View source for Method of steepest descent

{{Short description|Extension of Laplace's method for approximating integrals}}
{{for|the optimization algorithm|Gradient descent}}
In mathematics, the '''method of steepest descent''' or '''saddle-point method''' is an extension of [[Laplace's method]] for approximating an integral, where one deforms a contour integral in the complex plane to pass  near a stationary point ([[saddle point]]), in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals.

The integral to be estimated is often of the form
:<math>\int_Cf(z)e^{\lambda g(z)}\,dz,</math>
where ''C'' is a contour, and λ is large. One version of the method of steepest descent deforms the contour of integration ''C'' into a new path integration ''C′'' so that the following conditions hold:
# ''C′'' passes through one or more zeros of the derivative ''g''′(''z''),
# the imaginary part of ''g''(''z'') is constant on ''C′''.

The method of steepest descent was first published by {{harvtxt|Debye|1909}}, who used it to estimate [[Bessel functions]] and pointed out that it occurred in the unpublished note by {{harvtxt|Riemann|1863}} about [[hypergeometric functions]]. The contour of steepest descent has a minimax property, see {{harvtxt|Fedoryuk|2001}}. {{harvtxt|Siegel|1932}} described some other unpublished notes of Riemann, where he used this method to derive the [[Riemann–Siegel formula]].

== Basic idea ==
The method of steepest descent is a method to approximate a complex integral of the form<math display="block">I(\lambda) = \int_Cf(z)e^{\lambda g(z)}\,\mathrm{d}z</math>for large <math>\lambda \rightarrow \infty</math>, where <math>f(z)</math> and <math>g(z)</math> are [[analytic function]]s of <math>z</math>. Because the integrand is analytic, the contour <math>C</math> can be deformed into a new contour <math>C'</math> without changing the integral. In particular, one seeks a new contour on which the imaginary part, denoted <math>\Im (\cdot)</math>, of <math>g(z) = \Re [g(z)] + i \, \Im[g(z)]</math> is constant (<math>\Re (\cdot)</math> denotes the real part). Then <math display="block">I(\lambda) = e^{i \lambda \Im\{g(z)\}} \int_{C'}f(z)e^{\lambda \Re \{g(z)\}}\,\mathrm{d}z,</math>and the remaining integral can be approximated with other methods like [[Laplace's method]].<ref>{{Cite book|last1=Bender|first1=Carl M.|url=http://link.springer.com/10.1007/978-1-4757-3069-2|title=Advanced Mathematical Methods for Scientists and Engineers I|last2=Orszag|first2=Steven A.|date=1999|publisher=Springer New York|isbn=978-1-4419-3187-0|location=New York, NY|language=en|doi=10.1007/978-1-4757-3069-2}}</ref>

== Etymology ==
The method is called the method of '''steepest descent''' because for analytic <math>g(z)</math>, constant phase contours are equivalent to steepest descent contours.

If <math>g(z) = X(z) + i Y(z)</math> is an [[analytic function]] of <math>z = x + i y</math>, it satisfies the [[Cauchy–Riemann equations]]<math display="block">\frac{\partial X}{\partial x}
=
\frac{\partial Y}{\partial y}
\qquad \text{and} \qquad
\frac{\partial X}{\partial y}
=
- \frac{\partial Y}{\partial x}
.</math>Then <math display="block">\frac{\partial X}{\partial x} \frac{\partial Y}{\partial x} + \frac{\partial X}{\partial y} \frac{\partial Y}{\partial y} = \nabla X \cdot \nabla Y = 0,</math>so contours of constant phase are also contours of steepest descent.

==A simple estimate==
Let {{math|&thinsp;''f'', ''S'' : '''C'''<sup>''n''</sup> → '''C'''}} and {{math|''C'' ⊂  '''C'''<sup>''n''</sup>}}. If

:<math> M = \sup_{x \in C} \Re (S(x)) < \infty,</math>

where <math>\Re (\cdot)</math> denotes the real part, and there exists a positive real number {{math|''λ''<sub>0</sub>}} such that

:<math>\int_{C} \left| f(x) e^{\lambda_0 S(x)}  \right| dx < \infty,</math>

then the following estimate holds:<ref>A modified version of Lemma 2.1.1 on page 56 in {{harvtxt|Fedoryuk|1987}}.</ref>
:<math>\left| \int_{C} f(x) e^{\lambda S(x)}  dx \right| \leqslant \text{const}\cdot e^{\lambda M}, \qquad \forall \lambda \in \mathbb{R}, \quad \lambda \geqslant \lambda_0.</math>
Proof of the simple estimate:

:<math>\begin{align}
\left| \int_{C} f(x) e^{\lambda S(x)} dx \right| &\leqslant \int_C |f(x)| \left|e^{\lambda S(x)} \right| dx \\ 
&\equiv \int_{C} |f(x)| e^{\lambda M} \left | e^{\lambda_0 (S(x)-M)} e^{(\lambda-\lambda_0)(S(x)-M)} \right| dx  \\
&\leqslant \int_C |f(x)| e^{\lambda M} \left| e^{\lambda_0 (S(x)-M)} \right| dx && \left| e^{(\lambda-\lambda_0)(S(x) - M)} \right| \leqslant 1 \\
&= \underbrace{e^{-\lambda_0 M} \int_{C} \left| f(x) e^{\lambda_0 S(x)} \right| dx}_{\text{const}} \cdot e^{\lambda M}.
\end{align}</math>

== The case of a single non-degenerate saddle point ==

=== Basic notions and notation ===
Let {{mvar|x}} be a complex {{mvar|n}}-dimensional vector, and

:<math>S''_{xx}(x) \equiv \left( \frac{\partial^2 S(x)}{\partial x_i \partial x_j} \right), \qquad 1\leqslant i,\, j\leqslant n,</math>

denote the [[Hessian matrix]] for a function {{math|''S''(''x'')}}. If

:<math>\boldsymbol{\varphi}(x) = (\varphi_1(x), \varphi_2(x), \ldots, \varphi_k(x))</math>

is a vector function, then its [[Jacobian matrix and determinant|Jacobian matrix]] is defined as

:<math>\boldsymbol{\varphi}_x' (x) \equiv \left( \frac{\partial \varphi_i (x)}{\partial x_j} \right), \qquad 1 \leqslant i \leqslant k, \quad 1 \leqslant j \leqslant n.</math>

A '''non-degenerate saddle point''', {{math|''z''<sup>0</sup> ∈ '''C'''<sup>''n''</sup>}}, of a holomorphic function {{math|''S''(''z'')}} is a critical point of the function (i.e., {{math|∇''S''(''z''<sup>0</sup>) {{=}} 0}}) where the function's Hessian matrix has a non-vanishing determinant (i.e., <math>\det S''_{zz}(z^0) \neq 0</math>).

The following is the main tool for constructing the asymptotics of integrals in the case of a non-degenerate saddle point:

===Complex Morse lemma===
The [[Morse theory#Morse lemma|Morse lemma]] for real-valued functions generalizes as follows<ref>Lemma 3.3.2 on page 113 in {{harvtxt|Fedoryuk|1987}}</ref> for [[holomorphic function]]s: near a non-degenerate saddle point {{math|''z''<sup>0</sup>}} of a holomorphic function {{math|''S''(''z'')}}, there exist coordinates in terms of which {{math|''S''(''z'') − ''S''(''z''<sup>0</sup>)}} is exactly quadratic. To make this precise, let {{mvar|S}} be a holomorphic function with domain {{math|''W'' ⊂ '''C'''<sup>''n''</sup>}}, and let {{math|''z''<sup>0</sup>}} in {{mvar|W}} be a non-degenerate saddle point of {{mvar|S}}, that is, {{math|∇''S''(''z''<sup>0</sup>) {{=}} 0}} and <math>\det S''_{zz}(z^0) \neq 0</math>. Then there exist neighborhoods {{math|''U'' ⊂ ''W''}} of {{math|''z''<sup>0</sup>}} and {{math|''V'' ⊂ '''C'''<sup>''n''</sup>}} of {{math|''w'' {{=}} 0}}, and a [[One-to-one function|bijective]] holomorphic function {{math|'''''φ''''' : ''V'' → ''U''}} with {{math|'''''φ'''''(0) {{=}} ''z''<sup>0</sup>}} such that

:<math>\forall w \in V: \qquad S(\boldsymbol{\varphi}(w)) = S(z^0) + \frac{1}{2} \sum_{j=1}^n \mu_j w_j^2, \quad \det\boldsymbol{\varphi}_w'(0) = 1,
</math>

Here, the {{math|''μ<sub>j</sub>''}} are the [[Eigenvalues and eigenvectors|eigenvalues]] of the matrix <math>S_{zz}''(z^0)</math>.

[[File:Complex Morse Lemma Illustration.pdf|thumb|center|An illustration of Complex Morse lemma]]
{{math proof|title=Proof of complex Morse lemma|proof=
The following proof is a straightforward generalization of the proof of the real [[Morse theory#Morse lemma|Morse Lemma]], which can be found in.<ref>{{harvtxt|Poston|Stewart|1978}}, page 54; see also the comment on page 479 in {{harvtxt|Wong|1989}}.
</ref> We begin by demonstrating

:'''Auxiliary statement.''' Let {{math|&thinsp;''f''&thinsp; : '''C'''<sup>''n''</sup> → '''C'''}} be [[Holomorphic function|holomorphic]] in a neighborhood of the origin and {{math|&thinsp;''f''&thinsp;(0) {{=}} 0}}. Then in some neighborhood, there exist functions {{math|''g<sub>i</sub>'' : '''C'''<sup>''n''</sup> → '''C'''}} such that <math display="block">f(z) = \sum_{i=1}^n z_i g_i(z),</math> where each {{math|''g<sub>i</sub>''}} is [[Holomorphic function|holomorphic]] and <math display="block">g_i(0) = \left. \tfrac{\partial f(z)}{\partial z_i} \right|_{z=0}.</math>

From the identity

:<math>f(z) = \int_0^1 \frac{d}{dt} f \left (t z_1,\cdots, t z_n \right ) dt = \sum_{i=1}^n z_i \int_0^1 \left. \frac{\partial f(z)}{\partial z_i}\right|_{z=(t z_1, \ldots, t z_n)} dt,</math>

we conclude that

:<math>g_i(z) = \int_0^1 \left. \frac{\partial f(z)}{\partial z_i}\right|_{z=(t z_1, \ldots, t z_n)} dt</math>

and

:<math>g_i(0) = \left. \frac{\partial f(z)}{\partial z_i} \right|_{z=0}.</math>

Without loss of generality, we translate the origin to {{math|''z''<sup>0</sup>}}, such that {{math|''z''<sup>0</sup> {{=}} 0}} and {{math|''S''(0) {{=}} 0}}. Using the Auxiliary Statement, we have
:<math>S(z) = \sum_{i=1}^n z_i g_i (z).</math>
Since the origin is a saddle point,
:<math>\left. \frac{\partial S(z)}{\partial z_i} \right|_{z=0} = g_i(0) = 0,</math>
we can also apply the Auxiliary Statement to the functions {{math|''g<sub>i</sub>''(''z'')}} and obtain 
{{NumBlk|:|<math>S(z) = \sum_{i,j=1}^n z_i z_j h_{ij}(z).</math>|{{EquationRef|1}}}}
Recall that an arbitrary matrix {{mvar|A}} can be represented as a sum of symmetric {{math|''A''<sup>(''s'')</sup>}} and anti-symmetric {{math|''A''<sup>(''a'')</sup>}} matrices, 
:<math>A_{ij} = A_{ij}^{(s)} + A_{ij}^{(a)}, \qquad A_{ij}^{(s)} = \tfrac{1}{2}\left( A_{ij} + A_{ji} \right), \qquad A_{ij}^{(a)} = \tfrac{1}{2}\left( A_{ij} - A_{ji} \right).</math>
The contraction of any symmetric matrix ''B'' with an arbitrary matrix {{mvar|A}} is
{{NumBlk|:|<math>\sum_{i,j} B_{ij} A_{ij} = \sum_{i,j} B_{ij} A_{ij}^{(s)},</math>
|{{EquationRef|2}}}}
i.e., the anti-symmetric component of {{mvar|A}} does not contribute because
:<math>\sum_{i,j} B_{ij} C_{ij} = \sum_{i,j} B_{ji} C_{ji} = - \sum_{i,j} B_{ij} C_{ij} = 0.</math>
Thus, {{math|''h<sub>ij</sub>''(''z'')}} in equation (1) can be assumed to be symmetric with respect to the interchange of the indices {{mvar|i}} and {{mvar|j}}. Note that
:<math>\left. \frac{\partial^2 S(z)}{\partial z_i \partial z_j} \right|_{z=0} = 2h_{ij}(0);</math>
hence, {{math|det(''h<sub>ij</sub>''(0)) ≠ 0}} because the origin is a non-degenerate saddle point.

Let us show by [[Mathematical induction|induction]] that there are local coordinates {{math|''u'' {{=}} (''u''<sub>1</sub>, ... ''u<sub>n</sub>''), ''z'' {{=}} '''''ψ'''''(''u''), 0 {{=}} '''''ψ'''''(0)}}, such that
{{NumBlk|:|<math>S(\boldsymbol{\psi}(u)) = \sum_{i=1}^n u_i^2.</math>|{{EquationRef|3}}}}
First, assume that there exist local coordinates {{math|''y'' {{=}} (''y''<sub>1</sub>, ... ''y<sub>n</sub>''), ''z'' {{=}} '''''φ'''''(''y''), 0 {{=}} '''''φ'''''(0)}}, such that
{{NumBlk|:|<math>S(\boldsymbol{\phi}(y)) = y_1^2 + \cdots + y_{r-1}^2 + \sum_{i,j = r}^n y_i y_j H_{ij} (y),</math>|{{EquationRef|4}}}}
where {{math|''H<sub>ij</sub>''}} is symmetric due to equation (2). By a linear change of the variables {{math|(''y<sub>r</sub>'', ... ''y<sub>n</sub>'')}}, we can assure that {{math|''H<sub>rr</sub>''(0) ≠ 0}}. From the [[chain rule]], we have

:<math>\frac{\partial^2 S (\boldsymbol{\phi}(y))}{\partial y_i \partial y_j} = \sum_{l,k=1}^n \left. \frac{\partial^2 S (z)}{\partial z_k \partial z_l} \right|_{z=\boldsymbol{\phi}(y)} \frac{\partial \phi_k}{\partial y_i} \frac{\partial \phi_l}{\partial y_j} + \sum_{k=1}^n \left. \frac{\partial S (z)}{\partial z_k } \right|_{z=\boldsymbol{\phi}(y)} \frac{\partial^2 \phi_k}{\partial y_i \partial y_j}</math>

Therefore:

:<math>S''_{yy} (\boldsymbol{\phi}(0)) = \boldsymbol{\phi}'_y(0)^T S''_{zz}(0) \boldsymbol{\phi}'_y(0), \qquad \det \boldsymbol{\phi}'_y(0) \neq 0;</math>

whence,

:<math>0 \neq \det  S''_{yy} (\boldsymbol{\phi}(0)) = 2^{r-1} \det \left( 2H_{ij}(0) \right).</math>

The matrix {{math|(''H<sub>ij</sub>''(0))}} can be recast in the [[Jordan normal form]]: {{math|(''H<sub>ij</sub>''(0)) {{=}} ''LJL''<sup>−1</sup>}}, were {{mvar|L}} gives the desired non-singular linear transformation and the diagonal of {{mvar|J}} contains non-zero [[Eigenvalues and eigenvectors|eigenvalues]] of {{math|(''H<sub>ij</sub>''(0))}}. If {{math|''H<sub>ij</sub>''(0) ≠ 0}} then, due to continuity of {{math|''H<sub>ij</sub>''(''y'')}}, it must be also non-vanishing in some neighborhood of the origin. Having introduced <math>\tilde{H}_{ij}(y) = H_{ij}(y)/H_{rr}(y)</math>, we write

:<math>\begin{align}
S(\boldsymbol{\varphi}(y)) =& y_1^2 + \cdots + y_{r-1}^2 + H_{rr}(y) \sum_{i,j = r}^n y_i y_j \tilde{H}_{ij} (y) \\
=& y_1^2 + \cdots + y_{r-1}^2 + H_{rr}(y)\left[ y_r^2 + 2y_r \sum_{j=r+1}^n y_j \tilde{H}_{rj} (y) +  \sum_{i,j = r+1}^n y_i y_j \tilde{H}_{ij} (y) \right] \\
=& y_1^2 + \cdots + y_{r-1}^2 + H_{rr}(y)\left[ \left( y_r + \sum_{j=r+1}^n y_j \tilde{H}_{rj} (y)\right)^2 - \left( \sum_{j=r+1}^n y_j \tilde{H}_{rj} (y)\right)^2  \right] + H_{rr}(y) \sum_{i,j = r+1}^n y_i y_j \tilde{H}_{ij}(y) 
\end{align}</math>

Motivated by the last expression, we introduce new coordinates {{math|''z'' {{=}} '''''η'''''(''x''), 0 {{=}} '''''η'''''(0),}}

:<math>x_r = \sqrt{ H_{rr}(y) } \left( y_r + \sum_{j=r+1}^n y_j \tilde{H}_{rj} (y)\right), \qquad x_j = y_j, \quad \forall j \neq r.</math>

The change of the variables {{math|''y'' ↔ ''x''}} is locally invertible since the corresponding [[Jacobian matrix and determinant|Jacobian]] is non-zero,

:<math>\left. \frac{\partial x_r}{\partial y_k} \right|_{y=0} = \sqrt{H_{rr}(0)} \left[ \delta_{r,\, k} + \sum_{j=r+1}^n \delta_{j, \, k} \tilde{H}_{jr}(0) \right].</math>

Therefore,
{{NumBlk|:|<math>S(\boldsymbol{\eta}(x)) = {x}_1^2 + \cdots + {x}_r^2 + \sum_{i,j = r+1}^n {x}_i {x}_j W_{ij} (x).</math>|{{EquationRef|5}}}}

Comparing equations (4) and (5), we conclude that equation (3) is verified. Denoting the [[Eigenvalues and eigenvectors|eigenvalues]] of <math>S''_{zz}(0)</math> by {{math|''μ<sub>j</sub>''}}, equation (3) can be rewritten as

{{NumBlk|:|<math>S(\boldsymbol{\varphi}(w)) = \frac 12 \sum_{j=1}^n \mu_j w_j^2.</math>|{{EquationRef|6}}}}

Therefore,

{{NumBlk|:|<math>S''_{ww} (\boldsymbol{\varphi}(0)) = \boldsymbol{\varphi}'_w(0)^T S''_{zz}(0) \boldsymbol{\varphi}'_w(0),</math>|{{EquationRef|7}}}}

From equation (6), it follows that <math>\det S''_{ww} (\boldsymbol{\varphi}(0)) = \mu_1 \cdots \mu_n</math>. The [[Jordan normal form]] of <math>S''_{zz}(0)</math> reads <math>S''_{zz}(0) = P J_z P^{-1}</math>, where {{math|''J<sub>z</sub>''}} is an upper diagonal matrix containing the [[Eigenvalues and eigenvectors|eigenvalues]] and {{math|det ''P'' ≠ 0}}; hence, <math>\det S''_{zz} (0) = \mu_1 \cdots \mu_n</math>. We obtain from equation (7)

:<math>\det S''_{ww} (\boldsymbol{\varphi}(0)) = \left[\det \boldsymbol{\varphi}'_w(0) \right]^2 \det S''_{zz}(0) \Longrightarrow  \det \boldsymbol{\varphi}'_w(0) = \pm 1.</math>	

If <math>\det \boldsymbol{\varphi}'_w(0) = -1</math>, then interchanging two variables assures that <math>\det \boldsymbol{\varphi}'_w(0) = +1</math>. 
}}<!-- END OF COMPLEX MORSE LEMMA -->

=== The asymptotic expansion in the case of a single non-degenerate saddle point ===
Assume
# {{math|&thinsp;''f''&thinsp;(''z'')}} and {{math|''S''(''z'')}} are [[Holomorphic function|holomorphic]] functions in an [[Open set|open]], [[Bounded set (topological vector space)|bounded]], and [[Simply connected space|simply connected]] set {{math|Ω''<sub>x</sub>'' ⊂ '''C'''<sup>''n''</sup>}} such that the {{math|''I<sub>x</sub>'' {{=}} Ω''<sub>x</sub>'' ∩ '''R'''<sup>''n''</sup>}} is [[Connected space|connected]];
# <math>\Re(S(z))</math> has a single maximum: <math>\max_{z \in I_x} \Re(S(z)) = \Re(S(x^0))</math> for exactly one point {{math|''x''<sup>0</sup> ∈ ''I<sub>x</sub>''}};
# {{math|''x''<sup>0</sup>}} is a non-degenerate saddle point (i.e., {{math|∇''S''(''x''<sup>0</sup>) {{=}} 0}} and <math>\det S''_{xx}(x^0) \neq 0</math>).

Then, the following asymptotic holds
{{NumBlk|:|<math>I(\lambda) \equiv \int_{I_x} f(x) e^{\lambda S(x)} dx = \left( \frac{2\pi}{\lambda}\right)^{\frac{n}{2}} e^{\lambda S(x^0)} \left(f(x^0)+ O\left(\lambda^{-1}\right) \right) \prod_{j=1}^n (-\mu_j)^{-\frac{1}{2}}, \qquad \lambda \to \infty,</math>|{{EquationRef|8}}}}
where {{math|''μ<sub>j</sub>''}} are eigenvalues of the [[Hessian matrix|Hessian]] <math>S''_{xx}(x^0)</math> and <math>(-\mu_j)^{-\frac{1}{2}}</math> are defined with arguments
{{NumBlk|:|<math>\left | \arg\sqrt{-\mu_j} \right| < \tfrac{\pi}{4}.</math>
|{{EquationRef|9}}}}

This statement is a special case of more general results presented in Fedoryuk (1987).<ref>{{harvtxt|Fedoryuk|1987}}, pages 417-420.</ref>
{{math proof|title=Derivation of equation (8)|proof=
[[File:Illustration To Derivation Of Asymptotic For Saddle Point Integration.pdf|thumb|center|An illustration to the derivation of equation (8)]]

First, we deform the contour {{math|''I<sub>x</sub>''}} into a new contour <math>I'_x \subset \Omega_x</math> passing through the saddle point {{math|''x''<sup>0</sup>}} and sharing the boundary with {{math|''I<sub>x</sub>''}}. This deformation does not change the value of the integral {{math|''I''(''λ'')}}. We employ the [[Method of steepest descent#Complex Morse Lemma|Complex Morse Lemma]] to change the variables of integration. According to the lemma, the function {{math|'''''φ'''''(''w'')}} maps a neighborhood {{math|''x''<sup>0</sup> ∈ ''U'' ⊂ Ω''<sub>x</sub>''}} onto a neighborhood {{math|Ω''<sub>w</sub>''}} containing the origin.  The integral {{math|''I''(''λ'')}} can be split into two: {{math|''I''(''λ'') {{=}} ''I''<sub>0</sub>(''λ'') + ''I''<sub>1</sub>(''λ'')}}, where {{math|''I''<sub>0</sub>(''λ'')}} is the integral over <math>U\cap I'_x</math>, while {{math|''I''<sub>1</sub>(''λ'')}} is  over <math>I'_x \setminus (U\cap I'_x)</math> (i.e., the remaining part of the contour {{math|''I′<sub>x</sub>''}}). Since the latter region does not contain the saddle point {{math|''x''<sup>0</sup>}}, the value of {{math|''I''<sub>1</sub>(''λ'')}} is exponentially smaller than {{math|''I''<sub>0</sub>(''λ'')}} as {{math|''λ'' → ∞}};<ref>This conclusion follows from a comparison between the final asymptotic for {{math|''I''<sub>0</sub>(''λ'')}}, given by equation (8), and [[Method of steepest descent#A simple estimate .5B1.5D|a simple estimate]] for the discarded integral {{math|''I''<sub>1</sub>(''λ'')}}.</ref> thus, {{math|''I''<sub>1</sub>(''λ'')}} is ignored. Introducing the contour {{math|''I<sub>w</sub>''}} such that <math>U\cap I'_x = \boldsymbol{\varphi}(I_w)</math>, we have

{{NumBlk|:|<math>I_0(\lambda) = e^{\lambda S(x^0)} \int_{I_w} f[\boldsymbol{\varphi}(w)] \exp\left( \lambda \sum_{j=1}^n \tfrac{\mu_j}{2} w_j^2 \right) \left |\det\boldsymbol{\varphi}_w'(w) \right | dw.</math>|{{EquationRef|10}}}}

Recalling that {{math|''x''<sup>0</sup> {{=}} '''''φ'''''(0)}} as well as <math>\det \boldsymbol{\varphi}_w'(0) = 1</math>, we expand the pre-exponential function <math>f[\boldsymbol{\varphi}(w)]</math> into a Taylor series and keep just the leading zero-order term

{{NumBlk|:|<math>I_0(\lambda) \approx f(x^0) e^{\lambda S(x^0)} \int_{\mathbf{R}^n} \exp \left( \lambda \sum_{j=1}^n \tfrac{\mu_j}{2} w_j^2 \right) dw = f(x^0)e^{\lambda S(x^0)} \prod_{j=1}^n \int_{-\infty}^{\infty} e^{\frac{1}{2}\lambda \mu_j y^2} dy.</math>|{{EquationRef|11}}}}

Here, we have substituted the integration region {{math|''I<sub>w</sub>''}} by {{math|'''R'''<sup>''n''</sup>}} because both contain the origin, which is a saddle point, hence they are equal up to an exponentially small term.<ref>This is justified by comparing the integral asymptotic over {{math|'''R'''<sup>''n''</sup>}} [see equation (8)] with [[Method of steepest descent#A simple estimate .5B1.5D|a simple estimate]] for the altered part.</ref> The integrals in the r.h.s. of equation (11) can be expressed as

{{NumBlk|:|<math>\mathcal{I}_j = \int_{-\infty}^{\infty} e^{\frac{1}{2} \lambda \mu_j y^2} dy = 2\int_0^{\infty} e^{-\frac{1}{2} \lambda \left(\sqrt{-\mu_j} y\right)^2} dy = 2\int_0^{\infty} e^{-\frac{1}{2} \lambda \left |\sqrt{-\mu_j} \right|^2 y^2\exp\left(2i\arg\sqrt{-\mu_j}\right)} dy.</math>|{{EquationRef|12}}}}

From this representation, we conclude that condition (9) must be satisfied in order for the r.h.s. and l.h.s. of equation (12) to coincide. According to assumption 2, <math>\Re \left( S_{xx}''(x^0) \right)</math> is a [[Definite bilinear form|negatively defined quadratic form]] (viz., <math>\Re(\mu_j)<0</math>) implying the existence of the integral <math>\mathcal{I}_j</math>, which is readily calculated

:<math>\mathcal{I}_j = \frac{2}{\sqrt{-\mu_j}\sqrt{\lambda}} \int_0^{\infty} e^{-\frac{\xi^2}{2}} d\xi = \sqrt{\frac{2\pi}{\lambda}} (-\mu_j)^{-\frac{1}{2}}.</math>
}}
<!-- END OF PROOF -->

Equation (8) can also be written as

{{NumBlk|:|<math>I(\lambda) = \left( \frac{2\pi}{\lambda}\right)^{\frac{n}{2}} e^{\lambda S(x^0)} \left ( \det (-S_{xx}''(x^0)) \right )^{-\frac{1}{2}} \left (f(x^0) + O\left(\lambda^{-1}\right) \right),</math>|{{EquationRef|13}}}}

where the branch of

:<math>\sqrt{\det \left (-S_{xx}''(x^0) \right)}</math>

is selected as follows

:<math>\begin{align}
\left (\det \left (-S_{xx}''(x^0) \right ) \right)^{-\frac{1}{2}} &= \exp\left( -i \text{ Ind} \left (- S_{xx}''(x^0) \right ) \right) \prod_{j=1}^n \left| \mu_j \right|^{-\frac{1}{2}}, \\
\text{Ind} \left (-S_{xx}''(x^0) \right) &= \tfrac{1}{2} \sum_{j=1}^n \arg (-\mu_j), && |\arg(-\mu_j)| < \tfrac{\pi}{2}.
\end{align}</math>

Consider important special cases:

* If {{math|''S''(''x'')}} is real valued for real {{mvar|x}} and {{math|''x''<sup>0</sup>}} in {{math|'''R'''<sup>''n''</sup>}} (aka, the '''multidimensional Laplace method'''), then<ref>See equation (4.4.9) on page 125 in {{harvtxt|Fedoryuk|1987}}</ref> <math display="block">\text{Ind} \left(-S_{xx}''(x^0) \right ) = 0.</math>
* If {{math|''S''(''x'')}} is purely imaginary for real {{mvar|x}} (i.e., <math>\Re(S(x)) = 0</math> for all {{mvar|x}} in {{math|'''R'''<sup>''n''</sup>}}) and {{math|''x''<sup>0</sup>}} in {{math|'''R'''<sup>''n''</sup>}} (aka, the '''multidimensional stationary phase method'''),<ref>Rigorously speaking, this case cannot be inferred from equation (8) because [[Method of steepest descent#The asymptotic expansion in the case of a single non-degenerate saddle point|the second assumption]], utilized in the derivation, is violated. To include the discussed case of a purely imaginary phase function, condition (9) should be replaced by <math> \left | \arg\sqrt{-\mu_j} \right | \leqslant \tfrac{\pi}{4}.</math></ref> then<ref>See equation (2.2.6') on page 186 in {{harvtxt|Fedoryuk|1987}}</ref> <math display="block">\text{Ind} \left (-S_{xx}''(x^0) \right ) = \frac{\pi}{4} \text{sign }S_{xx}''(x_0),</math> where <math>\text{sign }S_{xx}''(x_0)</math> denotes [[Sylvester's law of inertia#Statement of the theorem|the signature of matrix]] <math>S_{xx}''(x_0)</math>, which equals to the number of negative eigenvalues minus the number of positive ones. It is noteworthy that in applications of the stationary phase method to the multidimensional WKB approximation in quantum mechanics (as well as in optics), {{math|Ind}} is related to the [[Maslov index]] see, e.g., {{harvtxt|Chaichian|Demichev|2001}} and {{harvtxt|Schulman|2005}}.

== The case of multiple non-degenerate saddle points ==
If the function {{math|''S''(''x'')}} has multiple isolated non-degenerate saddle points, i.e.,

:<math>\nabla S \left (x^{(k)} \right ) = 0, \quad \det S''_{xx} \left (x^{(k)} \right ) \neq 0, \quad x^{(k)} \in \Omega_x^{(k)},</math>

where

:<math>\left \{ \Omega_x^{(k)} \right \}_{k=1}^K</math>

is an [[open cover]] of {{math|Ω<sub>''x''</sub>}}, then the calculation of the integral asymptotic is reduced to the case of a single saddle point by employing the [[partition of unity]]. The [[partition of unity]] allows us to construct a set of continuous functions {{math|''ρ<sub>k</sub>''(''x'') : Ω<sub>''x''</sub> → [0, 1], 1 ≤ ''k'' ≤ ''K'',}} such that

:<math>\begin{align} 
\sum_{k=1}^K \rho_k(x) &= 1, && \forall x \in \Omega_x, \\
\rho_k(x) &= 0 && \forall x \in \Omega_x\setminus \Omega_x^{(k)}.
\end{align}</math>

Whence,

:<math>\int_{I_x \subset \Omega_x} f(x) e^{\lambda S(x)} dx \equiv \sum_{k=1}^K \int_{I_x \subset \Omega_x} \rho_k(x) f(x) e^{\lambda S(x)} dx.</math>

Therefore as {{math|''λ'' → ∞}} we have:

:<math>\sum_{k=1}^K  \int_{\text{a neighborhood of }x^{(k)}} f(x) e^{\lambda S(x)} dx = \left(\frac{2\pi}{\lambda}\right)^{\frac{n}{2}} \sum_{k=1}^K e^{\lambda S \left (x^{(k)} \right )} \left ( \det \left(-S_{xx}'' \left (x^{(k)} \right )\right) \right)^{-\frac{1}{2}} f \left (x^{(k)} \right ),</math>

where equation (13) was utilized at the last stage, and the pre-exponential function {{math|&thinsp;''f''&thinsp;(''x'')}} at least must be continuous.

== The other cases ==
When {{math|∇''S''(''z''<sup>0</sup>) {{=}} 0}} and <math>\det S''_{zz}(z^0) = 0</math>, the point {{math|''z''<sup>0</sup> ∈ '''C'''<sup>''n''</sup>}} is called a '''degenerate saddle point''' of a function {{math|''S''(''z'')}}.

Calculating the asymptotic of

:<math> \int f(x) e^{\lambda S(x)} dx,</math>

when {{math|''λ'' → ∞, &thinsp;''f''&thinsp;(''x'')}} is continuous, and {{math|''S''(''z'')}} has a degenerate saddle point, is a very rich problem, whose solution heavily relies on the [[catastrophe theory]]. Here, the catastrophe theory replaces the [[Method of steepest descent#Complex Morse Lemma|Morse lemma]], valid only in the non-degenerate case, to transform the function {{math|''S''(''z'')}} into one of the multitude of canonical representations. For further details see, e.g., {{harvtxt|Poston|Stewart|1978}} and {{harvtxt|Fedoryuk|1987}}.

Integrals with degenerate saddle points naturally appear in many applications including [[Caustic (optics)|optical caustics]] and the multidimensional [[WKB approximation]] in quantum mechanics.

The other cases such as, e.g., {{math|&thinsp;''f''&thinsp;(''x'')}} and/or {{math|''S''(''x'')}} are discontinuous or when an extremum of {{math|''S''(''x'')}} lies at the integration region's boundary, require special care (see, e.g., {{harvtxt|Fedoryuk|1987}} and {{harvtxt|Wong|1989}}).

==Extensions and generalizations==
An extension of the steepest descent method is the so-called ''nonlinear stationary phase/steepest descent method''. Here, instead of integrals, one needs to evaluate asymptotically solutions of [[Riemann&ndash;Hilbert factorization]] problems.

Given a contour ''C'' in the [[complex sphere]], a function ''f'' defined on that contour and a special point, say infinity, one seeks a function ''M'' holomorphic away from the contour ''C'', with prescribed jump across ''C'', and with a given normalization at infinity. If ''f'' and hence ''M'' are matrices rather than scalars this is a problem that in general does not admit an explicit solution.

An asymptotic evaluation is then possible along the lines of the linear stationary phase/steepest descent method. The idea is to reduce asymptotically the solution of the given Riemann&ndash;Hilbert problem to that of a simpler, explicitly solvable, Riemann&ndash;Hilbert problem. Cauchy's theorem is used to justify deformations  of the jump contour.

The nonlinear stationary phase was introduced by Deift and Zhou in 1993, based on earlier work of the Russian mathematician Alexander Its. A (properly speaking) nonlinear steepest descent method was introduced by Kamvissis, K. McLaughlin and P. Miller in 2003, based on previous work of Lax, Levermore, Deift, Venakides and Zhou. As in the linear case, steepest descent contours solve a min-max problem. In the nonlinear case they turn out to be "S-curves" (defined in a different context back in the 80s by Stahl, Gonchar and Rakhmanov).

The nonlinear stationary phase/steepest descent method has applications to the theory of [[soliton]] equations and [[integrable model]]s, [[random matrices]] and [[combinatorics]].

Another extension is the [[Method of Chester–Friedman–Ursell]] for coalescing saddle points and uniform asymptotic extensions.

==See also==
* [[Pearcey integral]]
* [[Stationary phase approximation]]
* [[Laplace's method]]

==Notes==
{{reflist}}

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[[Category:Asymptotic analysis]]
[[Category:Perturbation theory]]