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Description
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Lambert's W function is a mathematical function
with a <a href="https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf">diverse
range of applications in math and science</a>. It is an inverse function
to $x = w \times \mathrm{e}^w$. In other words, for a given $x$, the value
of the Lambert function $W(x)$ is defined as the real number satisfying
$$x = W(x) \times \mathrm{e}^{W(x)}.$$


For example, W(126) is approximately 3.5651 since 126 = 3.5651 × e3.5651.


Use binary search to define a static method <tt>W(double x)</tt> that
evaluates the Lambert function at x. You may assume $x \ge 0$.
To help get the binary search
started, you may use the fact that $0 \le W(x) \le \max(1, \ln(x))$.
Terminate the binary search when
the search endpoints differ by 10-8 or less.


Hint: you will need <a href="http://docs.oracle.com/javase/8/docs/api/java/lang/Math.html#exp-double-">several</a>
<a href="http://docs.oracle.com/javase/8/docs/api/java/lang/Math.html#log-double-">methods</a>
from the <tt>Math</tt> library.

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Choices
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Correct answer
Template / Reference solution

Template / Reference solution

#include <cmath>
#include <iostream>
using namespace std;

// evaluate Lambert's W function at x. Assumes x >= 0.
double W(double x) {
// we know that W(x) >= 0 and W(x) <= max(1, ln(x))
double lo = 0;
\[double hi = max(1.0, log(x));]\

// continue until search endpoints differ by 1E-8 or less
while (\[hi - lo >= 1E-8]\) {
// evaluate midpoint times e^midpoint
\[
double mid = (lo + hi) / 2;
double midval = mid * exp(mid);
]\

// if this value is smaller than x, search above, else search below
\[
if (midval < x)
lo = mid;
else
hi = mid;
]\
}
return lo; // return some point in final range
}

int main() {
double x;
cin >> x;
cout << W(x);
}

Java test suite
See manual

C++ test suite
json list of stdin/args tests
e.g. [{"stdin":"hi", "args":["4", "5"]}, {"stdin":"noargs"}]
to just run once with no input use [{}]

[
{"stdin": "126"},
{"stdin": "2.718281828459045"},
{"stdin": "23423.348"},
{"stdin": "1E25"},
{"stdin": "1E-5"}
]

C++ test suite
See manual

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Property	Value
Description html markup shown to student	<i>Lambert's W function</i> is a mathematical function with a <a href="https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf">diverse range of applications in math and science</a>. It is an inverse function to $x = w \times \mathrm{e}^w$. In other words, for a given $x$, the value of the Lambert function $W(x)$ is defined as the real number satisfying $$x = W(x) \times \mathrm{e}^{W(x)}.$$ <p> For example, <i>W</i>(126) is approximately 3.5651 since 126 = 3.5651 × e<sup>3.5651</sup>. <p> Use binary search to define a static method <tt>W(double x)</tt> that evaluates the Lambert function at <i>x</i>. You may assume $x \ge 0$. To help get the binary search started, you may use the fact that $0 \le W(x) \le \max(1, \ln(x))$. Terminate the binary search when the search endpoints differ by 10<sup>-8</sup> or less. <p> <i>Hint</i>: you will need <a href="http://docs.oracle.com/javase/8/docs/api/java/lang/Math.html#exp-double-">several</a> <a href="http://docs.oracle.com/javase/8/docs/api/java/lang/Math.html#log-double-">methods</a> from the <tt>Math</tt> library.
Epilogue html markup shown when solved	remove
Public permissions	remove
Remarks Comments, history, license, etc.	remove
Engine
Choices json list of string, bool pairs e.g. `[["good", true], ["bad", false]]`
Correct answer
Template / Reference solution
Template / Reference solution	#include <cmath> #include <iostream> using namespace std; // evaluate Lambert's W function at x. Assumes x >= 0. double W(double x) { // we know that W(x) >= 0 and W(x) <= max(1, ln(x)) double lo = 0; \[double hi = max(1.0, log(x));]\ // continue until search endpoints differ by 1E-8 or less while (\[hi - lo >= 1E-8]\) { // evaluate midpoint times e^midpoint \[ double mid = (lo + hi) / 2; double midval = mid * exp(mid); ]\ // if this value is smaller than x, search above, else search below \[ if (midval < x) lo = mid; else hi = mid; ]\ } return lo; // return some point in final range } int main() { double x; cin >> x; cout << W(x); }
Java test suite See manual
C++ test suite json list of stdin/args tests e.g. `[{"stdin":"hi", "args":["4", "5"]}, {"stdin":"noargs"}]` to just run once with no input use `[{}]`	[ {"stdin": "126"}, {"stdin": "2.718281828459045"}, {"stdin": "23423.348"}, {"stdin": "1E25"}, {"stdin": "1E-5"} ]
C++ test suite See manual
Forbidden substrings json list of strings e.g. `["for","while"]`	remove
Solution visibility	remove
Java imports json list of importables e.g. `["java.util.*"]`	remove
Override Java classname else defaults to websheet name	remove
Dependent on other websheets? json list of websheet names in this folder e.g. `["DataStructure"]`	remove
Is example? i.e., just a demo	remove
Add compiler flag(s) json list of flags e.g. `["-Wno-unused-variable"]`	remove
Remove default compiler flag(s) json list of flags see default_cppflags e.g. `["-Wno-write-strings"]`	remove

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