Exponential of an arbitrary mapping f: R^n -> R^n

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3 total messages Started by Daniel A. Asimov Fri, 28 Aug 1992 18:12

#47

Author: Daniel A. Asimov
Date: Fri, 28 Aug 1992 18:12

26 lines
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One way to generalize the exponential of a matrix to arbitrary
continuous functions f: R^n -> R^n is as follows:

exp(f): R^n -> R^n via

	exp(f)(x) = x + f(x) + f(f(x))/2! + f(f(f(x)))/3! +...

It is easy to see that this does not necessarily converge for all x.

Question:  Has this version of exp(f) been defined and discussed
previously in the literature?  Can someone supply references and/or
mention some of the elementary properties (if any) of this definition?

For example, where it exists is exp(f) continuous?  If f is differentiable,
or even smooth, then what about exp(f) ?  Are there elementary conditions 
under which exp(f) is invertible?   For which g can one find a
corresponding "log," i.e., an f such that exp(f) = g ?


--Dan Asimov
asimov@nas.nasa.gov

Mail Stop T045-1
NASA Ames Research Center
Moffett Field, CA 94035-1000
(415) 604-4799

Re: Exponential of an arbitrary mapping f: R^n -> R^n

#48

Author: Dan Sullivan
Date: Fri, 28 Aug 1992 20:40

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In article <1992Aug28.181224.14463@nas.nasa.gov> asimov@nas.nasa.gov (Daniel A. Asimov) writes:
>One way to generalize the exponential of a matrix to arbitrary

You may have these references already, but maybe not:

C.B. Moler and C.F. Van Loan (1978). "Nineteen Dubious Ways to
   Compute the Exponential of a Matrix," SIAM Review 20, 801.
Gene H. Golub and Charles F. Van Loan (1989). Matirx Computations,
   2nd ed., The Johns Hopkins University Press, Baltimore.  pp. 551-560.
B.W. Helton (1968). "Logarithms of Matrices," Proc. Amer. Math.
   Soc. 19, 733.

--Dan

---------------------------------------------------------------------------
Dan Sullivan | +1 919 515 7991 | Dan_Sullivan@ncsu.edu or sullivan@ncsuphys
---------------------------------------------------------------------------

Re: Exponential of an arbitrary mapping f: R^n -> R^n

#57

Author: Paul R. Chernoff
Date: Mon, 07 Sep 1992 20:40

31 lines
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In article <1992Aug28.181224.14463@nas.nasa.gov> asimov@nas.nasa.gov (Daniel A. Asimov) writes:
>One way to generalize the exponential of a matrix to arbitrary
>continuous functions f: R^n -> R^n is as follows:
>
>exp(f): R^n -> R^n via
>
>	exp(f)(x) = x + f(x) + f(f(x))/2! + f(f(f(x)))/3! +...
>

----------------------------------------
Comment: this is an unusual generalization of the exponential to nonlinear 
functions f.  I'd be curious as to its motivation.

The usual idea, of course, is to regard exp(tA)x, where A is a linear map,
as the solution of the differential equation
              dy/dt = Ay(t),  y(0) = x.

The standard generalization is the solution of
              dy/dt = f(y(t)), y(0) = x.

There is no "explicit" form of the solution, but there is a highly developed
theory, e.g. the theory of nonlinear semigroups.

----


-- 
# Paul R. Chernoff                       chernoff@math.berkeley.edu     #
# Department of Mathematics              ucbvax!math!chernoff           #
# University of California               chernoff%math@ucbvax.bitnet    #
# Berkeley, CA  94720                                                   #

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