Problem about RootOf in Singular Value Decomposition

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Erdem 2014-7-12

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https://ww2.mathworks.cn/matlabcentral/answers/141567-problem-about-rootof-in-singular-value-decomposition

评论： Star Strider 2014-7-13

Hello,

I have a problem about getting rid of the RootOf function that appears as a result of a singular value decomposition (SVD). I'd appreciate your help.

I obtain a vector of singular values of a 4x4 matrix L by "sigma=svd(L)". L has only one variable, which is "kx". The vector of singular values is given below. What Matlab does is to add an additional variable "z1" to the result. As a result, I have two variables, kx and z1. My question is how I can obtain a vector of singular values without a "z1" variable?

In addition, there is a "[1]^(1/2)" term at the end of each row. What does that point out?

The result of sigma=svd(L) is:

sigma=
RootOf(256*kx^2*z1^6*conj((- kx^2 + 9/4)^(1/2))^2)[1]^(1/2)
RootOf(256*kx^2*z1^6*conj((- kx^2 + 9/8)^(1/2))^2)[1]^(1/2)
RootOf(256*kx^2*z1^6*conj((- kx^2 + 9/16)^(1/2))^2)[1]^(1/2)
RootOf(256*kx^2*z1^6*conj((- kx^2 + 9/24)^(1/2))^2)[1]^(1/2)

Thanks in advance,

Erdem

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Answer 1

Star Strider 2014-7-12

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https://ww2.mathworks.cn/matlabcentral/answers/141567-problem-about-rootof-in-singular-value-decomposition#answer_144811

The z1 is a dummy variable the Symbolic Math Toolbox inserts in some situations. Without seeing your code, it is difficult to say what it could mean. The RootOf may have to do with the symbolic engine assuming that kx could be complex. Declaring it real in your syms or sym statement could clear that up. No guarantees.

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Star Strider 2014-7-12

编辑：Star Strider 2014-7-12

在 MATLAB Online 中打开

My pleasure!

Using fminsearch should be relatively easy. In fact, the Symbolic Math Toolbox function matlabFunction will convert your matrix to an anonmymous function for you:

Erdem = @(kx) reshape([0.0,1.0,1.0./sqrt(-kx.^2+1.0./4.0).*(-9.0./4.0),0.0,0.0,0.0,1.0,1.0,1.0./sqrt(-kx.^2+1.0./4.0).*(9.0./4.0),0.0,0.0,0.0,0.0,-1.0,1.0./sqrt(-kx.^2+1.0./4.0).*(9.0./4.0),exp(sqrt(-kx.^2+1.0./4.0).*5.0e-8),exp(sqrt(-kx.^2+1.0./4.0).*5.0e-8).*1.0./sqrt(-kx.^2+1.0./4.0).*(-9.0./4.0),0.0,0.0,-1.0,1.0./sqrt(-kx.^2+1.0./4.0).*(-9.0./4.0),exp(sqrt(-kx.^2+1.0./4.0).*(-5.0e-8)),exp(sqrt(-kx.^2+1.0./4.0).*(-5.0e-8)).*1.0./sqrt(-kx.^2+1.0./4.0).*(9.0./4.0),0.0,0.0,0.0,0.0,-exp(sqrt(-kx.^2+1.0./4.0).*5.0e-8),exp(sqrt(-kx.^2+1.0./4.0).*5.0e-8).*1.0./sqrt(-kx.^2+1.0./4.0).*(9.0./4.0),1.0,0.0,0.0,0.0,-exp(sqrt(-kx.^2+1.0./4.0).*(-5.0e-8)),exp(sqrt(-kx.^2+1.0./4.0).*(-5.0e-8)).*1.0./sqrt(-kx.^2+1.0./4.0).*(-9.0./4.0),0.0],[6,6]);
Q = Erdem(1)        % Test at kx = 1

I named it Erdem, name it whatever you want. (I used the Q assignment as a test to be sure it returns a matrix.) All you need to do is to create an anonymous function using it as an argument to svd and you can then use it with fminsearch. The optimisation functions want a single scalar output from the argument function, so since svd returns a vector of singular values, I’ll leave it to you to decide how you want to write that function. If you are matching it to a known vector and you want to solve for a value of kx that creates that vector, use the norm function on the difference. (I thank Matt J for that valuable insight.)

Star Strider 2014-7-12

编辑：Star Strider 2014-7-12

在 MATLAB Online 中打开

My pleasure!

This is my idea:

T = rand(6,1)*10;                           % Desired SVD of the matrix
ObjFn = @(kx) norm(T - svd(Erdem(kx)));     % Objective Function for ‘fminsearch’
[kx, fval] = fminsearch(ObjFn, 1)           % Do the fit
Chk = svd(Erdem(kx))                        % Check the result

When I ran it it worked, but didn’t converge on the actual value of T. When I plotted it, I couldn’t find a value of kx that resulted in a zero-crossing, so you will likely have to experiment with it to get it to produce the result you want, including complex-valued kx. (If you have access to the Optimization Toolbox, you may want to see if other functions give better results.) You can also experiment with optimset to tighten the tolerances on fminsearch and otherwise alter its behaviour. (If the optimisation is successful, fval should be close to zero.) I’ve never attempted to do curve-fitting (essentially what this is) with the svd function, so this is a first for us all.

EDIT — Forgot to mention norm. The norm function calculates the euclidean distance between T and the result of the svd. It returns a scalar, which is what fminsearch wants. I’ve used it before to ‘fit’ vector-valued functions in this sort of situation. (To give credit where credit is due, Matt J enlightened me to its use in these situations.)

Erdem 2014-7-13

Thank you so much. I achieved to obtain a converging solution. There are some problems in the solution, but these can be discussed under a different thread. I'll open a different thread for these.

Erdem