Solving a non-linear system where x appears as matrix and vector
I have the following problem. I am trying to solve the following non-linear system of equations in MATLAB
$mathbf{b}=mathrm{diag}left(mathbf{x}right)left(mathbf{I}+mathbf{A'A}right)^{-1}left(mathbf{A'1}-mathbf{x}right)$
where $mathbf{b}$ (known) and $mathbf{x}$ (to be found) are $Ntimes1$ column vectors, $mathbf{A'A}$ (known) is a $Ntimes N$ cosine similarity matrix (symmetric positive definite). $mathbf{I}$ is the identity matrix and $mathbf{1}$ is a $Ntimes1$ vector of ones.
The obvious reference would seem this one:
Solving Non Linear System of Equations with MATLAB
the problem is I'm not sure how to take the Jacobian. It seems to me that $mathrm{diag}(mathbf{x})$ and $(mathbf{I+A'A})^{-1}$ should commute, so perhaps using that
could simplify computation of the Jacobian.
But perhaps I'm wrong and that's not the way to go. Perhaps another possibility would be to write
$mathbf{b}=mathrm{diag}left(mathbf{x_{t}}right)left(mathbf{I}+mathbf{A'A}right)^{-1}left(mathbf{A'1}-mathbf{x_{t+1}}right)$
make a guess for $mathbf{x}_1$ and iterate till convergence.
Any ideas?
Thanks in advance!
linear-algebra systems-of-equations matlab matrix-equations nonlinear-system
asked Nov 23 '18 at 14:19
br1br1
85
add a comment |
I have the following problem. I am trying to solve the following non-linear system of equations in MATLAB
$mathbf{b}=mathrm{diag}left(mathbf{x}right)left(mathbf{I}+mathbf{A'A}right)^{-1}left(mathbf{A'1}-mathbf{x}right)$
where $mathbf{b}$ (known) and $mathbf{x}$ (to be found) are $Ntimes1$ column vectors, $mathbf{A'A}$ (known) is a $Ntimes N$ cosine similarity matrix (symmetric positive definite). $mathbf{I}$ is the identity matrix and $mathbf{1}$ is a $Ntimes1$ vector of ones.
The obvious reference would seem this one:
Solving Non Linear System of Equations with MATLAB
the problem is I'm not sure how to take the Jacobian. It seems to me that $mathrm{diag}(mathbf{x})$ and $(mathbf{I+A'A})^{-1}$ should commute, so perhaps using that
could simplify computation of the Jacobian.
But perhaps I'm wrong and that's not the way to go. Perhaps another possibility would be to write
$mathbf{b}=mathrm{diag}left(mathbf{x_{t}}right)left(mathbf{I}+mathbf{A'A}right)^{-1}left(mathbf{A'1}-mathbf{x_{t+1}}right)$
make a guess for $mathbf{x}_1$ and iterate till convergence.
Any ideas?
Thanks in advance!
linear-algebra systems-of-equations matlab matrix-equations nonlinear-system
asked Nov 23 '18 at 14:19
br1br1
85
add a comment |
I have the following problem. I am trying to solve the following non-linear system of equations in MATLAB
$mathbf{b}=mathrm{diag}left(mathbf{x}right)left(mathbf{I}+mathbf{A'A}right)^{-1}left(mathbf{A'1}-mathbf{x}right)$
where $mathbf{b}$ (known) and $mathbf{x}$ (to be found) are $Ntimes1$ column vectors, $mathbf{A'A}$ (known) is a $Ntimes N$ cosine similarity matrix (symmetric positive definite). $mathbf{I}$ is the identity matrix and $mathbf{1}$ is a $Ntimes1$ vector of ones.
The obvious reference would seem this one:
Solving Non Linear System of Equations with MATLAB
the problem is I'm not sure how to take the Jacobian. It seems to me that $mathrm{diag}(mathbf{x})$ and $(mathbf{I+A'A})^{-1}$ should commute, so perhaps using that
could simplify computation of the Jacobian.
But perhaps I'm wrong and that's not the way to go. Perhaps another possibility would be to write
$mathbf{b}=mathrm{diag}left(mathbf{x_{t}}right)left(mathbf{I}+mathbf{A'A}right)^{-1}left(mathbf{A'1}-mathbf{x_{t+1}}right)$
make a guess for $mathbf{x}_1$ and iterate till convergence.
Any ideas?
Thanks in advance!
linear-algebra systems-of-equations matlab matrix-equations nonlinear-system
asked Nov 23 '18 at 14:19
br1br1
85
I have the following problem. I am trying to solve the following non-linear system of equations in MATLAB
$mathbf{b}=mathrm{diag}left(mathbf{x}right)left(mathbf{I}+mathbf{A'A}right)^{-1}left(mathbf{A'1}-mathbf{x}right)$
where $mathbf{b}$ (known) and $mathbf{x}$ (to be found) are $Ntimes1$ column vectors, $mathbf{A'A}$ (known) is a $Ntimes N$ cosine similarity matrix (symmetric positive definite). $mathbf{I}$ is the identity matrix and $mathbf{1}$ is a $Ntimes1$ vector of ones.
The obvious reference would seem this one:
Solving Non Linear System of Equations with MATLAB
the problem is I'm not sure how to take the Jacobian. It seems to me that $mathrm{diag}(mathbf{x})$ and $(mathbf{I+A'A})^{-1}$ should commute, so perhaps using that
could simplify computation of the Jacobian.
But perhaps I'm wrong and that's not the way to go. Perhaps another possibility would be to write
$mathbf{b}=mathrm{diag}left(mathbf{x_{t}}right)left(mathbf{I}+mathbf{A'A}right)^{-1}left(mathbf{A'1}-mathbf{x_{t+1}}right)$
make a guess for $mathbf{x}_1$ and iterate till convergence.
Any ideas?
Thanks in advance!
linear-algebra systems-of-equations matlab matrix-equations nonlinear-system
linear-algebra systems-of-equations matlab matrix-equations nonlinear-system
asked Nov 23 '18 at 14:19
br1br1
85
asked Nov 23 '18 at 14:19
br1br1
85
asked Nov 23 '18 at 14:19
br1br1
85
asked Nov 23 '18 at 14:19
br1br1
85
asked Nov 23 '18 at 14:19
br1br1
85
85
add a comment |
add a comment |
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