Linear system with Non-square LU factors
$begingroup$
Consider the following linear system of equations:
$$
textbf{A}textbf{x} = textbf{b}
$$
Where $textbf{x}, textbf{b} in mathbb{R}^{n}$ and $textbf{A} in mathbb{R}^{n times n}$. We also have that $textbf{A}=textbf{L}textbf{U}$ where $textbf{L} in mathbb{R}^{n times m}$ and $ textbf{U} in mathbb{R}^{m times n}$ are non-square matrices ($m > n$). $textbf{L}$ is constructed from a square lower triangular matrix $textbf{L}_0 in mathbb{R}^{m times m}$ by removing some of its rows, and $textbf{U}$ is constructed from a square upper triangular matrix $textbf{U}_0 in mathbb{R}^{m times m}$ by removing some of its columns. The indices of the removed rows and columns are the same.
My questions are the following:
If $textbf{A}$ is full rank, how can I use $textbf{L}$ and $textbf{U}$ to solve the linear system in $mathcal{O}(n^2)$?
If $textbf{A}$ is NOT full rank, how can I use $textbf{L}$ and $textbf{U}$ to find the least squares soluton of the system in $mathcal{O}(n^2)$?
EDIT: Complexity $mathcal{O}(m^2)$ is also acceptable in both cases.
linear-algebra numerical-linear-algebra lu-decomposition
asked Dec 11 '18 at 16:59
Daniel TurizoDaniel Turizo
636
$endgroup$
add a comment |
$begingroup$
Consider the following linear system of equations:
$$
textbf{A}textbf{x} = textbf{b}
$$
Where $textbf{x}, textbf{b} in mathbb{R}^{n}$ and $textbf{A} in mathbb{R}^{n times n}$. We also have that $textbf{A}=textbf{L}textbf{U}$ where $textbf{L} in mathbb{R}^{n times m}$ and $ textbf{U} in mathbb{R}^{m times n}$ are non-square matrices ($m > n$). $textbf{L}$ is constructed from a square lower triangular matrix $textbf{L}_0 in mathbb{R}^{m times m}$ by removing some of its rows, and $textbf{U}$ is constructed from a square upper triangular matrix $textbf{U}_0 in mathbb{R}^{m times m}$ by removing some of its columns. The indices of the removed rows and columns are the same.
My questions are the following:
If $textbf{A}$ is full rank, how can I use $textbf{L}$ and $textbf{U}$ to solve the linear system in $mathcal{O}(n^2)$?
If $textbf{A}$ is NOT full rank, how can I use $textbf{L}$ and $textbf{U}$ to find the least squares soluton of the system in $mathcal{O}(n^2)$?
EDIT: Complexity $mathcal{O}(m^2)$ is also acceptable in both cases.
linear-algebra numerical-linear-algebra lu-decomposition
asked Dec 11 '18 at 16:59
Daniel TurizoDaniel Turizo
636
$endgroup$
$begingroup$
Can't you extend L and U back to square form and use standard results for LU? Btw, it looks like in L you can only drop columns (not rows as you mention)
$endgroup$
– VorKir
Dec 17 '18 at 6:13
add a comment |
$begingroup$
Consider the following linear system of equations:
$$
textbf{A}textbf{x} = textbf{b}
$$
Where $textbf{x}, textbf{b} in mathbb{R}^{n}$ and $textbf{A} in mathbb{R}^{n times n}$. We also have that $textbf{A}=textbf{L}textbf{U}$ where $textbf{L} in mathbb{R}^{n times m}$ and $ textbf{U} in mathbb{R}^{m times n}$ are non-square matrices ($m > n$). $textbf{L}$ is constructed from a square lower triangular matrix $textbf{L}_0 in mathbb{R}^{m times m}$ by removing some of its rows, and $textbf{U}$ is constructed from a square upper triangular matrix $textbf{U}_0 in mathbb{R}^{m times m}$ by removing some of its columns. The indices of the removed rows and columns are the same.
My questions are the following:
If $textbf{A}$ is full rank, how can I use $textbf{L}$ and $textbf{U}$ to solve the linear system in $mathcal{O}(n^2)$?
If $textbf{A}$ is NOT full rank, how can I use $textbf{L}$ and $textbf{U}$ to find the least squares soluton of the system in $mathcal{O}(n^2)$?
EDIT: Complexity $mathcal{O}(m^2)$ is also acceptable in both cases.
linear-algebra numerical-linear-algebra lu-decomposition
asked Dec 11 '18 at 16:59
Daniel TurizoDaniel Turizo
636
$endgroup$
Consider the following linear system of equations:
$$
textbf{A}textbf{x} = textbf{b}
$$
Where $textbf{x}, textbf{b} in mathbb{R}^{n}$ and $textbf{A} in mathbb{R}^{n times n}$. We also have that $textbf{A}=textbf{L}textbf{U}$ where $textbf{L} in mathbb{R}^{n times m}$ and $ textbf{U} in mathbb{R}^{m times n}$ are non-square matrices ($m > n$). $textbf{L}$ is constructed from a square lower triangular matrix $textbf{L}_0 in mathbb{R}^{m times m}$ by removing some of its rows, and $textbf{U}$ is constructed from a square upper triangular matrix $textbf{U}_0 in mathbb{R}^{m times m}$ by removing some of its columns. The indices of the removed rows and columns are the same.
My questions are the following:
If $textbf{A}$ is full rank, how can I use $textbf{L}$ and $textbf{U}$ to solve the linear system in $mathcal{O}(n^2)$?
If $textbf{A}$ is NOT full rank, how can I use $textbf{L}$ and $textbf{U}$ to find the least squares soluton of the system in $mathcal{O}(n^2)$?
EDIT: Complexity $mathcal{O}(m^2)$ is also acceptable in both cases.
linear-algebra numerical-linear-algebra lu-decomposition
linear-algebra numerical-linear-algebra lu-decomposition
asked Dec 11 '18 at 16:59
Daniel TurizoDaniel Turizo
636
asked Dec 11 '18 at 16:59
Daniel TurizoDaniel Turizo
636
edited Dec 11 '18 at 17:31
Daniel Turizo
asked Dec 11 '18 at 16:59
Daniel TurizoDaniel Turizo
636
asked Dec 11 '18 at 16:59
Daniel TurizoDaniel Turizo
636
asked Dec 11 '18 at 16:59
Daniel TurizoDaniel Turizo
636
636
$begingroup$
Can't you extend L and U back to square form and use standard results for LU? Btw, it looks like in L you can only drop columns (not rows as you mention)
$endgroup$
– VorKir
Dec 17 '18 at 6:13
add a comment |
$begingroup$
Can't you extend L and U back to square form and use standard results for LU? Btw, it looks like in L you can only drop columns (not rows as you mention)
$endgroup$
– VorKir
Dec 17 '18 at 6:13
$begingroup$
Can't you extend L and U back to square form and use standard results for LU? Btw, it looks like in L you can only drop columns (not rows as you mention)
$endgroup$
– VorKir
Dec 17 '18 at 6:13
$begingroup$
Can't you extend L and U back to square form and use standard results for LU? Btw, it looks like in L you can only drop columns (not rows as you mention)
$endgroup$
– VorKir
Dec 17 '18 at 6:13
add a comment |
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$begingroup$
Can't you extend L and U back to square form and use standard results for LU? Btw, it looks like in L you can only drop columns (not rows as you mention)
$endgroup$
– VorKir
Dec 17 '18 at 6:13