From polynomials to Chebyshev polynomials
I was wondering how they got from the polynomial to a Chebyshev polynomial as outlined here:
In order to obtain “strong” stability, we replace the condition ($2.7$) by
$$left|prod_{j=1}^N (1-tau_jlambda)right| leq K qquad forall lambda in [mu,lambda_max], tag{2.8}$$ where $mu$ is some number in the interval $(0, lambda_min]$, and $K$ is some number $0<K<1$. The problem of finding the “optimal” values for the $tau_j$’s can then be reformulated as
Find $tau_1$, $tau_2$, $ldots$, $tau_N$ such that $p_N(lambda) =prod_{j=1}^N (1-tau_jlambda)$ satisfies
begin{array}{l@{qquad}l}
|p_N(lambda)| leq K quad forall lambda in [mu, lambda_max] && text{(STABILITY),} \
displaystyle |p'_N(0)| =sum_{j=1}^N tau_j quad text{maximal} && text{(OPTIMALITY).}
end{array}
Using the remarkable optimality properties of the Chebyshev polynomials $T_N(cdot)$ of degree $N$, Markoff [$5$] ($1892!$), we have that if $K$ is given by $$K=1/T_N left(frac{lambda_max +mu}{lambda_max -mu}right)$$
Anyone knows?
The actual paper, I'm referring to is here.
polynomials numerical-methods chebyshev-polynomials
edited Dec 26 at 3:04
Rócherz
2,7612721
asked Feb 19 '15 at 12:45
unrealbot
111
add a comment |
I was wondering how they got from the polynomial to a Chebyshev polynomial as outlined here:
In order to obtain “strong” stability, we replace the condition ($2.7$) by
$$left|prod_{j=1}^N (1-tau_jlambda)right| leq K qquad forall lambda in [mu,lambda_max], tag{2.8}$$ where $mu$ is some number in the interval $(0, lambda_min]$, and $K$ is some number $0<K<1$. The problem of finding the “optimal” values for the $tau_j$’s can then be reformulated as
Find $tau_1$, $tau_2$, $ldots$, $tau_N$ such that $p_N(lambda) =prod_{j=1}^N (1-tau_jlambda)$ satisfies
begin{array}{l@{qquad}l}
|p_N(lambda)| leq K quad forall lambda in [mu, lambda_max] && text{(STABILITY),} \
displaystyle |p'_N(0)| =sum_{j=1}^N tau_j quad text{maximal} && text{(OPTIMALITY).}
end{array}
Using the remarkable optimality properties of the Chebyshev polynomials $T_N(cdot)$ of degree $N$, Markoff [$5$] ($1892!$), we have that if $K$ is given by $$K=1/T_N left(frac{lambda_max +mu}{lambda_max -mu}right)$$
Anyone knows?
The actual paper, I'm referring to is here.
polynomials numerical-methods chebyshev-polynomials
edited Dec 26 at 3:04
Rócherz
2,7612721
asked Feb 19 '15 at 12:45
unrealbot
111
add a comment |
I was wondering how they got from the polynomial to a Chebyshev polynomial as outlined here:
In order to obtain “strong” stability, we replace the condition ($2.7$) by
$$left|prod_{j=1}^N (1-tau_jlambda)right| leq K qquad forall lambda in [mu,lambda_max], tag{2.8}$$ where $mu$ is some number in the interval $(0, lambda_min]$, and $K$ is some number $0<K<1$. The problem of finding the “optimal” values for the $tau_j$’s can then be reformulated as
Find $tau_1$, $tau_2$, $ldots$, $tau_N$ such that $p_N(lambda) =prod_{j=1}^N (1-tau_jlambda)$ satisfies
begin{array}{l@{qquad}l}
|p_N(lambda)| leq K quad forall lambda in [mu, lambda_max] && text{(STABILITY),} \
displaystyle |p'_N(0)| =sum_{j=1}^N tau_j quad text{maximal} && text{(OPTIMALITY).}
end{array}
Using the remarkable optimality properties of the Chebyshev polynomials $T_N(cdot)$ of degree $N$, Markoff [$5$] ($1892!$), we have that if $K$ is given by $$K=1/T_N left(frac{lambda_max +mu}{lambda_max -mu}right)$$
Anyone knows?
The actual paper, I'm referring to is here.
polynomials numerical-methods chebyshev-polynomials
edited Dec 26 at 3:04
Rócherz
2,7612721
asked Feb 19 '15 at 12:45
unrealbot
111
I was wondering how they got from the polynomial to a Chebyshev polynomial as outlined here:
In order to obtain “strong” stability, we replace the condition ($2.7$) by
$$left|prod_{j=1}^N (1-tau_jlambda)right| leq K qquad forall lambda in [mu,lambda_max], tag{2.8}$$ where $mu$ is some number in the interval $(0, lambda_min]$, and $K$ is some number $0<K<1$. The problem of finding the “optimal” values for the $tau_j$’s can then be reformulated as
Find $tau_1$, $tau_2$, $ldots$, $tau_N$ such that $p_N(lambda) =prod_{j=1}^N (1-tau_jlambda)$ satisfies
begin{array}{l@{qquad}l}
|p_N(lambda)| leq K quad forall lambda in [mu, lambda_max] && text{(STABILITY),} \
displaystyle |p'_N(0)| =sum_{j=1}^N tau_j quad text{maximal} && text{(OPTIMALITY).}
end{array}
Using the remarkable optimality properties of the Chebyshev polynomials $T_N(cdot)$ of degree $N$, Markoff [$5$] ($1892!$), we have that if $K$ is given by $$K=1/T_N left(frac{lambda_max +mu}{lambda_max -mu}right)$$
Anyone knows?
The actual paper, I'm referring to is here.
polynomials numerical-methods chebyshev-polynomials
polynomials numerical-methods chebyshev-polynomials
edited Dec 26 at 3:04
Rócherz
2,7612721
asked Feb 19 '15 at 12:45
unrealbot
111
edited Dec 26 at 3:04
Rócherz
2,7612721
asked Feb 19 '15 at 12:45
unrealbot
111
edited Dec 26 at 3:04
Rócherz
2,7612721
edited Dec 26 at 3:04
Rócherz
2,7612721
edited Dec 26 at 3:04
Rócherz
2,7612721
2,7612721
asked Feb 19 '15 at 12:45
unrealbot
111
asked Feb 19 '15 at 12:45
unrealbot
111
asked Feb 19 '15 at 12:45
unrealbot
111
111
add a comment |
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