how to write fraction function in complex plane by using matlab

Hi

I worked in ploting function defined in complex plane by using matlab I know we can write the polynomial in matlab by using its coefficients

for example f(z)= (2-8i)Z^3+(2+3i)Z^2 + (1+5i)Z+ (3-4i) this equation can be written in matlab as p=[(2-8i) (2+3i) (1+5i) (3-4i) ],. But when we find this equation f(z)= (2-8i)/Z^3+(2+3i)/Z^2 + (1+5i)/Z+ (3-4i) how can I write this equation in matlab? I mean here the complex variables in the Denominator

Thanks

[Benjamin Thompson]

MATLAB supports complex numbers. See documentation on the topic "complex". Are you trying to write the equation symbolically using the Symbolic Toolbox, or numerically to calculate answers at specific values of Z or to plot over a range of the complex plane?

Answer 1

[John D'Errico]

MATLAB has a simple form to allow you to store only the coefficients of a polynomial. You aready know that. But there is no simple form to allow you to store some arbitrary function. In this case, you wish to store the coefficients of a function that uses inverse powers of a variable Z. Luckily, you can cheat, just a bit. That is, if you have the coefficients as a vector as you do here,

(2-8i)/Z^3+(2+3i)/Z^2 + (1+5i)/Z+ (3-4i)

So you have just the vector:

p = [(2-8i)  (2+3i)   (1+5i)   (3-4i)];

Then you can evaluate it using polyval as:

z = 1:3;
polyval(p,1./z)

ans = 
   8.0000 - 4.0000i   4.2500 - 1.7500i   3.6296 - 2.2963i

That is, we just think of this as a simple polynomial in powers of 1/z, so compute the inverse of z, THEN use polyval on those values.

[Aisha Mohamed]

Hi John

Thank you very much.

I am trying to understand your steps I think you ignored two important things

Firstly- I work in a whole complex plane not only z = 1:3;

Secondly, do you think that using polyval in this example makes sense as I know polyval gives the value of the polynomial at a specific point and it is more suitable for using to check the value of roots?

My question is How to write the polynomial in(1/z) by using Matlable make sence as I know polyval gives the value of polynomial at specfic point and it more suitable for using to check the value of roots.

My question is How to write polynomial in(1/z) by using matlab.

Could you please give me more help?

[Benjamin Thompson]

What are you trying to do with this complex valued function? Do you want a plot over a range of the complex plane, or trying to define the function symbolically to use it in some other way?

[Aisha Mohamed]

I am trying to find the zeros of this complex valued function and plot these zeros in the complex plane

[John D'Errico]

I did not ignore two things. I answered the question you asked. You did NOT ask how to find the zeros of a function. You asked how to write what is effectively a polynomial in 1/Z.

My choice of 1:3 is irrelevant, I picked it as an example.

You then say that polyval may not be appropriate. Why not? It evaluates the polynomial that you give it. For example, you gave these coefficients.

format long g
p = [(2-8i)  (2+3i)   (1+5i)   (3-4i)];
z = 1 + 3i;
polyval(p,1./z)

ans = 
                      4.712 -                 3.916i

Is that the correct value? Test it.

sum(p.*z.^(-3:1:0))

ans = 
                      4.712 -                 3.916i

Note that the above line of code applies ONLY for scalar values of z. However it clearly agrees with polyval.

As far as zeroes goes, please show me where in your question where you EXPLICITLY said even a single word about the zeros of this function? Here is the statement of your question, just in case you forgot:

"I worked in ploting function defined in complex plane by using matlab I know we can write the polynomial in matlab by using its coefficients

for example f(z)= (2-8i)Z^3+(2+3i)Z^2 + (1+5i)Z+ (3-4i) this equation can be written in matlab as p=[(2-8i) (2+3i) (1+5i) (3-4i) ],. But when we find this equation f(z)= (2-8i)/Z^3+(2+3i)/Z^2 + (1+5i)/Z+ (3-4i) how can I write this equation in matlab? I mean here the complex variables in the Denominator:"

That was your exact statement. Yes, you finally added the word zeros in a final comment. Sorry for expressing irritation, but should I be able to read your mind? SIGH. Maybe you were thinking about zeros. But don't tell me that I did not answer the question you asked.

Ok. You want to find the zeros of this function? Still trivial.

proots = 1./roots(p)

proots = 
          -1.05497409102958 +     0.115399209863617i
          0.705937495228178 +     0.539274174004756i
            1.0290365958014 -      1.41467338386837i

Now evaluate the function at those points. I will use polyval.

polyval(p,1./proots)

ans = 
      -8.88178419700125e-15 +  1.06581410364015e-14i
       2.39808173319034e-14 -  1.02140518265514e-14i
       8.88178419700125e-16 +  8.88178419700125e-16i

So zero, to within floating point trash. proots contains the zeros of the indicated polynomial in powers of 1/z.

[Aisha Mohamed]

Hi John

Sorry for bothering you, it was nice explaination, that encorge me to ask more help please,

I work in both function f(z) and f(1/z) and the coefficiens of both change with time t. and z is complex number.

For example time = 0:0.01:2*pi at every time t I have to find f(z) and f(1/z) and plot them in order to find the graphs of these functions.

This is what I did:

I used (for loop )to calculate the coefficient at every time ( time = 0:0.01:2*pi ) and the coefficients of f(z) as well. Then at each time t and its corresponding coefficients I plot p=f(z) and polyval(p,1./z)[by using your idea (Thank you very much)]but I have to choose spicific value of z to plot polyval(p,1./z) .

The problem is how to check these graphs are correct? and how can me plot both function in whole complex plane(arbitrary values of z)

Thank you very much

Answer 2

[Sphiwe]

num = [1]

den= conv([1 3 5], [1 3], [1 5])

g = tf (num,den)

[z,p,k] = tf2zp(num,den)

pzmap(g)