoHundred
}
ZDR = ZR
ZDI = ZI
if IFLAG == 1 {
goto TwoSeventy
}
goto TwoFourty
TwoHundred:
// COMPUTE P1/P2=(P1/CABS(P2)*CONJG(P2)/CABS(P2) FOR SCALING.
TM = cmplx.Abs(complex(P2R, P2I))
PTR = 1.0e0 / TM
P1R = P1R * PTR
P1I = P1I * PTR
P2R = P2R * PTR
P2I = -P2I * PTR
tmp = complex(P1R, P1I) * complex(P2R, P2I)
PTR = real(tmp)
PTI = imag(tmp)
STR = DNU + 0.5e0 - PTR
STI = -PTI
tmp = complex(STR, STI) / complex(ZR, ZI)
STR = real(tmp)
STI = imag(tmp)
STR = STR + 1.0e0
tmp = complex(STR, STI) * complex(S1R, S1I)
S2R = real(tmp)
S2I = imag(tmp)
// FORWARD RECURSION ON THE THREE TERM RECURSION WITH RELATION WITH
// SCALING NEAR EXPONENT EXTREMES ON KFLAG=1 OR KFLAG=3
TwoTen:
STR = DNU + 1.0e0
CKR = STR * RZR
CKI = STR * RZI
if N == 1 {
INU = INU - 1
}
if INU > 0 {
goto TwoTwenty
}
if N > 1 {
goto TwoFifteen
}
S1R = S2R
S1I = S2I
TwoFifteen:
ZDR = ZR
ZDI = ZI
if IFLAG == 1 {
goto TwoSeventy
}
goto TwoFourty
TwoTwenty:
INUB = 1
if IFLAG == 1 {
goto TwoSixtyOne
}
TwoTwentyFive:
P1R = CSRR[KFLAG]
ASCLE = BRY[KFLAG]