Re: Solution of Transcendental Equation: Exp(a.X) + b.X + c

monir <monirg@xxxxxxxxxxxx> writes:

On Nov 27, 3:32 am, ... <....@xxxxxxxxxxxxxx> wrote:
On Mon, 26 Nov 2007 22:26:39 -0800 (PST), monir <mon...@xxxxxxxxxxxx>
wrote:

[snip]

1)"W(X)...if X=[a/b exp(-ac/b)] is in the interval (-1/e <= X < 0)
there are two real
values of W(X), one using the principal branch and the other using
the "-1" branch."
Then, W(-0.1) should have 2 values. The 32-term series expansion with
X=-0.10 gives
W(-0.1) =+0.09127653 ...
Could it be that mine is one value and yours W(-0.1) = -0.1118325592
is the other value??

W(0, -0.1) = W(-0.1) ~ -0.1118325592

W(-1, -0.1) ~ -3.57715206396



2)"W(X)...if X=[a/b exp(-ac/b)] = -1/e or is in the interval (0 <= X <
infty) there is
one value, using the principal branch."
So, W(5.0) should have one real value!! The approximate value (read
from the published
W-X plot) is W(5.0) = ~ 1.35

W(5) ~ 1.32672466524



3)"The Taylor series of the principal branch is as above:
W(X) = SUM(n=1 to n=infty) (-1)^(n-1)*(n)^(n-2)/(n-1)!*X^n
its radius of convergence is 1/e, which is the distance from 0 to
the closest
singularity, the branch point at -1/e."
"... the series diverges for the others W(5.0) and ..."
How would you reconsile 2) and 3) above ??

Thank you and I appreciate your patience.
Monir

FWIW, there is an online calculator that can compute W(x) for both
the 0 and -1 branches.
<http://www.geocities.com/scroussette/calc.html>

I do not know if all digits displayed are accurate for all x >= -1/e.

The W0 button handles the principal real (or 0) branch.
The W1 button handles the non-principal real (or -1) branch.


1) OK. So we've established that the value of Lambert function W(X)
depends on where its argument X lies:
1.i LambertW(X) : one real value if -1/e <= X < infty (in the "0" or
principal branch)
1.ii LambertW(X) : two real values if -1/e <= X < 0 (one in the
principal branch and one in the "-1" branch)
1.iii LambertW(X) : imaginary if -infty <= X < -1/e

2) The link <http://mathworld.wolfram.com/LambertW-Function.html>
states that The Lambert W(X) function is real for X >= -1./e, and it
has the series expansion:
W(X) = SUM(n=1 to n=infty) (-1)^(n-1)*(n)^(n-2)/(n-1)!*X^n

It has that series expansion, which works for |X| < 1/e. That's not
the only way to calculate W(X), though, and obviously can't be the way to
calculate it when |X| > 1/e.

3) For X= -0.1, it follows 1.ii above, and you calculated the two
values of W(-0.1) as:
W(0, -0.1) = W(-0.1) ~ -0.1118325592 (for the principal branch)
W(-1, -0.1) ~ -3.57715206396 (for the -1 branch)
The first value (in the principal branch) can be calculated by the
series expansion 2) above for X= -0.1; W(-0.1) = ~ -0.1118325...
But, how did you calculate the 2nd value in the "-1" branch ??
What series expansion did you use??

What I used was Maple, and I don't know what method Maple used.
But you might look at the following iteration methods from David Cantrell's
article in the thread " Lamberts W-function revealed: a converging sequence
for all 0 < x <infinity!" from August 2002
<http://groups.google.ca/group/sci.math/msg/0607167131e93b88>:

Definition: f(x,y) = (x+y^2*e^y)/((1+y)*e^y)

(1) The sequence x_0 = ln|x|, x_(n+1) = f(x,x_n) converges reasonably well
to the Lambert W function's principal branch, W(0,x), if x >= 1 and to its
other real branch, W(-1,x), if -1/e < x < 0.

(2) The sequence x_0 = |x|, x_(n+1) = f(x,x_n) converges reasonably well
to the Lambert W function's principal branch, W(0,x), if -1/e < x < 1.

For lots more information on the Lambert W function, see
<http://kong.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/>.
In particular, if you're interested in series expansions, try
the paper of Corless, Jeffrey and Knuth, "A Sequence of Series for the
Lambert W Function".

...and What X value did you apply in the series for the 2nd value
W(-1,-0.1)??

4) For X=5.0, it follows 1.i above, and you calculated the W(5.0) as:
W(5.0) ~ 1.32672466524 which agrees well with the value read from the
published W(X) vs X plot
Again, how did you calculate W(5.0)??
What series expansion of the principal branch did you use here ??
and What X value did you apply in the series for W(5.0), realizing
that the series radius of convergence is limited to 1/e from 0 ??

5) Unfortunately, the online W(X) calculator can't be plugged in the
program.

Any thoughts ?? Thank you kindly for your help.
Monir
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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