Method of steepest descent

Etymology

The method is called the method of steepest descent because for analytic ${\displaystyle g(z)}$, constant phase contours are equivalent to steepest descent contours.

If ${\displaystyle g(z)=X(z)+iY(z)}$ is an analytic function of ${\displaystyle z=x+iy}$, it satisfies the Cauchy–Riemann equations$${\displaystyle \frac{\mathrm{\partial }X}{\mathrm{\partial }x}=\frac{\mathrm{\partial }Y}{\mathrm{\partial }y}\text{and}\frac{\mathrm{\partial }X}{\mathrm{\partial }y}=-\frac{\mathrm{\partial }Y}{\mathrm{\partial }x}.}$$Then $${\displaystyle \frac{\mathrm{\partial }X}{\mathrm{\partial }x}\frac{\mathrm{\partial }Y}{\mathrm{\partial }x}+\frac{\mathrm{\partial }X}{\mathrm{\partial }y}\frac{\mathrm{\partial }Y}{\mathrm{\partial }y}=\mathrm{\nabla }X\cdot \mathrm{\nabla }Y=0,}$$so contours of constant phase are also contours of steepest descent.

A simple estimate

Let f, S : Cⁿ → C and C ⊂ Cⁿ. If

${\displaystyle M=\underset{x\in C}{sup}\mathrm{\Re }(S(x))<\mathrm{\infty },}$

where ${\displaystyle \mathrm{\Re }(\cdot )}$ denotes the real part, and there exists a positive real number λ₀ such that

${\displaystyle {\int }_C\left|f(x)e^{{\lambda }_{0}S(x)}\right|dx<\mathrm{\infty },}$

then the following estimate holds:

${\displaystyle \left|{\int }_{C}f(x)e^{\lambda S(x)}dx\right|⩽\text{const}\cdot e^{\lambda M},\mathrm{\forall }\lambda \in \mathbb{R},\lambda ⩾{\lambda }_0.}$

Proof of the simple estimate:

${\displaystyle \begin{array}{cc}\left|{\int }_{C}f(x)e^{\lambda S(x)}dx\right|& ⩽{\int }_C|f(x)|\left|e^{\lambda S(x)}\right|dx\\ & \equiv {\int }_C|f(x)|e^{\lambda M}\left|e^{{\lambda }_0(S(x)-M)}{e}^{(\lambda -{\lambda }_0)(S(x)-M)}\right|dx\\ & ⩽{\int }_C|f(x)|e^{\lambda M}\left|e^{{\lambda }_0(S(x)-M)}\right|dx& & \left|e^{(\lambda -{\lambda }_0)(S(x)-M)}\right|⩽1\\ & =\underset{\text{const}}{\underbrace{e^{-{\lambda }_{0}M}{\int }_C\left|f(x)e^{{\lambda }_{0}S(x)}\right|dx}}\cdot e^{\lambda M}.\end{array}}$

The case of a single non-degenerate saddle point

Basic notions and notation

Let x be a complex n-dimensional vector, and

${\displaystyle S_{xx}^{″}(x)\equiv \left(\frac{{\mathrm{\partial }}^{2}S(x)}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}\right),1⩽i,j⩽n,}$

denote the Hessian matrix for a function S(x). If

${\displaystyle \mathit{\phi }(x)=({\phi }_1(x),{\phi }_2(x),\dots ,{\phi }_k(x))}$

is a vector function, then its Jacobian matrix is defined as

${\displaystyle {\mathit{\phi }}_x^{\prime }(x)\equiv \left(\frac{\mathrm{\partial }{\phi }_i(x)}{\mathrm{\partial }{x}_j}\right),1⩽i⩽k,1⩽j⩽n.}$

A non-degenerate saddle point, z⁰ ∈ Cⁿ, of a holomorphic function S(z) is a critical point of the function (i.e., ∇S(z⁰) = 0) where the function's Hessian matrix has a non-vanishing determinant (i.e., ${\displaystyle det{S}_{zz}^{″}(z^0)\ne 0}$).

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 lemma for real-valued functions generalizes as follows for holomorphic functions: near a non-degenerate saddle point z⁰ of a holomorphic function S(z), there exist coordinates in terms of which S(z) − S(z⁰) is exactly quadratic. To make this precise, let S be a holomorphic function with domain W ⊂ Cⁿ, and let z⁰ in W be a non-degenerate saddle point of S, that is, ∇S(z⁰) = 0 and ${\displaystyle det{S}_{zz}^{″}(z^0)\ne 0}$. Then there exist neighborhoods U ⊂ W of z⁰ and V ⊂ Cⁿ of w = 0, and a bijective holomorphic function φ : V → U with φ(0) = z⁰ such that

${\displaystyle \mathrm{\forall }w\in V:S(\mathit{\phi }(w))=S(z^0)+\frac{1}{2}\sum _{j=1}^n{\mu }_j{w}_j^2,det{\mathit{\phi }}_w^{\prime }(0)=1,}$

Here, the μ_j are the eigenvalues of the matrix ${\displaystyle S_{zz}^{″}(z^0)}$.

Proof of complex Morse lemma

The following proof is a straightforward generalization of the proof of the real Morse Lemma, which can be found in. We begin by demonstrating

Auxiliary statement. Let f : Cⁿ → C be holomorphic in a neighborhood of the origin and f (0) = 0. Then in some neighborhood, there exist functions g_i : Cⁿ → C such that $${\displaystyle f(z)=\sum _{i=1}^n{z}_i{g}_i(z),}$$ where each g_i is holomorphic and $${\displaystyle g_i(0)={\frac{\mathrm{\partial }f(z)}{\mathrm{\partial }{z}_i}|}_{z=0}.}$$

From the identity

${\displaystyle f(z)={\int }_0^1\frac{d}{dt}f\left(t{z}_1,\cdots ,t{z}_n\right)dt=\sum _{i=1}^n{z}_i{\int }_0^1{\frac{\mathrm{\partial }f(z)}{\mathrm{\partial }{z}_i}|}_{z=(t{z}_1,\dots ,t{z}_n)}dt,}$

we conclude that

${\displaystyle g_i(z)={\int }_0^1{\frac{\mathrm{\partial }f(z)}{\mathrm{\partial }{z}_i}|}_{z=(t{z}_1,\dots ,t{z}_n)}dt}$

and

${\displaystyle g_i(0)={\frac{\mathrm{\partial }f(z)}{\mathrm{\partial }{z}_i}|}_{z=0}.}$

Without loss of generality, we translate the origin to z⁰, such that z⁰ = 0 and S(0) = 0. Using the Auxiliary Statement, we have

${\displaystyle S(z)=\sum _{i=1}^n{z}_i{g}_i(z).}$

Since the origin is a saddle point,

${\displaystyle {\frac{\mathrm{\partial }S(z)}{\mathrm{\partial }{z}_i}|}_{z=0}=g_i(0)=0,}$

we can also apply the Auxiliary Statement to the functions g_i(z) and obtain

Recall that an arbitrary matrix A can be represented as a sum of symmetric A^(s) and anti-symmetric A^(a) matrices,

${\displaystyle A_{ij}=A_{ij}^{(s)}+A_{ij}^{(a)},A_{ij}^{(s)}=\frac{1}{2}\left(A_{ij}+A_{ji}\right),A_{ij}^{(a)}=\frac{1}{2}\left(A_{ij}-A_{ji}\right).}$

The contraction of any symmetric matrix B with an arbitrary matrix A is

i.e., the anti-symmetric component of A does not contribute because

${\displaystyle \sum _{i,j}{B}_{ij}{C}_{ij}=\sum _{i,j}{B}_{ji}{C}_{ji}=-\sum _{i,j}{B}_{ij}{C}_{ij}=0.}$

Thus, h_ij(z) in equation (1) can be assumed to be symmetric with respect to the interchange of the indices i and j. Note that

${\displaystyle {\frac{{\mathrm{\partial }}^{2}S(z)}{\mathrm{\partial }{z}_i\mathrm{\partial }{z}_j}|}_{z=0}=2h_{ij}(0);}$

hence, det(h_ij(0)) ≠ 0 because the origin is a non-degenerate saddle point.

Let us show by induction that there are local coordinates u = (u₁, ... u_n), z = ψ(u), 0 = ψ(0), such that

First, assume that there exist local coordinates y = (y₁, ... y_n), z = φ(y), 0 = φ(0), such that

where H_ij is symmetric due to equation (2). By a linear change of the variables (y_r, ... y_n), we can assure that H_rr(0) ≠ 0. From the chain rule, we have

${\displaystyle \frac{{\mathrm{\partial }}^{2}S(\mathit{\varphi }(y))}{\mathrm{\partial }{y}_i\mathrm{\partial }{y}_j}=\sum _{l,k=1}^n{\frac{{\mathrm{\partial }}^{2}S(z)}{\mathrm{\partial }{z}_k\mathrm{\partial }{z}_l}|}_{z=\mathit{\varphi }(y)}\frac{\mathrm{\partial }{\varphi }_k}{\mathrm{\partial }{y}_i}\frac{\mathrm{\partial }{\varphi }_l}{\mathrm{\partial }{y}_j}+\sum _{k=1}^n{\frac{\mathrm{\partial }S(z)}{\mathrm{\partial }{z}_k}|}_{z=\mathit{\varphi }(y)}\frac{{\mathrm{\partial }}^2{\varphi }_k}{\mathrm{\partial }{y}_i\mathrm{\partial }{y}_j}}$

Therefore:

${\displaystyle S_{yy}^{″}(\mathit{\varphi }(0))={\mathit{\varphi }}_y^{\prime }(0{)}^T{S}_{zz}^{″}(0){\mathit{\varphi }}_y^{\prime }(0),det{\mathit{\varphi }}_y^{\prime }(0)\ne 0;}$

whence,

${\displaystyle 0\ne det{S}_{yy}^{″}(\mathit{\varphi }(0))=2^{r-1}det\left(2H_{ij}(0)\right).}$

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

${\displaystyle \begin{array}{cc}S(\mathit{\phi }(y))=& y_1^2+\cdots +y_{r-1}^2+H_{rr}(y)\sum _{i,j=r}^n{y}_i{y}_j{\tilde{H}}_{ij}(y)\\ =& y_1^2+\cdots +y_{r-1}^2+H_{rr}(y)\left[y_r^2+2y_r\sum _{j=r+1}^n{y}_j{\tilde{H}}_{rj}(y)+\sum _{i,j=r+1}^n{y}_i{y}_j{\tilde{H}}_{ij}(y)\right]\\ =& y_1^2+\cdots +y_{r-1}^2+H_{rr}(y)\left[{\left(y_r+\sum _{j=r+1}^n{y}_j{\tilde{H}}_{rj}(y)\right)}^2-{\left(\sum _{j=r+1}^n{y}_j{\tilde{H}}_{rj}(y)\right)}^2\right]+H_{rr}(y)\sum _{i,j=r+1}^n{y}_i{y}_j{\tilde{H}}_{ij}(y)\end{array}}$

Motivated by the last expression, we introduce new coordinates z = η(x), 0 = η(0),

${\displaystyle x_r=\sqrt{H_{rr}(y)}\left(y_r+\sum _{j=r+1}^n{y}_j{\tilde{H}}_{rj}(y)\right),x_j=y_j,\mathrm{\forall }j\ne r.}$

The change of the variables y ↔ x is locally invertible since the corresponding Jacobian is non-zero,

${\displaystyle {\frac{\mathrm{\partial }{x}_r}{\mathrm{\partial }{y}_k}|}_{y=0}=\sqrt{H_{rr}(0)}\left[{\delta }_{r,k}+\sum _{j=r+1}^n{\delta }_{j,k}{\tilde{H}}_{jr}(0)\right].}$

Therefore,

Comparing equations (4) and (5), we conclude that equation (3) is verified. Denoting the eigenvalues of ${\displaystyle S_{zz}^{″}(0)}$ by μ_j, equation (3) can be rewritten as

Therefore,

From equation (6), it follows that ${\displaystyle det{S}_{ww}^{″}(\mathit{\phi }(0))={\mu }_1\cdots {\mu }_n}$. The Jordan normal form of ${\displaystyle S_{zz}^{″}(0)}$ reads ${\displaystyle S_{zz}^{″}(0)=P{J}_z{P}^{-1}}$, where J_z is an upper diagonal matrix containing the eigenvalues and det P ≠ 0; hence, ${\displaystyle det{S}_{zz}^{″}(0)={\mu }_1\cdots {\mu }_n}$. We obtain from equation (7)

${\displaystyle det{S}_{ww}^{″}(\mathit{\phi }(0))={\left[det{\mathit{\phi }}_w^{\prime }(0)\right]}^{2}det{S}_{zz}^{″}(0)⟹det{\mathit{\phi }}_w^{\prime }(0)=\pm 1.}$

If ${\displaystyle det{\mathit{\phi }}_w^{\prime }(0)=-1}$, then interchanging two variables assures that ${\displaystyle det{\mathit{\phi }}_w^{\prime }(0)=+1}$.

The asymptotic expansion in the case of a single non-degenerate saddle point

Assume

1. f (z) and S(z) are holomorphic functions in an open, bounded, and simply connected set Ω_x ⊂ Cⁿ such that the I_x = Ω_x ∩ Rⁿ is connected;
2. ${\displaystyle \mathrm{\Re }(S(z))}$ has a single maximum: ${\displaystyle \underset{z\in I_x}{max}\mathrm{\Re }(S(z))=\mathrm{\Re }(S(x^0))}$ for exactly one point x⁰ ∈ I_x;
3. x⁰ is a non-degenerate saddle point (i.e., ∇S(x⁰) = 0 and ${\displaystyle det{S}_{xx}^{″}(x^0)\ne 0}$).

Then, the following asymptotic holds

where μ_j are eigenvalues of the Hessian ${\displaystyle S_{xx}^{″}(x^0)}$ and ${\displaystyle (-{\mu }_j{)}^{-\frac{1}{2}}}$ are defined with arguments

This statement is a special case of more general results presented in Fedoryuk (1987).

Derivation of equation (8)

First, we deform the contour I_x into a new contour ${\displaystyle I_x^{\prime }\subset {\mathrm{\Omega }}_x}$ passing through the saddle point x⁰ and sharing the boundary with I_x. This deformation does not change the value of the integral I(λ). We employ the Complex Morse Lemma to change the variables of integration. According to the lemma, the function φ(w) maps a neighborhood x⁰ ∈ U ⊂ Ω_x onto a neighborhood Ω_w containing the origin. The integral I(λ) can be split into two: I(λ) = I₀(λ) + I₁(λ), where I₀(λ) is the integral over ${\displaystyle U\cap I_x^{\prime }}$, while I₁(λ) is over ${\displaystyle I_x^{\prime }\setminus (U\cap I_x^{\prime })}$ (i.e., the remaining part of the contour I′_x). Since the latter region does not contain the saddle point x⁰, the value of I₁(λ) is exponentially smaller than I₀(λ) as λ → ∞; thus, I₁(λ) is ignored. Introducing the contour I_w such that ${\displaystyle U\cap I_x^{\prime }=\mathit{\phi }(I_w)}$, we have

Recalling that x⁰ = φ(0) as well as ${\displaystyle det{\mathit{\phi }}_w^{\prime }(0)=1}$, we expand the pre-exponential function ${\displaystyle f[\mathit{\phi }(w)]}$ into a Taylor series and keep just the leading zero-order term

Here, we have substituted the integration region I_w by Rⁿ because both contain the origin, which is a saddle point, hence they are equal up to an exponentially small term. The integrals in the r.h.s. of equation (11) can be expressed as

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, ${\displaystyle \mathrm{\Re }\left(S_{xx}^{″}(x^0)\right)}$ is a negatively defined quadratic form (viz., ${\displaystyle \mathrm{\Re }({\mu }_j)<0}$) implying the existence of the integral ${\displaystyle {\mathcal{I}}_j}$, which is readily calculated

${\displaystyle {\mathcal{I}}_j=\frac{2}{\sqrt{-{\mu }_j}\sqrt{\lambda }}{\int }_0^{\mathrm{\infty }}{e}^{-\frac{{\xi }^2}{2}}d\xi =\sqrt{\frac{2\pi }{\lambda }}(-{\mu }_j{)}^{-\frac{1}{2}}.}$

Equation (8) can also be written as

where the branch of

${\displaystyle \sqrt{det\left(-S_{xx}^{″}(x^0)\right)}}$

is selected as follows

${\displaystyle \begin{array}{cc}{\left(det\left(-S_{xx}^{″}(x^0)\right)\right)}^{-\frac{1}{2}}& =\exp \left(-i\text{ Ind}\left(-S_{xx}^{″}(x^0)\right)\right)\prod _{j=1}^n{\left|{\mu }_j\right|}^{-\frac{1}{2}},\\ \text{Ind}\left(-S_{xx}^{″}(x^0)\right)& =\frac{1}{2}\sum _{j=1}^n\arg (-{\mu }_j),& & |\arg (-{\mu }_j)|<\frac{\pi }{2}.\end{array}}$

Consider important special cases:

• If S(x) is real valued for real x and x⁰ in Rⁿ (aka, the multidimensional Laplace method), then $${\displaystyle \text{Ind}\left(-S_{xx}^{″}(x^0)\right)=0.}$$
• If S(x) is purely imaginary for real x (i.e., ${\displaystyle \mathrm{\Re }(S(x))=0}$ for all x in Rⁿ) and x⁰ in Rⁿ (aka, the multidimensional stationary phase method), then $${\displaystyle \text{Ind}\left(-S_{xx}^{″}(x^0)\right)=\frac{\pi }{4}\text{sign }{S}_{xx}^{″}(x_0),}$$ where ${\displaystyle \text{sign }{S}_{xx}^{″}(x_0)}$ denotes the signature of matrix ${\displaystyle S_{xx}^{″}(x_0)}$, 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), Ind is related to the Maslov index see, e.g., Chaichian & Demichev (2001) and Schulman (2005).

The case of multiple non-degenerate saddle points

If the function S(x) has multiple isolated non-degenerate saddle points, i.e.,

${\displaystyle \mathrm{\nabla }S\left(x^{(k)}\right)=0,det{S}_{xx}^{″}\left(x^{(k)}\right)\ne 0,x^{(k)}\in {\mathrm{\Omega }}_x^{(k)},}$

where

${\displaystyle {\left\{{\mathrm{\Omega }}_x^{(k)}\right\}}_{k=1}^K}$

is an open cover of Ω_x, 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 ρ_k(x) : Ω_x → [0, 1], 1 ≤ k ≤ K, such that

${\displaystyle \begin{array}{cccc}\sum _{k=1}^K{\rho }_k(x)& =1,& & \mathrm{\forall }x\in {\mathrm{\Omega }}_x,\\ {\rho }_k(x)& =0& & \mathrm{\forall }x\in {\mathrm{\Omega }}_x\setminus {\mathrm{\Omega }}_x^{(k)}.\end{array}}$

Whence,

${\displaystyle {\int }_{I_x\subset {\mathrm{\Omega }}_x}f(x)e^{\lambda S(x)}dx\equiv \sum _{k=1}^K{\int }_{I_x\subset {\mathrm{\Omega }}_x}{\rho }_k(x)f(x)e^{\lambda S(x)}dx.}$

Therefore as λ → ∞ we have:

${\displaystyle \sum _{k=1}^K{\int }_{\text{a neighborhood of }{x}^{(k)}}f(x)e^{\lambda S(x)}dx={\left(\frac{2\pi }{\lambda }\right)}^{\frac{n}{2}}\sum _{k=1}^K{e}^{\lambda S\left(x^{(k)}\right)}{\left(det\left(-S_{xx}^{″}\left(x^{(k)}\right)\right)\right)}^{-\frac{1}{2}}f\left(x^{(k)}\right),}$

where equation (13) was utilized at the last stage, and the pre-exponential function f (x) at least must be continuous.

The other cases

When ∇S(z⁰) = 0 and ${\displaystyle det{S}_{zz}^{″}(z^0)=0}$, the point z⁰ ∈ Cⁿ is called a degenerate saddle point of a function S(z).

Calculating the asymptotic of

${\displaystyle \int f(x)e^{\lambda S(x)}dx,}$

when λ → ∞, f (x) is continuous, and 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 Morse lemma, valid only in the non-degenerate case, to transform the function S(z) into one of the multitude of canonical representations. For further details see, e.g., Poston & Stewart (1978) and Fedoryuk (1987).

Integrals with degenerate saddle points naturally appear in many applications including optical caustics and the multidimensional WKB approximation in quantum mechanics.

The other cases such as, e.g., f (x) and/or S(x) are discontinuous or when an extremum of S(x) lies at the integration region's boundary, require special care (see, e.g., Fedoryuk (1987) and 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–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–Hilbert problem to that of a simpler, explicitly solvable, Riemann–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 models, random matrices and combinatorics.

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