Mixing
Chebyshev polynomial: recursive formula error estimate
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
I am trying to solve Problem 3 & 4 from Numerical methods of Bakhvalov, Zhidkov, Kobelkov from section 2.8 on Chebyshev polynomials.
If in the recursive formula
$$
T_n(x) = 2xT_n-1(x)-T_n-2(x)
$$
each calculation contains a round-off error:
$$
T^*_0=1,quad T_1^*=T_1+delta_1,quad T_n^*(x)=2xT_n-1^*(x) - T_n-2^*(x) + delta_n
$$
then
- To develop
$$
T_N^*(x)-T_N(x) = sum_k=1^Ndelta_kfracsin((N+1-k)arccosx)sqrt1-x^2
$$ - And
$$
|T_N^*(x)-T_N(x)|leq max|delta_k|cdot Ncdot minleftN,frac1sqrt1-x^2right
$$
With the first problem, by calculating the first few terms I can see that the cumulative round-off is the sum of derivatives divided by the power:
$$
fracT'_11+fracT'_22+cdots+fracT'_NN
$$
and of course the derivative of $T_n(x)$ is
$$
T'_n(x)=nfracsin(narccosx)sqrt1-x^2
$$
So, then I can prove the formula by induction. But I can't claim that I have properly developed this formula.
For the second problem, I can easily prove
$$
|T^*_N(x)-T_N(x)|leq fraccdot Nsqrt1-x^2
$$
since $sin(narccosx)leq 1$. But I can't get
$$
|T^*_N(x)-T_N(x)|leq max|delta_k|cdot N^2.
$$
numerical-methods chebyshev-polynomials rounding-error
asked Jul 31 at 15:29
mobiuseng
23619
add a comment |Â
up vote
0
down vote
favorite
I am trying to solve Problem 3 & 4 from Numerical methods of Bakhvalov, Zhidkov, Kobelkov from section 2.8 on Chebyshev polynomials.
If in the recursive formula
$$
T_n(x) = 2xT_n-1(x)-T_n-2(x)
$$
each calculation contains a round-off error:
$$
T^*_0=1,quad T_1^*=T_1+delta_1,quad T_n^*(x)=2xT_n-1^*(x) - T_n-2^*(x) + delta_n
$$
then
- To develop
$$
T_N^*(x)-T_N(x) = sum_k=1^Ndelta_kfracsin((N+1-k)arccosx)sqrt1-x^2
$$ - And
$$
|T_N^*(x)-T_N(x)|leq max|delta_k|cdot Ncdot minleftN,frac1sqrt1-x^2right
$$
With the first problem, by calculating the first few terms I can see that the cumulative round-off is the sum of derivatives divided by the power:
$$
fracT'_11+fracT'_22+cdots+fracT'_NN
$$
and of course the derivative of $T_n(x)$ is
$$
T'_n(x)=nfracsin(narccosx)sqrt1-x^2
$$
So, then I can prove the formula by induction. But I can't claim that I have properly developed this formula.
For the second problem, I can easily prove
$$
|T^*_N(x)-T_N(x)|leq fraccdot Nsqrt1-x^2
$$
since $sin(narccosx)leq 1$. But I can't get
$$
|T^*_N(x)-T_N(x)|leq max|delta_k|cdot N^2.
$$
numerical-methods chebyshev-polynomials rounding-error
asked Jul 31 at 15:29
mobiuseng
23619
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am trying to solve Problem 3 & 4 from Numerical methods of Bakhvalov, Zhidkov, Kobelkov from section 2.8 on Chebyshev polynomials.
If in the recursive formula
$$
T_n(x) = 2xT_n-1(x)-T_n-2(x)
$$
each calculation contains a round-off error:
$$
T^*_0=1,quad T_1^*=T_1+delta_1,quad T_n^*(x)=2xT_n-1^*(x) - T_n-2^*(x) + delta_n
$$
then
- To develop
$$
T_N^*(x)-T_N(x) = sum_k=1^Ndelta_kfracsin((N+1-k)arccosx)sqrt1-x^2
$$ - And
$$
|T_N^*(x)-T_N(x)|leq max|delta_k|cdot Ncdot minleftN,frac1sqrt1-x^2right
$$
With the first problem, by calculating the first few terms I can see that the cumulative round-off is the sum of derivatives divided by the power:
$$
fracT'_11+fracT'_22+cdots+fracT'_NN
$$
and of course the derivative of $T_n(x)$ is
$$
T'_n(x)=nfracsin(narccosx)sqrt1-x^2
$$
So, then I can prove the formula by induction. But I can't claim that I have properly developed this formula.
For the second problem, I can easily prove
$$
|T^*_N(x)-T_N(x)|leq fraccdot Nsqrt1-x^2
$$
since $sin(narccosx)leq 1$. But I can't get
$$
|T^*_N(x)-T_N(x)|leq max|delta_k|cdot N^2.
$$
numerical-methods chebyshev-polynomials rounding-error
asked Jul 31 at 15:29
mobiuseng
23619
I am trying to solve Problem 3 & 4 from Numerical methods of Bakhvalov, Zhidkov, Kobelkov from section 2.8 on Chebyshev polynomials.
If in the recursive formula
$$
T_n(x) = 2xT_n-1(x)-T_n-2(x)
$$
each calculation contains a round-off error:
$$
T^*_0=1,quad T_1^*=T_1+delta_1,quad T_n^*(x)=2xT_n-1^*(x) - T_n-2^*(x) + delta_n
$$
then
- To develop
$$
T_N^*(x)-T_N(x) = sum_k=1^Ndelta_kfracsin((N+1-k)arccosx)sqrt1-x^2
$$ - And
$$
|T_N^*(x)-T_N(x)|leq max|delta_k|cdot Ncdot minleftN,frac1sqrt1-x^2right
$$
With the first problem, by calculating the first few terms I can see that the cumulative round-off is the sum of derivatives divided by the power:
$$
fracT'_11+fracT'_22+cdots+fracT'_NN
$$
and of course the derivative of $T_n(x)$ is
$$
T'_n(x)=nfracsin(narccosx)sqrt1-x^2
$$
So, then I can prove the formula by induction. But I can't claim that I have properly developed this formula.
For the second problem, I can easily prove
$$
|T^*_N(x)-T_N(x)|leq fraccdot Nsqrt1-x^2
$$
since $sin(narccosx)leq 1$. But I can't get
$$
|T^*_N(x)-T_N(x)|leq max|delta_k|cdot N^2.
$$
numerical-methods chebyshev-polynomials rounding-error
asked Jul 31 at 15:29
mobiuseng
23619
asked Jul 31 at 15:29
mobiuseng
23619
asked Jul 31 at 15:29
mobiuseng
23619
asked Jul 31 at 15:29
mobiuseng
23619
23619
add a comment |Â
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2868180%2fchebyshev-polynomial-recursive-formula-error-estimate%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password