MATLAB Central Newsreader - problem with complex numbers

problem with complex numbers

Yuriy — Tue, 08 Jan 2008 23:30:10 -0500

can anybody explain me why this code gives different results
for F1 and F2

clear all
A=5*rand(5,5);

z=2.5;

F1=exp(-j*A).^z;
F2=exp(-j*A*z);

Re: problem with complex numbers

James Tursa — Wed, 09 Jan 2008 01:26:02 -0500

"Yuriy " <aaa@yahoo.com> wrote in message <fm1121
$9dg$1@fred.mathworks.com>...
> can anybody explain me why this code gives different
results
> for F1 and F2
>
> clear all
> A=5*rand(5,5);
>
> z=2.5;
>
> F1=exp(-j*A).^z;
> F2=exp(-j*A*z);

Raising complex numbers to powers is a multi-valued
operation. Depending on how you formulate the calculation
you can get different results, all valid. For example,
consider the very simple calculation of the square root
of -1. MATLAB will give you one answer, i, by default when
you do sqrt(-1), but an equally valid answer is -i.
Depending on how you do the calculation you can get MATLAB
to give you the other answer as well. Anytime you raise
complex numbers to powers you need to be careful. Other
functions are multi-valued for complex arguments also.

James Tursa

Re: problem with complex numbers

Roger Stafford — Wed, 09 Jan 2008 02:08:02 -0500

"Yuriy " <aaa@yahoo.com> wrote in message <fm1121$9dg
$1@fred.mathworks.com>...
> can anybody explain me why this code gives different results
> for F1 and F2
>
> clear all
> A=5*rand(5,5);
>
> z=2.5;
>
> F1=exp(-j*A).^z;
> F2=exp(-j*A*z);
-------
In the mathematics of complex numbers it is understood that the quantity
w^z, where w is complex-valued, must satisfy the identity

w^z = exp(z*log(w)).

It is this logarithm function that gives rise to the ambiguity of possible values
for w^z.

  If one travels around the zero value of w in a counterclockwise circular
direction, keeping log(w) a continuous function, it will be increased by 2*pi*j
after each complete revolution. Thus the logarithm function is regarded as
an analytic function but with infinitely many branches, and it has a
discontinuity at w = 0.

  The effect on w^2.5 is to reverse sign after each revolution. This is the
cause of the sign reversals you see between F1 and F2. Matlab follows the
convention that the imaginary part of the logarithm is restricted to the
interval from -pi*j to +pi*j, and this leads to the difference between F1 and
F2. If you had selected z = 2.75, you would see four different kinds of
inequalities occurring between F1 and F2. They would differ by various
multiples of a pi/2 rotation.

  This is an inevitable ambiguity in the mathematics of complex numbers and
not a fault of matlab's convention. It is the same problem that prevails when
taking square roots or higher roots. The square root of 9 can be +3 or -3.
The fourth root of 16 can be any one of the quantities 2, 2*j, -2, or -2*j. All
are equally valid from the point of view of complex numbers.

Roger Stafford

Re: problem with complex numbers

Vasia — Wed, 09 Jan 2008 16:26:02 -0500

Thanks for the answer.