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{ ******************************************************************
Lambert's function
Translated from Fortran code by Barry et al.
(http://www.netlib.org/toms/743)
****************************************************************** }
unit ulambert;
interface
uses
utypes, umath;
function LambertW(X : Float; UBranch, Offset : Boolean) : Float;
{ ----------------------------------------------------------------------
Lambert's W function: Y = W(X) ==> X = Y * Exp(Y) X >= -1/e
----------------------------------------------------------------------
X = Argument
UBranch = TRUE for computing the upper branch (X >= -1/e, W(X) >= -1)
FALSE for computing the lower branch (-1/e <= X < 0, W(X) <= -1)
Offset = TRUE for computing W(X - 1/e), X >= 0
FALSE for computing W(X)
---------------------------------------------------------------------- }
implementation
{$IFDEF SINGLEREAL}
const
NBITS = 23; { MachEp = 2^(-NBITS) }
X0 = 0.03507693900966790567; { MachEp^(1/6) / 2 }
X1 = -0.30212011943278473033; { - Exp(-1) * (1 - 17 * MachEp^(2/7) }
{$ELSE}
{$IFDEF EXTENDEDREAL}
const
NBITS = 63;
X0 = 0.0003452669830012439084;
X1 = -0.36785558424357094358;
{$ELSE}
const
NBITS = 52;
X0 = 0.001230391650287962075;
X1 = -0.36766871970031223379;
{$ENDIF}
{$ENDIF}
const
EM = -0.36787944117144232160; { - Exp(-1) }
EM9 = -0.0001234098040866795495; { - Exp(-9) }
C13 = 1.0 / 3.0;
C23 = 2.0 * C13;
EM2 = 2.0 / EM;
D12 = - EM2;
AN3 = 8.0 / 3.0;
AN4 = 135.0 / 83.0;
AN5 = 166.0 / 39.0;
AN6 = 3167.0 / 3549.0;
S21 = 2.0 * Sqrt2 - 3.0;
S22 = 4.0 - 3.0 * Sqrt2;
S23 = Sqrt2 - 2.0;
function LambertW(X : Float; UBranch, Offset : Boolean) : Float;
var
I, NITER : Integer;
AN2, DELX, ETA, RETA, T, TEMP, TEMP2, TS, WAPR, XX, ZL, ZN : Float;
begin
SetErrCode(FOk);
if Offset then
begin
DELX := X;
if DELX < 0.0 then
begin
LambertW := DefaultVal(FDomain, 0.0);
Exit;
end;
XX := X + EM;
end
else
begin
if X < EM then
begin
LambertW := DefaultVal(FDomain, 0.0);
Exit;
end;
if X = EM then
begin
LambertW := - 1.0;
Exit;
end;
XX := X;
DELX := XX - EM;
end;
if UBranch then
begin
if Abs(XX) <= X0 then
begin
LambertW := XX / (1.0 + XX / (1.0 + XX / (2.0 + XX / (0.6 + 0.34 * XX))));
Exit;
end;
if XX <= X1 then
begin
RETA := Sqrt(D12 * DELX);
LambertW := RETA / (1.0 + RETA / (3.0 + RETA / (RETA / (AN4 +
RETA / (RETA * AN6 + AN5)) + AN3))) - 1.0;
Exit;
end;
if XX <= 20.0 then
begin
RETA := Sqrt2 * Sqrt(1.0 - XX / EM);
AN2 := 4.612634277343749 * Sqrt(Sqrt(RETA + 1.09556884765625));
WAPR := RETA / (1.0 + RETA / (3.0 + (S21 * AN2 + S22) * RETA / (S23 * (AN2 + RETA)))) - 1.0;
end
else
begin
ZL := Ln(XX);
WAPR := Ln(XX / Ln(XX / Power(ZL, Exp(- 1.124491989777808 / (0.4225028202459761 + ZL)))));
end
end
else
begin
if XX >= 0.0 then
begin
LambertW := DefaultVal(FDomain, 0.0);
Exit;
end;
if XX <= X1 then
begin
RETA := Sqrt(D12 * DELX);
LambertW := RETA / (RETA / (3.0 + RETA / (RETA / (AN4 +
RETA / (RETA * AN6 - AN5)) - AN3)) - 1.0) - 1.0;
Exit;
end;
ZL := Ln(- XX);
if XX <= EM9 then
begin
T := - 1.0 - ZL;
TS := Sqrt(T);
WAPR := ZL - (2.0 * TS) / (SQRT2 + (C13 - T / (2.7E2 + TS * 127.0471381349219)) * TS);
end
else
begin
ETA := 2.0 - EM2 * XX;
WAPR := Ln(XX / Ln(- XX / ((1.0 - 0.5043921323068457 * (ZL + 1.0)) * (Sqrt(ETA) + ETA / 3.0) + 1.0)));
end
end;
if NBITS < 56 then NITER := 1 else NITER := 2;
for I := 1 to NITER do
begin
ZN := Ln(XX / WAPR) - WAPR;
TEMP := 1.0 + WAPR;
TEMP2 := TEMP + C23 * ZN;
TEMP2 := 2.0 * TEMP * TEMP2;
WAPR := WAPR * (1.0 + (ZN / TEMP) * (TEMP2 - ZN) / (TEMP2 - 2.0 * ZN));
end;
LambertW := WAPR;
end;
end.
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