Chunjing-Sheng-Wang an Effective Method to Compute Frechet Derivative of Matrix Exponential and Its Error Analys

8/4/2019 Chunjing-Sheng-Wang an Effective Method to Compute Frechet Derivative of Matrix Exponential and Its Error Analys

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Journal of Information & Computational Science 7: 9 (2010) 18541859Available at http://www.joics.com

An Effective Method to Compute Frechet Derivative of

Matrix Exponential and Its Error Analysis

Chunjing Li a, Yanyan Sheng a, Minsi Wang b,

aDept. of Mathematics, Tongji University, Shanghai 200092, China

bDept. of Environmental Engineering, Tongji University, Shanghai 200092, China

Abstract

Lots of research has been conducted on matrix exponential due to its kinds of application in practices.As the development of computer technology, nowadays more and more attention have been put to thestability of exp(A). Frechet Derivative of matrix exponential is hereby invented to describes the first-order sensitivity of exp(A) to perturbations in A and its norm directly determines a condition numberfor exp(A). It has performed as a criterion of exp(A) stability. This paper applies scaling and squaringmethod by approximating matrix exponential with a well-studied function and iterates to approximateexp(A) and its Frechet Derivative. Finally we adopt an extension of the existing backward error analysisfor the scaling and squaring method to make choice of algorithmic parameters.

Keywords : Matrix Exponential; Frechet Derivative; Condition Number; Scaling and Squaring Method;

Backward Error

1 Introduction of Frechet Derivative

Matrix Functions play an important role in science and engineering. They arise mostly fromcontrol system, nuclear magnetic resonance and kinds of differential equations extracted fromthe applied mathematics. Among those basic matrix functions, Matrix Exponential has beenwell studied for its widely applications in solving differential equations and control system prob-lems. Therefore, more and more attention have been put to the stability of exp(A), then FrechetDerivative [1] of matrix exponential is invented to describe the first-order sensitivity of exp(A) toperturbations in A.

Definition 1 Frechet derivative representes the sensitivity of a matrix function f : Cmn Cmn to small perturbations at a pointA Cmn.it is a linear mapping, which is defined as

L(A) : E Cmn L(A,E) Cmn

Project supported by the Research and Industrialization of Embedded Software Platform for Digital TV(No. 2009ZX01039-003-003).

Corresponding author.Email address: [email protected] (Minsi Wang).

15487741/ Copyright 2010 Binary Information PressSeptember 2010


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1855 C. Li et al. /Journal of Information & Computational Science 7: 9 (2010) 18541859

for allE Cmn

f(A + E) f(A)L(A,E) = o(E

) (1.1)

therefore it describes the first-order effect on f of perturbations in A.

To show the dependence ofL on f, we will write Lf(A,E) [1].In order to show the perturbations off on A, we define:

Definition 2

cond(f, A) := lim0

supEA

f(A + E) f(A)

f(A)

(1.2)

and

L(A) := max

z=0

L(A,Z)Z

(1.3)

Combine with the definition in (1.1), (1.2), (1.3), we get

cond(f, A) := lim0

supEA

f(A + E) f(A)

f(A)

= lim0

supEA

L(A,E) + o(E)

f(A)

=

L(A)A

f(A)

.

(1.4)

So we set up an effective map between L(A) and cond(f, A). L(A) can be an effective tool thatreflects modulo ofcond(f, A).There are three properties of Frechet Derivative:

Lemma 1 (Sum Rule) If g and h are Frechet differentiable at A and f = g + h, then so is fandLf(A,E) = Lg(A,E) + Lh(A,E).

Lemma 2 (Product Rule) Ifg andh are Frechet differentiable atA andf = gh, then so is f andLf(A,E) = Lg(A,E)h(A) + g(A)Lh(A,E).

Lemma 3 (Chain Rule) If g and h are Frechet differentiable at A respectively. Let f(A) =g(h(A)), thenf is Frechet differentiable at A andLf(A,E) = Lg(h(A), Lh(A,E)).


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2 Previous Research

In Nick Higham papers [2], he uses Pade approximation to exp(A) and iterates with squaring andscaling method to approximate exp(A). This method has got sufficient theory and computationachievement on applications. Now we will give a brief introduction about this method.Frist Nick Higham expresses exp(A) as below,

eA = (eA

2 )2, (2.1)

As we know thatLx2(A,E) = AE+ EA (2.2)

combining with the chain rule, we get

Lexp(A,E) = Lx2

(

A

2 , Lexp(

A

2 ,

E

2 ))

= eA

2 Lexp(A

2,E

2) + Lexp(

A

2,E

2)e

A

2 .

(2.3)

Define Ls = Lexp(2sA, 2sE) and

Li1 = e2iALi + Lie

2iA i = s : 1 : 1 (2.4)

then use Pade approximation to e2iA as well as Frechet derivation of Pade approximation to that

ofe2iA and iterate with squaring and scaling method, repeating use (2.4) of this relation s times

leads to the recurrenceL0 = Lexp(A,E) (2.5)

3 New Method to Compute Matrix Exponential and Its

Frechet Derivative

Now we introduce a more effective and convenience function instead of Pade approximation.

H(A) = A coth(A) = Ae2A + I

e2A I(3.1)

H(A) [3] is a matrix function that depends on only even power of A and radius of A satisfies

(A) , the continued fraction expansion of H(A) is very simple.

H(A) = I+A2

3I+A2

5I+A2

7I+

(3.2)

Truncating the first m terms ofHm(A), we get

Hm(A) = I+A2

3I+

A2

5I+A2

+A2

(2m + 1)I

(3.3)


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and it satisfies that H(A) Hm(A) = o(A2m+2).

From expression (3.1) we get

e2A =H(A) + A

H(A) A (3.4)

In (3.4) we mark the numerator and denominator as p(A), q(A) respectively, then p(A) = q(A).u(A), v(A) stands for the even, odd power of A in p(A) respectively,

we get p(A) = u(A) + v(A), q(A) = u(A) v(A) and define W(A) =p(A)

q(A)

(u(A) v(A))W(A) = u(A) + v(A) (3.5)

applying product rule of Frechet Derivative to the equation, we can get equations as follow:

(u(A) v(A))LW(A,E) = Lu(A,E) + Lv(A,E) + (Lv(A,E) Lu(A,E))W(A) (3.6)

Through solving matrix equations (3.5), (3.6), we can get W(A) and LW(A,E).Now we use LWm to approximate Lexp and Wm(2

s1A) to approximate e2sA, so replace Lexp

with LWm and e2sA by Wm(2

s1A) for approximation. Li.

Xs = Wm(2s1A), (3.7)

Ls = LWm(2s1A, 2s1E) (3.8)

Li1 = XiLi + LiXi

Xi1

= Xi

2

}i = s : 1 : 1 (3.9)

In this article, we just take the computation of multiplications into our considerations. In

this method, the approximation of h(x) is in (3.3) and its computation is 2n3 +8

3n3 (m 1).

The approximation of exp(x) is in step (3.4) and its computation is8

3n3. According to the

principle that the overhead of evaluating the Frechet Derivative is at most twice of the cost forapproximation for exp(x) [2]. Therefore in the scaling process, the total computation is at most

3 (2n3 +8

3n3 m). In the squaring process in (3.9), the total computation is 3 2n3 s.Then the

total computation for approximate exp(x) and its Frechet Derivative is at most 3 (2n3 +8

3n3

m) + 3 2n3 s. Comparing with Nick Higham method, this method carries out the same targetwith less computation and more convenience.

4 Backward Error Analysis

How to choose the appropriate s and measure the approximate error, theory below is used toidentify those questions.This article ignores the rounding errors and pays attention to errors caused by approximation, itappears when H

mapproximates H, finally it affects the W

mto approximation to matrix expo-

nential.We denote log as the principal logarithm ofA, which has no eigenvalues on R. log is the uniquelogarithm whose eigenvalues all have imaginary parts in (, ).


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C. Li et al. /Journal of Information & Computational Science 7: 9 (2010) 18541859 1858

Theorem 1 Suppose thateAWm(

A

2) I

< 1,A < min{

t : qm(

t

2) = 0} for some con-

sistent matrix norm. So that gm(A) = log(eAWm(A

2 )) is guaranteed to exist. ThenWm(

A

2 ) =

eA+gm(A) and gm(A) log(1

eAWm(A

2) I

) (4.1)

Proof gm(A) = log(eAWm(

A

2)) = log(I+ (eAWm(

A

2) I))

Define G := eAWm(A

2) I

gm(A) =

log(I+ G)

j=1

Gj

j= log(1

G) = log(1

eAWm(A

2) I

)

In particular that

Wm(2s1A) = e2

sA+gm(2sA) (4.2)

Combining with (4.1), we get

2sgm(2sA)

A log(1

e2sAWm(2

sA) I

)

2sA(4.3)

Set expression (4.3) less than an appropriate precision such as 253 1.1 1016, then we get s.In our method, from expression (3.8), (4.2), we have

Ls = LWm(2s1A, 2s1E) = Lexp(2

sA + gm(2sA), 2sE+ Lgm(2

sA, 2sE)) (4.5)

Substitute expression (3.7), (4.2), (4.5) into (3.9) when i = s,

Ls1 = Wm(2s1A)Ls + LsWm(2

s1A)

= e2sA+gm(2sA)

Lexp(2s

A + gm(2s

A), 2s

E+ Lgm(2s

A, 2s

E))+Lexp(2

sA + gm(2sA), 2sELgm(2

sA, 2sE))e2sA+gm(2sA)

= Lexp(2(s1)A + 2gm(2

sA), 2(s1)E+ Lgm(2sA, 2(s1)E))

(4.6)

and

Xi = Xs2si = (e2

sA+gm(2sA))2si

= e2iA+2sigm(2sA) (4.7)

Iterate the recurrence (3.9) for s times, we finally get the backward error expression of our method

L0 = Lexp(A + 2sgm(2

sA), E+ Lgm(2sA,E)) (4.8)

Algorithm:

(1) Choose appropriate m for Hm(A) approximates H(A).


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(2) Set minimum integer s to satisfies

2sgm(2sA)

A (reserved bound).

(3) Use rational expression Hm(2s1A) to approximate the continued fraction H(2s1A).

(4) Evaluation ofe2sA by function ofHm(2

s1A).

(5) Solve the matrix equation (3.5) and (3.6) to get Wm(2s1A) and LWm(2

s1A, 2s1E)marked as Xs and Ls.

(6) Iterate with the recurrence (3.9) until L0.

References

[1] Nicholas J. Higham, Functions of Matrices Theoty and Computation, SIAM, Philadelphia, 2008

[2] Awad H. Al-Mohy and Nicholas J.Higham, Computing the Frechrt Derivative of the Matrix ex-ponential with an application to Condition Number Estimation, SIAM J. Matrix Anal.Appl. Vol.30, No. 4, (2009) 1639-1657

[3] Igor Najfeld and Timothy F. Havel, Derivatives of the Matrix Exponential and Their Computation,Advances in applied mathematics 16, (1995) 321-375

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Chunjing-Sheng-Wang an Effective Method to Compute Frechet Derivative of Matrix Exponential and Its Error Analys