Condition number of a square matrix with entry $A_{ij}$ being $j^{2i-1}$ and numerical stability for $Ax=b$
$begingroup$
$i,j geq 1$ (that is row $i$ and column $j$ counting start from $1$), and let entry $A_{ij}$ of the $n times n$ square matrix $A$ be defined as $A_{ij} = j^{2i-1}$. Without fixing $n$, would there be an easy way to calculate condition number of this matrix $A$? (And would there be a special name for such a matrix?)
Also, suppose that with a such matrix, I would like to solve $Ax=b$. What would be computational complexity of a reasonably good numerical algorithm, with condition number in mind? For ease, one can fix $b$ as $[1,0,..,0]^T$.
linear-algebra numerical-linear-algebra
asked Jan 9 at 10:24
Jacob MacherovJacob Macherov
212
$endgroup$
add a comment |
$begingroup$
$i,j geq 1$ (that is row $i$ and column $j$ counting start from $1$), and let entry $A_{ij}$ of the $n times n$ square matrix $A$ be defined as $A_{ij} = j^{2i-1}$. Without fixing $n$, would there be an easy way to calculate condition number of this matrix $A$? (And would there be a special name for such a matrix?)
Also, suppose that with a such matrix, I would like to solve $Ax=b$. What would be computational complexity of a reasonably good numerical algorithm, with condition number in mind? For ease, one can fix $b$ as $[1,0,..,0]^T$.
linear-algebra numerical-linear-algebra
asked Jan 9 at 10:24
Jacob MacherovJacob Macherov
212
$endgroup$
$begingroup$
Looks like a scaled version of a Vandermonde matrix
$endgroup$
– daw
Jan 9 at 15:36
add a comment |
$begingroup$
$i,j geq 1$ (that is row $i$ and column $j$ counting start from $1$), and let entry $A_{ij}$ of the $n times n$ square matrix $A$ be defined as $A_{ij} = j^{2i-1}$. Without fixing $n$, would there be an easy way to calculate condition number of this matrix $A$? (And would there be a special name for such a matrix?)
Also, suppose that with a such matrix, I would like to solve $Ax=b$. What would be computational complexity of a reasonably good numerical algorithm, with condition number in mind? For ease, one can fix $b$ as $[1,0,..,0]^T$.
linear-algebra numerical-linear-algebra
asked Jan 9 at 10:24
Jacob MacherovJacob Macherov
212
$endgroup$
$i,j geq 1$ (that is row $i$ and column $j$ counting start from $1$), and let entry $A_{ij}$ of the $n times n$ square matrix $A$ be defined as $A_{ij} = j^{2i-1}$. Without fixing $n$, would there be an easy way to calculate condition number of this matrix $A$? (And would there be a special name for such a matrix?)
Also, suppose that with a such matrix, I would like to solve $Ax=b$. What would be computational complexity of a reasonably good numerical algorithm, with condition number in mind? For ease, one can fix $b$ as $[1,0,..,0]^T$.
linear-algebra numerical-linear-algebra
linear-algebra numerical-linear-algebra
asked Jan 9 at 10:24
Jacob MacherovJacob Macherov
212
asked Jan 9 at 10:24
Jacob MacherovJacob Macherov
212
asked Jan 9 at 10:24
Jacob MacherovJacob Macherov
212
asked Jan 9 at 10:24
Jacob MacherovJacob Macherov
212
asked Jan 9 at 10:24
Jacob MacherovJacob Macherov
212
212
$begingroup$
Looks like a scaled version of a Vandermonde matrix
$endgroup$
– daw
Jan 9 at 15:36
add a comment |
$begingroup$
Looks like a scaled version of a Vandermonde matrix
$endgroup$
– daw
Jan 9 at 15:36
$begingroup$
Looks like a scaled version of a Vandermonde matrix
$endgroup$
– daw
Jan 9 at 15:36
$begingroup$
Looks like a scaled version of a Vandermonde matrix
$endgroup$
– daw
Jan 9 at 15:36
add a comment |
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$begingroup$
Looks like a scaled version of a Vandermonde matrix
$endgroup$
– daw
Jan 9 at 15:36