Analytic Combinatorics/Saddle-point Method

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Theorem[edit | edit source]

Theorem due to Flajolet and Sedgewick[1].

If is an admissible function (we will discuss what this means later) then


where and is such that .

Proof[edit | edit source]

Proof due to Flajolet and Sedgewick[2].

By Cauchy's coefficient formula:

We can visualise this as a 3D graph whose and axes are the real and imaginary parts of respectively and the axis is the real part of .

For the generating functions we are interested in, has a saddle-point on the positive real axis[3]. This is the highest altitude of the green path in the above graph. Call this .

Being an analytic function (except at ), we can deform the contour to go through the saddle-point. The biggest contributor to the integral is made by the part of the contour near the saddle-point (call this , which is the red part of the path in the graph below). The rest of the contour (call this , the green part of the path) contributes relatively little.

We can deform the contour even more to make the part of the contour near the saddle-point a straight line (in the complex plane). is deformed to a straight, vertical line, perpendicular to the real axis, crossing the saddle-point, starting from to . is chosen such that and as . This is so that the Taylor series expansion around


can be reduced to just .

is real-valued because is real and , being a generating function, has real coefficients. , so is real. Therefore, is real-valued.

So, any imaginary part of can be moved outside the integral, leaving just a real-valued integrand:

This also means that the real-valued surface we discussed in the beginning is a valid estimate of the potentially complex-valued .

We change variables to remove the imaginary part of the interval and turn it into a real-valued integrand over the real line. Setting :

Because is very small for large :

Due to our choice of before, as , .

Therefore, the integral can be estimated by a Gaussian integral which we know how to calculate:

Putting it all together:

Theorem[edit | edit source]

Theorem due to Flajolet and Sedgewick[4].

If is an admissible function then


where and is the solution to the equation .

Admissibility[edit | edit source]

Definition from Flajolet and Sedgewick[5] and Wilf[6]. Also known as Hayman-admissibility[7].

The function is admissible if:

  • is a function with radius of convergence
  • There exists an such that
  • H1: and
  • H2: There exists a function defined for such that and as
uniformly for
  • H3: Uniformly for and as

Intuitive explanation[edit | edit source]

To find the coefficients of a function, we can use the Cauchy coefficient formula. This requires us to find the integral of a path in the complex plane. Imagine trying to estimate this integral, displayed as the red and green line below.

The biggest contribution to the integral comes from around the saddle-point (displayed in red) and the tail's contribution (displayed in green) is negligible (by H3).

Therefore, to estimate the integral of the entire path you can estimate the integral of just the red part of the path. This is the asymptotic relation described in H2.

Proof[edit | edit source]

Proof due to Flajolet and Sedgewick[8]

By Cauchy's coefficient formula

Converting to polar form so we can apply our admissibility conditions

Split the integral into a central approximation crossing the saddle-point and a tail

By H3



and this allows us to ignore the tail in our estimate.

By H2

because by assumption in the theorem.

Changing variables with


because as .


Putting it all together

Example[edit | edit source]

Example from Wilf[12] and Flajolet and Sedgewick[13].

Say we want to estimate the coefficients of :

  1. has radius of convergence and is positive for all .
  2. H1: . Therefore, .
  3. H1: . Therefore, .
  4. Find the saddle-point by solving the equation . Therefore, and the path of integration is the circle of radius centered at the origin.
  5. Choose .
  6. H3: For as
  7. H2: For (due to the power series expansion of ).
  8. Apply the theorem: .

Saddle-points of higher order[edit | edit source]

Theorem from Flajolet and Sedgewick[14].

If has a saddle-point of order :

where .

Notes[edit | edit source]

  1. Flajolet and Sedgewick 2009, pp. 553.
  2. Flajolet and Sedgewick 2009, pp. 551-554.
  3. Flajolet and Sedgewick 2009, pp. 549.
  4. Flajolet and Sedgewick 2009, pp. 565.
  5. Flajolet and Sedgewick 2009, pp. 565.
  6. Wilf 2006, pp. 199.
  7. After Walter Hayman.
  8. Flajolet and Sedgewick 2009, chapter VIII.
  9. Flajolet and Sedgewick 2009, pp. 546-549.
  10. This is Laplace's Method. See w:Laplace's_method or Widder 1941, pp. 277.
  11. See, for example, w:List_of_integrals_of_exponential_functions#Definite_integrals.
  12. Wilf 2006, pp. 198.
  13. Flajolet and Sedgewick 2009, pp. 555-557.
  14. Flajolet and Sedgewick 2009, pp. 603.

References[edit | edit source]

  • Flajolet, Philippe; Sedgewick, Robert (2009). Analytic Combinatorics (PDF). Cambridge University Press.
  • Wilf, Herbert S. (2006). Generatingfunctionology (PDF) (3rd ed.). A K Peters, Ltd.
  • Widder, David Vernon (1941). The Laplace Transform. Princeton University Press.