An undefined index in Kress's Numerical Analysis
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I am reading Kress's Numerical Analysis.
There is a paragraph on page 84 which is discussing what will happen if we "perturb" an equation $Ax=y$. It says:
If for some $deltain Bbb{C}$ we perturb the right-hand side by setting $y^{delta}=y+delta v_j$,
we obtain a perturbed solution $x^{delta}=x+delta u_j/mu_j$.
Hence,
the ratio $||x^{delta}-x||_2/||y^{delta}-y||_2=1/mu_j$ becomes large if $A$ possesses small singular values.
where $mu_1, ..., u_r$ are singular values of $A$,
$A$ has a singular value decomposition $VDU^*$,
$V=(v_1, ..., v_m)$ and $U=(u_1, ..., u_n)$.
My Quesiton:
The definition $y^{delta}=y+delta v_j$ means I can choose any one of $v_1, ..., v_m$ to define $y^{delta}$?
Or I have to choose some $j$ which satisfies some condition? Like $||y^{delta}-y||_2leq delta$?
numerical-methods
asked Nov 22 at 5:55
bfhaha
1,5091024
add a comment |
up vote
1
down vote
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I am reading Kress's Numerical Analysis.
There is a paragraph on page 84 which is discussing what will happen if we "perturb" an equation $Ax=y$. It says:
If for some $deltain Bbb{C}$ we perturb the right-hand side by setting $y^{delta}=y+delta v_j$,
we obtain a perturbed solution $x^{delta}=x+delta u_j/mu_j$.
Hence,
the ratio $||x^{delta}-x||_2/||y^{delta}-y||_2=1/mu_j$ becomes large if $A$ possesses small singular values.
where $mu_1, ..., u_r$ are singular values of $A$,
$A$ has a singular value decomposition $VDU^*$,
$V=(v_1, ..., v_m)$ and $U=(u_1, ..., u_n)$.
My Quesiton:
The definition $y^{delta}=y+delta v_j$ means I can choose any one of $v_1, ..., v_m$ to define $y^{delta}$?
Or I have to choose some $j$ which satisfies some condition? Like $||y^{delta}-y||_2leq delta$?
numerical-methods
asked Nov 22 at 5:55
bfhaha
1,5091024
$j$ is an arbitrary index. The ratio (...) is large, if $mu_j$ is a small singular value of $A$.
– daw
Nov 22 at 7:36
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am reading Kress's Numerical Analysis.
There is a paragraph on page 84 which is discussing what will happen if we "perturb" an equation $Ax=y$. It says:
If for some $deltain Bbb{C}$ we perturb the right-hand side by setting $y^{delta}=y+delta v_j$,
we obtain a perturbed solution $x^{delta}=x+delta u_j/mu_j$.
Hence,
the ratio $||x^{delta}-x||_2/||y^{delta}-y||_2=1/mu_j$ becomes large if $A$ possesses small singular values.
where $mu_1, ..., u_r$ are singular values of $A$,
$A$ has a singular value decomposition $VDU^*$,
$V=(v_1, ..., v_m)$ and $U=(u_1, ..., u_n)$.
My Quesiton:
The definition $y^{delta}=y+delta v_j$ means I can choose any one of $v_1, ..., v_m$ to define $y^{delta}$?
Or I have to choose some $j$ which satisfies some condition? Like $||y^{delta}-y||_2leq delta$?
numerical-methods
asked Nov 22 at 5:55
bfhaha
1,5091024
I am reading Kress's Numerical Analysis.
There is a paragraph on page 84 which is discussing what will happen if we "perturb" an equation $Ax=y$. It says:
If for some $deltain Bbb{C}$ we perturb the right-hand side by setting $y^{delta}=y+delta v_j$,
we obtain a perturbed solution $x^{delta}=x+delta u_j/mu_j$.
Hence,
the ratio $||x^{delta}-x||_2/||y^{delta}-y||_2=1/mu_j$ becomes large if $A$ possesses small singular values.
where $mu_1, ..., u_r$ are singular values of $A$,
$A$ has a singular value decomposition $VDU^*$,
$V=(v_1, ..., v_m)$ and $U=(u_1, ..., u_n)$.
My Quesiton:
The definition $y^{delta}=y+delta v_j$ means I can choose any one of $v_1, ..., v_m$ to define $y^{delta}$?
Or I have to choose some $j$ which satisfies some condition? Like $||y^{delta}-y||_2leq delta$?
numerical-methods
numerical-methods
asked Nov 22 at 5:55
bfhaha
1,5091024
asked Nov 22 at 5:55
bfhaha
1,5091024
asked Nov 22 at 5:55
bfhaha
1,5091024
asked Nov 22 at 5:55
bfhaha
1,5091024
asked Nov 22 at 5:55
bfhaha
1,5091024
1,5091024
$j$ is an arbitrary index. The ratio (...) is large, if $mu_j$ is a small singular value of $A$.
– daw
Nov 22 at 7:36
add a comment |
$j$ is an arbitrary index. The ratio (...) is large, if $mu_j$ is a small singular value of $A$.
– daw
Nov 22 at 7:36
$j$ is an arbitrary index. The ratio (...) is large, if $mu_j$ is a small singular value of $A$.
– daw
Nov 22 at 7:36
$j$ is an arbitrary index. The ratio (...) is large, if $mu_j$ is a small singular value of $A$.
– daw
Nov 22 at 7:36
add a comment |
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$j$ is an arbitrary index. The ratio (...) is large, if $mu_j$ is a small singular value of $A$.
– daw
Nov 22 at 7:36