New version 16.2
Main change:
- Branch k in stack level 2
- Argument x in stack level 1
(previously it was a list)
Code:
\<< "Lambert (ver 16.2)
2 inputs
k & x with branch k
=0, \1771, \1772, \1773, \1774
Or 1 input x possible
for k = 0 by default
when stack level 2
.\=/0, \1771, \1772, \1773, \1774
.or empty
.With x=W(x)*e^W(x),
W(x) unknown
.x may be a complex #
(a,b), even ext.big
complex, i.e a or b
=\1779.99999999999E499
Output(s)
-1 output Wk(x):
. w
. or '\oo \177 in\pi/2'
for ext.big comp.
(\=/Wolfram\-> \oo)
. or '-\oo \177 in\pi'
for x\->-0 or x\->0+
(\=/Wolfram\-> -\oo)
-Or 2 outputs,
if -1/e \<=x \<=0
& 'k=0 or k=-1':
. W0 (x): w1
. W-1(x): w2
Formulae initial guess
.Albert Chan
.WIKIP: W-1 & -1/e\<=x\<=0
For branch k of Wk(x):
https://www.johndcook.com/blog/2021/11/15/lambert-w-branch-number/
" DROP
IFERR OVER
THEN 0
ELSE \-> k
\<< k TYPE 0 == k TYPE 28 == OR
IF
THEN k FP 0 == { SWAP } 0 IFTE
ELSE 0
END
\>>
END SWAP DUP \->NUM 0 RCLF 0 0 0 0 .367879441171 4.42321595524E-13 0 0 0 0 0 \-> k x0 x f fg xR xi t x0\175 R R2 H X Y Z h
\<< x IM DUP 'xi' STO 0 ==
IF
THEN x RE 'x' STO
END x 0 \<= x R NEG \>= AND k -1 == AND
IF
THEN 0 'k' STO
END -103 SF x RE DUP 'xR' STO ABS \->NUM \oo \->NUM == xi ABS \->NUM \oo \->NUM == OR
IF
THEN "W" k R\->I + "((" +
CASE xi \oo \->NUM NEG ==
THEN xR ABS \oo \->NUM ==
IF
THEN xR 0 > "\oo" "-\oo" IFTE
ELSE xR DUP FP 0 == { R\->I } IFT
END + ",-\oo))" + 4 k * 1 - DUP 0 > "+i\183" "-i\183" IFTE SWAP ABS R\->I + "\pi/2" +
END xi \oo \->NUM ==
THEN xR ABS \oo \->NUM ==
IF
THEN xR 0 > "\oo" "-\oo" IFTE
ELSE xR DUP FP 0 == { R\->I } IFT
END + ",-\oo))" + 4 k * 1 + DUP 0 > "+i\183" "-i\183" IFTE SWAP ABS R\->I + "\pi/2" +
END xR \oo \->NUM NEG == xi 0 < AND
THEN "-\oo," + xi DUP FP 0 == { R\->I } IFT + "))" + 2 k * 1 - DUP 0 > "+i\183" "-i\183" IFTE SWAP ABS R\->I + "\pi" +
END xR \oo \->NUM NEG ==
THEN "-\oo," + xi DUP FP 0 == { R\->I } IFT + "))" + 2 k * 1 + DUP 0 > "+i\183" "-i\183" IFTE SWAP ABS R\->I + "\pi" +
END "\oo," + xi DUP FP 0 == { R\->I } IFT + "))" + 2 k * DUP 0 > "+i\183" "-i\183" IFTE SWAP ABS R\->I + "\pi" +
END "i\1830" "" SREPL 1 ==
IF
THEN DROP '\oo'
ELSE "\oo " SWAP + "\pi" "\183\pi" SREPL DROP "\1831\183" "\183" SREPL DROP " " " <U>" SREPL DROP "<U>" + "<U>" 19 CHR 3 CHR OVER + + SREPL DROP
END SWAP \->TAG
ELSE -105 SF RAD '2*k*\pi' x ARG + i * \->NUM 't' STO
CASE xi 0 == xR R NEG < AND xR -.38 > AND
THEN '\v/(2*R*(x+R+R2))*(k+k+1)' \->NUM 'Y' STO 'R+Y*\v/(1+Y/(3*R))' \->NUM 'Y' STO
DO Y 'Y+x' '(Y-R-R2)/R' \->NUM DUP TYPE 1 ==
IF
THEN 1 + LN
ELSE LNP1
END / - \->NUM 'H' STO 'Y-H' \->NUM 'Y' STO Y 'Y+H*.0001' \->NUM ==
UNTIL
END x Y / "W" k R\->I + "(" + x0 TYPE 9 ==
IF
THEN x0 + "=" +
END x DUP FP 0 == { R\->I } IFT + ")" + \->TAG fg STOF KILL
END k 1 == xi 0 < AND k -1 == xi 0 \>= AND OR
THEN x NEG LN x ABS .5 <
IF
THEN 0
ELSE t 2 /
END +
END k 0 ==
THEN xi 0 == xR 0 > AND
IF
THEN x 2 * LNP1
ELSE x 2 * 1 + DUP 0 ==
IF
THEN DROP 1.6 DUP R\->C
ELSE LN
END
END 2 /
END t
END 1 \->ARRY xi 0 == xR 0 \<= AND xR e INV NEG \>= AND k 0 == k -1 == OR AND
IF
THEN 'W*e^W' x - DUP 'f' STO 'W' ROT 1 GET ROOT
CASE x e INV NEG \->NUM == x0 TYPE 9 == AND
THEN DROP -1
END x -.3615 \<= x e NEG INV > AND
THEN DROP '(x+R+R2)/R' \->NUM 'H' STO '\v/(2*H)*(2*k+1)' \->NUM 'X' STO 'X*\v/(1+X/3)' \->NUM 'X' STO 'R+R*X' \->NUM 'Y' STO
DO X DUP TYPE 1 ==
IF
THEN 1 + LN
ELSE LNP1
END 'Z' STO 'X-(X-Z+H)/Z' \->NUM 'Z' STO X Z - 'X' STO X 'X+Z*.000001' \->NUM ==
UNTIL
END X DUP TYPE 1 ==
IF
THEN 1 + LN
ELSE LNP1
END 1 -
END
END DUP FP 0 == { R\->I } IFT "W0(" x0 TYPE 9 ==
IF
THEN x0 + x R NEG == "~" "=" IFTE +
END x DUP FP 0 == { R\->I } IFT + ")" + \->TAG
CASE x 0 < x e INV NEG \>= AND
THEN x 0 < x ABS 1.E-495 < AND
IF
THEN x NEG LN 1 3
START 'x0\175' STO '(1-(LN(-x0\175)-LN(-x)))/(1+1/x0\175)' \->NUM
NEXT
ELSE f 'W'
IF x 4 INV NEG \<=
THEN '-1-\v/(2*(1+e*x))'
ELSE 'LN(-x)-LN(-LN(-x))'
END \->NUM ROOT x e NEG INV \->NUM == x0 TYPE 9 == AND
IF
THEN DROP -1
END
END
END x 0 ==
THEN -105 CF "-\oo (-\oo<U>-i\pi<U>)" "<U>" 19 CHR 3 CHR OVER + + SREPL DROP
END -1
END x -.3636 \<= x e NEG INV > AND
IF
THEN -1 'k' STO DROP '(x+R+R2)/R' \->NUM 'H' STO '\v/(2*H)*(2*k+1)' \->NUM 'X' STO 'X*\v/(1+X/3)' \->NUM 'X' STO 'R+R*X' \->NUM 'Y' STO
DO X DUP TYPE 1 ==
IF
THEN 1 + LN
ELSE LNP1
END 'Z' STO 'X-(X-Z+H)/Z' \->NUM 'Z' STO X Z - 'X' STO X 'X+Z*.000001' \->NUM ==
UNTIL
END X DUP TYPE 1 ==
IF
THEN 1 + LN
ELSE LNP1
END 1 -
END "W-1(" x0 TYPE 9 ==
IF
THEN x0 + x R NEG == "~" "=" IFTE +
END x DUP FP 0 == { R\->I } IFT x 0 ==
IF
THEN DROP "-0 (0+)"
END + ")" + \->TAG 'W' PURGE
ELSE x 0 \=/
IF
THEN 'k*2*\pi*i+LN(x)' 'W+LN(W)' - 1 \->ARRY [ 'W' ] ROT MSLV 1 GET DUP IM 0 == { RE } IFT UNROT DROP2 DUP \-> r
\<< TYPE 1 ==
IF
THEN r C\->R DUP ABS DUP .0001 + \pi 2 / \>= SWAP \pi 2 / / \->NUM DUP 0 RND - ABS .0001 \<= AND ROT ABS .00000000001 < AND
IF
THEN 0 SWAP R\->C
ELSE DROP r
END
ELSE r
END
\>>
ELSE DROP "-\oo<U>" 2 k * k SIGN - DUP 0 >
IF
THEN "+i"
ELSE "-i"
END \-> kk i
\<< i + kk ABS k 0 > 1 -1 IFTE + R\->I + "\pi<U>" + " (-\oo<U>" + i + kk ABS + "\pi<U>)" + "i1\pi" "i\pi" SREPL DROP "<U>" 19 CHR 3 CHR OVER + + SREPL DROP
\>>
END "W" k R\->I + "(" + x0 TYPE 9 ==
IF
THEN x0 + "=" +
END x DUP 0 ==
IF
THEN DROP "\->-0 (\->0+)"
ELSE DUP FP 0 == { R\->I } IFT
END + ")" + \->TAG
END
END fg STOF
IFERR 0 DOERR
THEN
END
\>>
\>
>
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HP49-50G,VER16.2 Lambert Funktion Wk(x), k=k= 0, ±1, ±2, ±3, ±4..., x real or complex
