Chebyshev polynomial: recursive formula error estimate

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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



  1. To develop
    $$
    T_N^*(x)-T_N(x) = sum_k=1^Ndelta_kfracsin((N+1-k)arccosx)sqrt1-x^2
    $$

  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




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    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



    1. To develop
      $$
      T_N^*(x)-T_N(x) = sum_k=1^Ndelta_kfracsin((N+1-k)arccosx)sqrt1-x^2
      $$

    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




    share|cite|improve this question





















      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



      1. To develop
        $$
        T_N^*(x)-T_N(x) = sum_k=1^Ndelta_kfracsin((N+1-k)arccosx)sqrt1-x^2
        $$

      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




      share|cite|improve this question











      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



      1. To develop
        $$
        T_N^*(x)-T_N(x) = sum_k=1^Ndelta_kfracsin((N+1-k)arccosx)sqrt1-x^2
        $$

      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





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