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I have a quadratic matrix equation

$$ \mathbf{C}\mathbf{S}\mathbf{C}+\mathbf{C}+\mathbf{A}=\mathbf{0} $$

Where $\mathbf{C}$ is a covariance matrix (symmetric, positive semi definite), $\mathbf{S}$ is a singular matrix and $\mathbf{A}$ is some matrix of the form $\mathbf{A}=a*a^{T}$ (Product of a column vector with its transpose). I need to iteratively find $\mathbf{C}$

1. If I try to set up the problem using Bernoulli iteration, in order to find the dominant solvent I would have to go for

$$ \mathbf{S}+\{\mathbf{I}+\mathbf{C}_{k}^{-1}\mathbf{A}\}*\mathbf{C}_{k+1}^{-1} = \mathbf{0} $$

$$ \Longrightarrow \mathbf{C}_{k+1}^{-1} = \{\mathbf{I}+\mathbf{C}_{k}^{-1}\mathbf{A}\}^{-1}*(-\mathbf{S}) $$

Which won't make sense since $\mathbf{S}$ is singular

2. If I try to to find the minimal solvent, I set it up as

$$ (\mathbf{C}_{k}\mathbf{S}+\mathbf{I})*\mathbf{C}_{k+1}+\mathbf{A}=\mathbf{0} $$

$$ \Longrightarrow \mathbf{C}_{k+1}=(\mathbf{C}_{k}\mathbf{S}+\mathbf{I})^{-1}*(-\mathbf{A}) $$

But this doesn't seem to be working either.

3. A last, non-Bernoulli iteration I set up was

$$ \mathbf{C}_{k}\mathbf{S}\mathbf{C}_{k}+\mathbf{C}_{k+1}+\mathbf{A}=\mathbf{0} $$

$$ \Longrightarrow \mathbf{C}_{k+1}=-\mathbf{A}-\mathbf{C}_{k}\mathbf{S}\mathbf{C}_{k} $$

Which seems to be working but I'm not sure which root it converges to. Also the final estimate $\mathbf{C}_{k}$ converges to, somehow appears to be twice the actual value (upon performing a ground truth test).

My queries are

1- Am I on the right track on dealing with this problem? If so, where am I making mistakes?

2- Is there any procedure to find the dominant and minimal solvent when the coefficient of the square term is singular?

3- Is the last, non-Bernoulli iteration valid? If so, why is the result twice the actual one? Which root does it converge to?

Any guidance will be highly appreciated. Thank you.

I have the following quadratic matrix equation in symmetric positive semidefinite covariance matrix $\mathbf{C}$

$$\mathbf{C}\mathbf{S}\mathbf{C}+\mathbf{C}+\mathbf{A}=\mathbf{0}$$

where $\mathbf{S}$ is a singular matrix and $\mathbf{A} = \mathbf{a} \mathbf{a}^{T}$. I need to iteratively find $\mathbf{C}$.

1. If I try to set up the problem using Bernoulli iteration, in order to find the dominant solvent I would have to go for

$$ \mathbf{S}+\{\mathbf{I}+\mathbf{C}_{k}^{-1}\mathbf{A}\}*\mathbf{C}_{k+1}^{-1} = \mathbf{0} $$

$$ \Longrightarrow \mathbf{C}_{k+1}^{-1} = \{\mathbf{I}+\mathbf{C}_{k}^{-1}\mathbf{A}\}^{-1}*(-\mathbf{S}) $$

Which won't make sense since $\mathbf{S}$ is singular

2. If I try to to find the minimal solvent, I set it up as

$$ (\mathbf{C}_{k}\mathbf{S}+\mathbf{I})*\mathbf{C}_{k+1}+\mathbf{A}=\mathbf{0} $$

$$ \Longrightarrow \mathbf{C}_{k+1}=(\mathbf{C}_{k}\mathbf{S}+\mathbf{I})^{-1}*(-\mathbf{A}) $$

But this doesn't seem to be working either.

3. A last, non-Bernoulli iteration I set up was

$$ \mathbf{C}_{k}\mathbf{S}\mathbf{C}_{k}+\mathbf{C}_{k+1}+\mathbf{A}=\mathbf{0} $$

$$ \Longrightarrow \mathbf{C}_{k+1}=-\mathbf{A}-\mathbf{C}_{k}\mathbf{S}\mathbf{C}_{k} $$

Which seems to be working but I'm not sure which root it converges to. Also the final estimate $\mathbf{C}_{k}$ converges to, somehow appears to be twice the actual value (upon performing a ground truth test).

My queries are

1. Am I on the right track on dealing with this problem? If so, where am I making mistakes?

2. Is there any procedure to find the dominant and minimal solvent when the coefficient of the square term is singular?

3. Is the last, non-Bernoulli iteration valid? If so, why is the result twice the actual one? Which root does it converge to?

Any guidance will be highly appreciated. Thank you.

I have a quadratic matrix equation

$$ \mathbf{C}\mathbf{S}\mathbf{C}+\mathbf{C}+\mathbf{A}=\mathbf{0} $$

Where $\mathbf{C}$ is a covariance matrix (symmetric, positive semi definite), $\mathbf{S}$ is a singular matrix and $\mathbf{A}$ is some matrix of the form $\mathbf{A}=a*a^{T}$ (Product of a column vector with its transpose). I need to iteratively find $\mathbf{C}$

1. If I try to set up the problem using Bernoulli iteration, in order to find the dominant solvent I would have to go for

$$ \mathbf{S}+\{\mathbf{I}+\mathbf{C}_{k}^{-1}\mathbf{S}\}*\mathbf{C}_{k+1}^{-1} = \mathbf{0} $$

$$ \Longrightarrow \mathbf{C}_{k+1}^{-1} = \{\mathbf{I}+\mathbf{C}_{k}^{-1}\mathbf{S}\}^{-1}*(-\mathbf{S}) $$

Which won't make sense since $\mathbf{S}$ is singular

2. If I try to to find the minimal solvent, I set it up as

$$ (\mathbf{C}_{k}\mathbf{S}+\mathbf{I})*\mathbf{C}_{k+1}+\mathbf{A}=\mathbf{0} $$

$$ \Longrightarrow \mathbf{C}_{k+1}=(\mathbf{C}_{k}\mathbf{S}+\mathbf{I})^{-1}*(-\mathbf{A}) $$

But this doesn't seem to be working either.

3. A last, non-Bernoulli iteration I set up was

$$ \mathbf{C}_{k}\mathbf{S}\mathbf{C}_{k}+\mathbf{C}_{k+1}+\mathbf{A}=\mathbf{0} $$

$$ \Longrightarrow \mathbf{C}_{k+1}=-\mathbf{A}-\mathbf{C}_{k}\mathbf{S}\mathbf{C}_{k} $$

Which seems to be working but I'm not sure which root it converges to. Also the final estimate $\mathbf{C}_{k}$ converges to, somehow appears to be twice the actual value (upon performing a ground truth test).

My queries are

1- Am I on the right track on dealing with this problem? If so, where am I making mistakes?

2- Is there any procedure to find the dominant and minimal solvent when the coefficient of the square term is singular?

3- Is the last, non-Bernoulli iteration valid? If so, why is the result twice the actual one? Which root does it converge to?

Any guidance will be highly appreciated. Thank you.

I have a quadratic matrix equation

$$ \mathbf{C}\mathbf{S}\mathbf{C}+\mathbf{C}+\mathbf{A}=\mathbf{0} $$

Where $\mathbf{C}$ is a covariance matrix (symmetric, positive semi definite), $\mathbf{S}$ is a singular matrix and $\mathbf{A}$ is some matrix of the form $\mathbf{A}=a*a^{T}$ (Product of a column vector with its transpose). I need to iteratively find $\mathbf{C}$

1. If I try to set up the problem using Bernoulli iteration, in order to find the dominant solvent I would have to go for

$$ \mathbf{S}+\{\mathbf{I}+\mathbf{C}_{k}^{-1}\mathbf{A}\}*\mathbf{C}_{k+1}^{-1} = \mathbf{0} $$

$$ \Longrightarrow \mathbf{C}_{k+1}^{-1} = \{\mathbf{I}+\mathbf{C}_{k}^{-1}\mathbf{A}\}^{-1}*(-\mathbf{S}) $$

Which won't make sense since $\mathbf{S}$ is singular

2. If I try to to find the minimal solvent, I set it up as

$$ (\mathbf{C}_{k}\mathbf{S}+\mathbf{I})*\mathbf{C}_{k+1}+\mathbf{A}=\mathbf{0} $$

$$ \Longrightarrow \mathbf{C}_{k+1}=(\mathbf{C}_{k}\mathbf{S}+\mathbf{I})^{-1}*(-\mathbf{A}) $$

But this doesn't seem to be working either.

3. A last, non-Bernoulli iteration I set up was

$$ \mathbf{C}_{k}\mathbf{S}\mathbf{C}_{k}+\mathbf{C}_{k+1}+\mathbf{A}=\mathbf{0} $$

$$ \Longrightarrow \mathbf{C}_{k+1}=-\mathbf{A}-\mathbf{C}_{k}\mathbf{S}\mathbf{C}_{k} $$

Which seems to be working but I'm not sure which root it converges to. Also the final estimate $\mathbf{C}_{k}$ converges to, somehow appears to be twice the actual value (upon performing a ground truth test).

My queries are

1- Am I on the right track on dealing with this problem? If so, where am I making mistakes?

2- Is there any procedure to find the dominant and minimal solvent when the coefficient of the square term is singular?

3- Is the last, non-Bernoulli iteration valid? If so, why is the result twice the actual one? Which root does it converge to?

Any guidance will be highly appreciated. Thank you.

I have a quadratic matrix equation

CSC+C+A=0

Where C is a covariance matrix (symmetric, positive semi definite), S is a singular matrix and A is some matrix of the form A=a*a' (Product of a column vector with its transpose). I need to iteratively find C

1. If I try to set up the problem using Bernoulli iteration, in order to find the dominant solvent I would have to go for

S+{I+inv(C|k|)*A}*inv(C|k+1|) = 0

=> inv(C|k+1|) = -inv[{I+inv(C|k|)*A}]*S

=> C|k+1| = inv[-inv[{I+inv(C|k|)*A}]*S]

Which won't make sense since S is singular

2. If I try to to find the minimal solvent, I set it up as

(C|k|S+I)*C|k+1|+A = 0

=> C|k+1| = inv{C|k|S+I}*(-A)

But this doesn't seem to be working as well.

3. A last, non-Bernoulli iteration I set up was

C|k|SC|k|+C|k+1|+A = 0

=>C|k+1| = -A-C|k|SC|k|

Which seems to be working but I'm not sure which root it converges to. Also the final estimate C|k| converges to, somehow appears to be twice the actual value (upon performing a ground truth test).

My queries are

1- Am I on the right track on dealing with this problem? If so, where am I making mistakes?

2- Is there any procedure to find the dominant and minimal solvent when the coefficient of the square term is singular?

3- Is the last, non-Bernoulli iteration valid? If so, why is the result twice the actual one? Which root does it converge to?

Any guidance will be highly appreciated. Thank you.

I have a quadratic matrix equation

$$ \mathbf{C}\mathbf{S}\mathbf{C}+\mathbf{C}+\mathbf{A}=\mathbf{0} $$

Where $\mathbf{C}$ is a covariance matrix (symmetric, positive semi definite), $\mathbf{S}$ is a singular matrix and $\mathbf{A}$ is some matrix of the form $\mathbf{A}=a*a^{T}$ (Product of a column vector with its transpose). I need to iteratively find $\mathbf{C}$

1. If I try to set up the problem using Bernoulli iteration, in order to find the dominant solvent I would have to go for

$$ \mathbf{S}+\{\mathbf{I}+\mathbf{C}_{k}^{-1}\mathbf{S}\}*\mathbf{C}_{k+1}^{-1} = \mathbf{0} $$

$$ \Longrightarrow \mathbf{C}_{k+1}^{-1} = \{\mathbf{I}+\mathbf{C}_{k}^{-1}\mathbf{S}\}^{-1}*(-\mathbf{S}) $$

Which won't make sense since $\mathbf{S}$ is singular

2. If I try to to find the minimal solvent, I set it up as

$$ (\mathbf{C}_{k}\mathbf{S}+\mathbf{I})*\mathbf{C}_{k+1}+\mathbf{A}=\mathbf{0} $$

$$ \Longrightarrow \mathbf{C}_{k+1}=(\mathbf{C}_{k}\mathbf{S}+\mathbf{I})^{-1}*(-\mathbf{A}) $$

But this doesn't seem to be working either.

3. A last, non-Bernoulli iteration I set up was

$$ \mathbf{C}_{k}\mathbf{S}\mathbf{C}_{k}+\mathbf{C}_{k+1}+\mathbf{A}=\mathbf{0} $$

$$ \Longrightarrow \mathbf{C}_{k+1}=-\mathbf{A}-\mathbf{C}_{k}\mathbf{S}\mathbf{C}_{k} $$

Which seems to be working but I'm not sure which root it converges to. Also the final estimate $\mathbf{C}_{k}$ converges to, somehow appears to be twice the actual value (upon performing a ground truth test).

My queries are

1- Am I on the right track on dealing with this problem? If so, where am I making mistakes?

2- Is there any procedure to find the dominant and minimal solvent when the coefficient of the square term is singular?

3- Is the last, non-Bernoulli iteration valid? If so, why is the result twice the actual one? Which root does it converge to?

Any guidance will be highly appreciated. Thank you.