non-linear equations Jacobian

I have the following system of non-linear equations:

f(x)={(x(1)^5 + x(2)^3*x(3)^4 + 1)

      (x(1)^2*x(2)*x(3))

      (X(3)^4 - 1)}

My problem says: Compute the Jacobian, J(x) (Note that the Jacobian is singular for x(3)=0)

So far, I have used:

syms x y z jacobian([x^5 + y^3*z^4 + 1; x^2*y*z; z^4-1],[x;y;z])

and a have following matrix

ans =

[ 5*x^4, 3*y^2*z^4, 4*y^3*z^3]

[ 2*x*y*z, x^2*z, x^2*y]

[ 0, 0, 4*z^3]

Now, what is my next step to compute Jacobian?

Answer 1

[Torsten]

The matrix you got is the Jacobian.

You just have to replace x by x(1), y by x(2) and z by x(3).

Best wishes

Torsten.

[Ana Garvanlieva]

Thank you for your responce... I thought there was more... i will replace them, but how can i note that Jacobian is singular for x(3)=0.... According to what?

And my next question is how can I solve this:

Consider two initial quesses: 1) x0=(-.01, -.01, -.01)T and 2) x0=(.1, .1, .1)T

AND Compute both |J(x0)^-1|

I will appreciate your help Torsten, i am begginer in Matlab...

[Torsten]

1. The third row of the Jacobian becomes zero if you insert x(3)=0. This means that the Jacobian becomes singular if x(3)=0.

2. Insert x(1)=-0.01, x(2)=-0.01, x(3)=-0.01 and let MATLAB calculate inv(J)

(if |J(x0)^(-1)| means "determinant of J", then let MATLAB calculate det(inv(J)) for you).

Best wishes

Torsten.

[Ana Garvanlieva]

So i Insert theese comands

>> syms x1 x2 x3

>> J1 = jacobian([x1^5 + x2^3*x3^4 + 1; x1^2*x2*x3; x3^4-1],[x1;x2;x3])

>> x3=0

How to call the matrix now?

When I call it with

>> J1 = jacobian([x1^5 + x2^3*x3^4 + 1; x1^2*x2*x3; x3^4-1],[x1;x2;x3])

i get

J1 =

[ 5*x1^4, 0, diff(x1^5 + 1, 0)]

[ 0, 0, diff(0, 0)]

[ 0, 0, diff(-1, 0)]

And why do I need inv(J) for x0=(-.01, -.01, -.01)^T???

Thank you verry much for your responces...

[Ana Garvanlieva]

@Torsten, can you help me?

[Torsten]

Use MATLAB's "subs"-command:

syms x1 x2 x3
J1 = jacobian([x1^5 + x2^3*x3^4 + 1; x1^2*x2*x3; x3^4-1],[x1;x2;x3]);
J2 = subs(J1,x3,0);
J3 = subs(J1,[x1,x2,x3],[-.01, -.01, -.01]);
J4 = subs(J1,[x1,x2,x3],[.1, .1, .1]);
val1=1/det(J3);   %calculates |J(x0)^-1| for the first initial guess
val2=1/det(J4);   %calculates |J(x0)^-1| for the second initial guess

Best wishes

Torsten.

[Ana Garvanlieva]

@Torsten Do you know, and can you help me with the code for the Newton method. As help I have instructions to note some difficulties with convergence and "As a remedy implement a damped Newton modification using the Armijo-Goldstein criterion."