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Numerical mathematics - Coggle Diagram

- - - - \[\sum_{i=1}^nf(x_i)\Delta x\]
    - - \[\sum_{i=1}^nf\left(\frac{x_{i-1}+x_i}{2}\right)\Delta x\]
    - - \[\sum_{i=1}^nf(x_{i-1})\Delta x\]
  - - - \[\sum_{i=1}^n\frac{f(x_{i-1})+f(x_i)}{2}\Delta x\]
  - - - \[\sum_{i=1}^n\frac{f(x_{i-1})+4f\left(\frac{x_{i-1}+x_i}{2}\right)+f(x_i)}{6}\Delta x\]
- - - - The filters are independent to each other, banana filters only capture banana, adding mango does not affect it.
      - The ingredients is blendable and the order of blending does not matter so that we always end up with the same smoothie.
      - The set of filters is complete, every ingredient is captured.
  - - - Start with a time-based signal.
      - Apply filters to measure each possible "circular ingredient".
      - Collect the full recipe, listing the amount of each "circular ingredient".
    - - Frequency (speed)
      - Phase angle (starting point)
      - Amplitude (size)
- - - - A measure how sensitive a function is to changes in the input, and how much change in the output results from a change in the input.
    - - \(\mathrm{cond}(A)=|A^{-1}||A|\)
    - - \(\left|x\frac{f'(x)}{f(x)}\right|\)
    - - \(|x|\frac{|J(x)|}{|f(x)|}\)
- - - - How? Use FFT to find \((d+1)\) data points for each polynomials.
      - Why? It is easier to multiply data points then algebraic multiplication
      - How do we get the product? Just use the IFFT on the data points.
      - Given two polynomials, find the product in an efficient way.
    - - Let \(P(x)=x^3+x^2-x-1\) which has degree \(d\)
      - So we need \(N=d+1\) data points in positive negative pairs
      - Let the pairs be \(x_1, -x_1, x_2, -x_2\)
      - Now we start from the beginning with evaluating \(x_1\cdot(-x_1)=x_1^2\) and \(x_2\cdot(-x_2)=x_2^2\) instead.
      - We need those two points to be a positive-negative pair s.t. \(x_1^4=x_1^2\cdot x_2^2\)
      - Remember we can choose the points, so let \(x_1=1\).
      - Then \((x_1^2=1)\Rightarrow(-x_1^2=-1)\Rightarrow(x_2^2=-1)\Rightarrow(x_2=i,-x_2=-i)\)
      - Which means \(x_1^4=1\)
    - - Now split the polynimial in an od d and an even polynomial and repeat
      - Let \(x_1=1\) and solve for \(x_1^{N+1}=1\)
      - Let the polynomial of degree \(d\) have \(2^k\ge d+1\) positive-negative data pairs