for C99
An implementation of
Lambert W function and exponent of Lambert W function
for C99

Nov. 7 2020
Takayuki HOSODA

Lambert W function exponent of Lambert W function
Japanese Japanese edition is here.

The Lambert W function is also known as the omega function or product logarithm, named after Johann Heinrich Lambert in France (26 or 28 August 1728-25 September 1777). It is a multi-valued function and is a branch of the inverse function of the function f (w)= w e w. Where w is any complex number and e w is an exponential function. A principal branch with a Lambert W function value greater than or equal to -1 is represented as W 0.

eW0 — Exponent of the principal branch of the Lambert W function

eW0(x) — the exponent of the principal branch of the Lambert W function W0(x) — can be calculated by using Newton-Raphson method,

y_\mathrm{n+1} \leftarrow y_\mathrm{n} -  \frac{y_\mathrm{n}\ln(y_\mathrm{n}) - x}{\ln(y_\mathrm{n}) + 1}
or Halley's method shown below.
y_\mathrm{n+1} \leftarrow y_\mathrm{n} + \frac
{2y_\mathrm{n}(1 + \ln y_\mathrm{n}) \,(x - y_\mathrm{n} \ln y_\mathrm{n})}
{2y_\mathrm{n}(1 + \ln y_\mathrm{n})^2 + x - y_\mathrm{n} \ln y_\mathrm{n}}

When finding eW using an iterative method such as Newton's Rapson method, giving a bad initial value, especially when it is very close to the singularity x = -1 / e, can be terrible in terms of convergence. Therefore, it is desirable that the initial value asymptotically approaches eW at the singularity. For that reason, I chose the following formula as an approximate expression that gives the initial value. This equation replaces the coefficient 1/3 of the third term of the Piaseux series expansion near x = -1 / e of e W0 with 0.3 and ignores the fourth and subsequent terms.

Initial approximation :

y_0 = \frac{1}{\mathrm{e}} + \sqrt{\frac{2}{\mathrm{e}} \, \bigg(\frac{1}{\mathrm{e}} + x\bigg)}  + 0.3 \,\bigg(\frac{1}{\mathrm{e}} + x\bigg)
Series expansion of eW at z = -1 / e (Puiseux series) :
\frac{1}{\mathrm{e}} 
+ \sqrt{\frac{2}{\mathrm{e}}\left(z + \frac{1}{\mathrm{e}}\right)}}
+\frac{1}{3} \left(z + \frac{1}{\mathrm{e}}\right)
- \frac{1}{18} \sqrt{\frac{\mathrm{e}}{2}} \left(z + \frac{1}{\mathrm{e}}\right)^{\big{\frac{3}{2}}} +O(z^2)
Instead of the coefficient of 0.3 in the formula above, it can be (e - √2 - 1) which make it zero at x = 0,
y_0 = \frac{1}{\mathrm{e}} + \sqrt{\frac{2}{\mathrm{e}} \, \bigg(\frac{1}{\mathrm{e}} + x\bigg)}  + (\mathrm{e} -  \sqrt{2} - 1) \,\bigg(\frac{1}{\mathrm{e}} + x\bigg)
or "1 / 3" from Puiseux series, but 0.3 is quite good enough for this purpose.

From the relation below, the Lambert W can be calculated by using eW — the exponent of Lambert W function.

W(x) = \frac{x}{\mathrm{e}^{W(x)}} = \ln\big(\mathrm{e}^{W(x)}}\big\,)
cexplambertw.c (C99)
NAME
     cexplambertw -- exponent of the principal branch of the complex Lambert W function.

LIBRARY
     Math library (libm, -lm)

SYNOPSIS
     #include <math.h>
     #include <complex.h>

     double complex
     cexplambertw(complex double x);

RETURN VALUE
     cexplambertw(x) returns requested exponent of the principal branch of the complex Lambert W function.

Recurrence formula (Halleys method) :

    y_\mathrm{n+1} \leftarrow y_\mathrm{n} + \frac
{2y_\mathrm{n}(1 + \ln y_\mathrm{n}) \,(x - y_\mathrm{n} \ln y_\mathrm{n})}
{2y_\mathrm{n}(1 + \ln y_\mathrm{n})^2 + x - y_\mathrm{n} \ln y_\mathrm{n}}

Initial approximation :

    y_0 = \frac{1}{\mathrm{e}} + \sqrt{\frac{2}{\mathrm{e}} \, \bigg(\frac{1}{\mathrm{e}} + x\bigg)}  + 0.3 \,\bigg(\frac{1}{\mathrm{e}} + x\bigg)

    
/* cexplambertw : exponent of Lambert W0 function
    Rev.1.52 (Dec. 4, 2020) (c) Takayuki HOSODA (aka Lyuka)
    http://www.finetune.co.jp/~lyuka/technote/lambertw/
    Acknowledgments: Thanks to Albert Chan for his informative suggestions. 
 */
#include <math.h>
#include <complex.h>

#ifdef HAVE_NO_CLOG
double complex clog(double complex);
double complex clog(double complex x) {
    return log(cabs(x)) + atan2(cimag(x), creal(x)) * I;
}
#endif

double complex cexplambertw(double complex);
double complex cexplambertw(double complex x) {
    double complex y, p, s, t;
    double q, r, m;

    r = 1 / M_E;
    q = M_E - M_SQRT2 - 1;
    if ((fabs(creal(x)) <= (DBL_MAX / 1024.0)) && (fabs(cimag(x)) <= (DBL_MAX / 1024.0))) {
        m = 1.0;
        s = x + r;
        if (s == 0) return r;
        y = (r + csqrt((2.0 * r) * s) + q * s); // approximation near x=-1/e
    } else {
        m = (1.0 / 1024.0);
        x *= m; // scaling
        s = x * m;
        y = r * m + csqrt(r * (s * 64.0)) + q * s;
        y *= 1024.0;
    }
    r = DBL_MAX;
    do {
        q = r; p = y;
        t = clog(y);
        s =  0.5 * (x / y - t * m) / (1 + t) + (1 + t) * m;
        t = (x - y * (t * m));
        y += t / s;
        r = cabs(y - p); // correction radius
    } while (r != 0 && q > r);// convergence check by Urabe's theorem
    return p;
}
Calculation example of cexplambertw()
cexplambertw(1 - i2) = 1.963022024795711 - i1.157904818620495 (i2.220446049250313e-16)
cexplambertw(1) = 1.763222834351897
cexplambertw(i) = 1.219531415904638 + i0.7927604805362661
cexplambertw(0) = 1
cexplambertw(-0.36) = 0.4466034047150879 (5.551115123125783e-17)
cexplambertw(-0.3678794411714423) = 0.3678794411714423
cexplambertw(-0.37) = 0.3671727690033079 + i0.03950593783033725 (0 - i1.52655665885959e-16)
cexplambertw(-1) = 0.168376379087223 + i0.7077541887847276 (-5.551115123125783e-17 + i0)
cexplambertw(-1.78) = -0.05991375474182515 + i1.091676160700024 (2.081668171172169e-17 + i0)
cexplambertw(-6 + i8) = 0.5264016089780164 + i4.672167782982316 (-1.110223024625157e-16 + i0)
cexplambertw(-1e+40 + i1e+40) = -1.105839096727962e+38 + i1.166002733393464e+38
cexplambertw(1e+99) = 4.493356750426822e+96
cexplambertw(1.797693134862316e+308 + i0) = 2.556348163871691e+305 + i0 (-3.89812560456e+289 + i0)
cexplambertw(-1.797693134862316e+308 + i0) = -2.556297327179282e+305 + i1.140377281032545e+303
cexplambertw(1.797693134862316e+308 + i1.797693134862316e+308) = 2.557935739687931e+305 + i2.552239351260207e+305
cexplambertw(-1.797693134862316e+308 + i1.797693134862316e+308) = -2.546517666521172e+305 + i2.563606662249022e+305 (3.89812560456e+289 + i0)
cexplambertw(0 + i1.797693134862316e+308) = 5.701971348523801e+302 + i2.556335454509411e+305 (7.613526571406249e+286 + i0)
W0 — The principal branch of the Lambert W function

It is necessary to consider the singularity, but since eW does not become zero, it was relatively easy to find eW by Newton-Raphson method or Harrey's method. However, when trying to find W0 - Lambert's W of the primary branch by such an iterative method, since W0 becomes zero when x is zero, so it is necessary to consider zero as well as the singularity. In other words, it is strongly desired that the approximation formula of the initial value of the iterative method has an asymptotic to zero error at the singular point and zero. Also, since W0 is, from the singularity, a monotonically increasing function, its approximation must also be a monotonically increasing function. To meet these requirements, the initial value is calculated using the following approximation function.

y_0 = \frac{1}{\mathrm{e}} + \sqrt{\frac{2}{\mathrm{e}} \, \bigg(\frac{1}{\mathrm{e}} + x\bigg)}  + (\mathrm{e} -  \sqrt{2} - 1) \,\bigg(\frac{1}{\mathrm{e}} + x\bigg)

Since W0 has a large derivative around the singularity, Newton's method can cause catastrophic behavior near the singularity if the initial approximation is poor or due to computational errors. Therefore, we will use the Halley method or Householder's method of order 4, which is more stable than the Newton method, to obtain W0.

 Newton-Raphson method :
y_\mathrm{n+1} \leftarrow y_\mathrm{n} + \frac
{x - y_\mathrm{n}\, \mathrm{e}^{\, \displaystyle y_\mathrm{n}}}
{ \mathrm{e}^{\,\displaystyle y_\mathrm{n}} + y_\mathrm{n}\,\mathrm{e}^{\, \displaystyle y_\mathrm{n}}
 Halley's method :
y_\mathrm{n+1} \leftarrow y_\mathrm{n} - \frac
{2 (y_\mathrm{n} + 1) \, (y_\mathrm{n}\,\mathrm{e}^{\, \displaystyle y_\mathrm{n}} - x)}
{ ({ y_\mathrm{n}}^2 + 2 y_\mathrm{n} + 2) \, 
\mathrm{e}^{\,\displaystyle y_\mathrm{n}} + (y_\mathrm{n} + 2) x}
 Householder's method of order 4 :
y_{\mathrm n+1} \leftarrow y_{\mathrm n} - 
\frac{3y_{\mathrm n}^3(\mathrm{e}^{\, \displaystyle y_\mathrm{n}})^2+6y_{\mathrm n}^2(\mathrm{e}^{\, \displaystyle y_\mathrm{n}})^2-3y_{\mathrm n}{x}^2+6y_{\mathrm n}(\mathrm{e}^{\, \displaystyle y_\mathrm{n}})^2-6{x}^2-6{x}\mathrm{e}^{\, \displaystyle y_\mathrm{n}}}
{{6y_{\mathrm n}}^2{x}\mathrm{e}^{\, \displaystyle y_\mathrm{n}}+{y_{\mathrm n}}^2\mathrm{e}^{\, \displaystyle y_\mathrm{n}}+18y_{\mathrm n}{x}\mathrm{e}^{\, \displaystyle y_\mathrm{n}}+6y_{\mathrm n}(\mathrm{e}^{\, \displaystyle y_\mathrm{n}})^2+5y_{\mathrm n}\mathrm{e}^{\, \displaystyle y_\mathrm{n}}+12{x}\mathrm{e}^{\, \displaystyle y_\mathrm{n}}+6(\mathrm{e}^{\, \displaystyle y_\mathrm{n}})^2+6\mathrm{e}^{\, \displaystyle y_\mathrm{n}}}
clambertw.c (C99)
NAME
     clambertw -- the principal branch of the complex Lambert W function.

LIBRARY
     Math library (libm, -lm)

SYNOPSIS
     #include <math.h>
     #include <complex.h>

     double complex
     clambertw(complex double x);

RETURN VALUE
     clambertw(x) returns complex value of the principal branch
     of the Lambert W function.

Recurrence formula (Halley's method) :

    y_\mathrm{n+1} \leftarrow y_\mathrm{n} - \frac
{2 (y_\mathrm{n} + 1) \, (y_\mathrm{n}\,\mathrm{e}^{\, \displaystyle y_\mathrm{n}} - x)}
{ ({ y_\mathrm{n}}^2 + 2 y_\mathrm{n} + 2) \, 
\mathrm{e}^{\,\displaystyle y_\mathrm{n}} + (y_\mathrm{n} + 2) x}

Initial approximation :

    y_0 = \ln\left(\frac{1}{\mathrm{e}} + \sqrt{\frac{2}{\mathrm{e}} \, \left(\frac{1}{\mathrm{e}} + x\right)}  + (\mathrm{e} -  \sqrt{2} - 1) \,\left(\frac{1}{\mathrm{e}} + x\right)\right)
/* clambertw : Lambert W0 function 
    Rev.1.52 (Dec. 4, 2020) (c) Takayuki HOSODA (aka Lyuka)
    http://www.finetune.co.jp/~lyuka/technote/lambertw/
    Acknowledgments: Thanks to Albert Chan for his informative suggestions.
 */
#include <math.h>
#include <complex.h>
#include <float.h>

#ifdef HAVE_NO_CLOG
double complex clog(double complex);
double complex clog(double complex x) {
    return log(cabs(x)) + atan2(cimag(x), creal(x)) * I;
}
#endif

double complex clambertw(double complex);
double complex clambertw(double complex x) {
    double complex y, p, s, t;
    double q, r, m;

    r = 1 / M_E;
    q = M_E - M_SQRT2 - 1;
    if ((fabs(creal(x)) <= (DBL_MAX / 1024.0)) && (fabs(cimag(x)) <= (DBL_MAX / 1024.0))) {
        m = 1.0;
        s = x + r;
        if (s == 0) return -1;
        y = clog(r + csqrt((2.0 * r) * s) + q * s); // approximation near x=-1/e
    } else {
        m = (1.0 / 1024.0);
        x *= m; // scaling
        s = x * m;
        y = clog(r * m + csqrt(r * (s * 64.0)) + q * s);
        y += (M_LN2 * 10.0);
    }
    r = DBL_MAX;
    do {
        q = r; p = y;
        t = cexp(y) * m;
        s = y * t - x;
        u = (0.5 * y) / (y + 1.0) * s + t + x;
        y -= s / u;// Halley's method
        r = cabs(y - p); // correction radius
    } while (r != 0 && q > r); // convergence check by Urabe's theorem
    return p;
}
Calculation example of clambertw()
clambertw(2.718281828459045 + i0) = 0.9999999999999999 + i0 (1.110223024625157e-16 + i0)
clambertw(1 - i2) = 0.8237712167092305 - i0.5329289867954415 (0 - i1.110223024625157e-16)
clambertw(1 + i0) = 0.5671432904097838 + i0
clambertw(0 + i1) = 0.3746990207371175 + i0.5764127230314353 (-5.551115123125783e-17 + i0)
clambertw(0 + i0) = 0 + i0
clambertw(-0.36 + i0) = -0.8060843159708175 + i0 (-2.220446049250313e-16 + i0)
clambertw(-0.3678794411714423 + i0) = -1 + i0
clambertw(-0.37 + i0) = -0.996167692712445 + i0.1071826188083488 (3.33066907387547e-16 + i1.956768080901838e-15)
clambertw(-1 + i0) = -0.3181315052047642 + i1.337235701430689 (1.110223024625157e-16 + i2.220446049250313e-16)
clambertw(-1.78 + i0) = 0.08921804985620939 + i1.625623674427768 (-6.938893903907228e-17 + i0)
clambertw(-6 + i8) = 1.547930197079636 + i1.458601930168348
clambertw(-1e+40 + i1e+40) = 87.9726013585729 + i2.329718360883123
clambertw(1e+99 + i0) = 222.5507689557502 + i0
clambertw(1.797693134862316e+308 + i0) = 703.2270331047702 + i0
clambertw(-1.797693134862316e+308 + i0) = 703.2270231685106 + i3.137131632158036
clambertw(0 + i1.797693134862316e+308) = 703.2270306206868 + i1.568565805021136
clambertw(-1.797693134862316e+308 + i1.797693134862316e+308) = 703.5731090989279 + i2.352850357853284
clambertw(1.797693134862316e+308 + i1.797693134862316e+308) = 703.5731140622003 + i0.7842834489371958
Calculation results by WolframAlpha Appendix
4D plot of complex LambertW using gnuplot splot with pm3d -- Plot scripts, data and reference images.
Lambert W function
Download : lambertw-gnuplot-pm3d.zip [5MB zip]
Reference
  1. "Lambert W-function", Wolfram MathWorld
  2. "Why W?", Hayes, B. (2005). American Scientist. 93 (2): 104‐108.
  3. "Lambert W function", Wikipedia
  4. "The Lambert W function poster", Ontario Research Centre for Computer Algebra.
  5. "Puiseux series", Wikipedia
  6. "Newton's Method", Wolfram MathWorld
  7. "Householder's Method", Wolfram MathWorld
  8. "Convergence of Numerical Iteration in Solution of Equations.", Urabe Minoru, J. Sci. Hiroshima Univ. Ser. A 19 (1956), no. 3, 479--489. doi:10.32917/hmj/1556071264.
  9. "Improved Complex Division", Michael Baudin, Robert Smith, 2011
  10. GSL - GNU Scientific Library
  11. gnuplot demo - 3D plotting complex functions with color mapping using pm3d
  12. "Mathematics Written in Sand", W. Kahan, University of California @ Berkeley, Nov. 1983
See Also
www.finetune.co.jp [Mail] © 2000 Takayuki HOSODA.