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Rational approximation to the exponential in a complex region

Yuji Nakatsukasa and Stefan Güttel, 23rd July 2012

(Chebfun example approx/ScalingAndSquaring.m)
[Tags: #rational, #exponential, #pade]

Approximation of the exponential function has many important applications, some of which have been described in several previous Chebfun examples. Here we consider approximating $e^z$ via a rational function in a region in the complex plane, in the context of computing the matrix exponential $e^A$ via the scaling and squaring method, the most commonly used method for this purpose, cf. chapter 10 in [2]. Constructions of rational approximations to $e^z$ on a real interval are described in [3].

In brief, the scaling and squaring method is based on a type $(2^sm,2^sm)$ rational approximation to the exponential in a region in the complex plane that contains the eigenvalues, where $s$ and $m$ are suitably chosen integer parameters. It first computes $e^A$ by first choosing an integer $s$ such that $A/2^s$ has norm of order 1, then taking a rational (normally type $(m,m)$ Padé) approximation $r(z)$ to $e^z$ where $m$ is chosen so that $e^{A/2^s}\approx r(A/2^s)$. Then we compute $e^A\approx (r(A/2^s))^{2^s}$ via repeated squaring of the matrix.

Suppose $A$ is diagonalizable and $A=X\mbox{diag}(\lambda_i)X^{-1}$. Then the method is approximating $e^A=X\mbox{diag}(e^{\lambda_i})X^{-1}$ by $Y:=X\mbox{diag}((r(\lambda_i/2^s))^{2^s})X^{-1}\approx e^A,$ and the error satisfies $\|e^A- Y\|_2 \leq \kappa_2(X)\max_{z\in {\cal D}}\left|(r(z/2^s))^{2^s}-e^{z}\right|,$ where $\cal{D}$ is a region in the complex plane that contains the eigenvalues of $A$, and $\kappa_2(X)=\|X\|_2 \|X^{-1}\|_2$ is the 2-norm condition number of the eigenvectors. The term $\max_i\left|(r(\lambda_i/2^s))^{2^s}-e^{\lambda_i}\right|$ represents the exact error if $A$ is a normal matrix: the term $\kappa_2(X)$ can have nontrivial effects on the method for nonnormal $A$, but this is outside the scope of this example. Below we investigate the error $\left|(r(z/2^s))^{2^s}-e^{z}\right|$ where $z$ takes points in the complex plane. If this is small in the whole region $\cal D$, then $Y$ is a good approximation to $e^A$.

The type $(k,m)$ Padé approximant of the exponential is explicitly known to be $r(z)=p_{k,m}(z)/q_{k,m}(z)$, where

$$p_{k,m}(z)=\sum_{j=0}^{k}\frac{(k+m-j)!k!}{(k+m)!(k-j)!}\frac{z^j}{j!}, \quad q_{k,m}(z)=\sum_{j=0}^{m}\frac{(k+m-j)!m!}{(k+m)!(m-j)!}\frac{(-z)^j}{j!}.$$

Alternatively (and in particular for general functions for which Padé approximants are not explicitly known), Padé approximants can be obtained simply and robustly via the Chebfun command padeapprox, see [1].

Here is a contour plot of the logarithm of $\max_{z\in {\cal D}}\left|(r(z/2^s))^{2^s}-e^{z}\right|$ for the case $s=2$ and $m=8$:

s = 2; m = 8;
f = [1 , 1./cumprod(1:50)]; % or f = @(z)exp(z);
[r,a,b] = padeapprox(f,m,m,0);
[x,y] = meshgrid(linspace(-100,100),linspace(-100,100));
z = x + 1i*y;
contourf(x,y,log10(abs(exp(z) - r(z/(2^s)).^(2^s))+eps),[-16:2:0])
colorbar

Since we are using a Padé approximation centered at the origin, the error is zero at the origin and is expected to grow with $|z|$. However, notice that the plot is highly nonsymmetric about the imaginary axis: the error is large for large $\mbox{Re}(z)>0$ and small for $\mbox{Re}(z)<0$. In particular, the region in which the error is $O(10^{-14})$ stretches much farther into the left complex plane than the right, reaching $\mbox{Re}(z)\approx -65$. This is perhaps not surprising because $e^z$ (literally) grows exponentially with $\mbox{Re}(z)$, so for $\mbox{Re}(z)\ll 0$ we have $|e^z|=e^{\mbox{Re}(z)}\approx 0$ and the error is essentially just $\left|(r(z/2^s))^{2^s}\right|$, which is itself small for moderately large $s$ provided that $|(r(z/2^s))| < 1$.

The relative error $\frac{1}{|e^{z}|}\left|(r(z/2^s))^{2^s}-e^{z}\right|$, on the other hand, looks completely different:

contourf(x,y,log10(abs(exp(z) - r(z/(2^s)).^(2^s))./abs(exp(z))),[-16:2:16])
colorbar

The graph shows that $(r(z/2^s))^{2^s}$ has no digits of relative accuracy except in the blue region.

The two plots illustrate for example that around the point $z = -50$, the relative accuracy is terrible but the absolute accuracy is of order unit roundoff. Hence $(r(z/2^s))^{2^s}$ is a good approximant near $z=-50$ if we are concerned with absolute accuracy instead of relative accuracy.

In the context of the matrix exponential, the forward error is measured by the quantity $\|e^A- Y\|_2/\|e^A\|_2$, which for normal $A$ reduces to

$$\max_i\left|(r(\lambda_i/2^s))^{2^s}-e^{\lambda_i}\right|/e^{\max_i\{\mbox{Re}(\lambda_i)\}}\leq \max_{z\in {\cal D}}\left|(r(z/2^s))^{2^s}-e^{z}\right|/e^{\max_i\{\mbox{Re}(\lambda_i)\}}.$$

Since the denominator is constant (when $z$ varies in $\cal D$), this means the absolute error is indeed the appropriate metric of error in the $e^A$ context. The implication is that $e^A$ is well-approximated by $Y$ if the eigenvalues of $A$ lie inside the dark-blue region in the first plot.

References:

[1] P. Gonnet, S. Güttel, and L. N. Trefethen: Robust Padé approximation via SVD. To appear in SIAM Review.

[2] N. J. Higham. Functions of Matrices: Theory and Computation. SIAM, Philadelphia, PA, USA, 2008.

[3] L. N. Trefethen, J. A. C. Weideman, and T. Schmelzer: Talbot quadratures and rational approximations. BIT Numer. Math. (46), pp. 653-670, 2006.


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