week 12


Conjugate gradient method

Material:Guler, ch. 14.7-9

Conjugate directions. Gram-Schmidt algorithm. Conjugate directions algorithm. Performance estimates via spectral data.

Sample problems

  • Consider the 4-dimensional space spanned by the polynomials
    of degree at most \(3\).
    Run the Gram-Schmidt orthogonalization procedure on the vectors
    \(\{1,x,x^2,x^3\}\), if the scalar product is given by
    p_1’Qp_2=\int_{-1}^1 p_1(x)p_2(x) dx.
  • Consider quadratic form in \(\Real^d\) given by
    Run two iterations of the conjugate gradient method for the system
    where \(b=(1,0)’\).

2 Responses to week 12

  1. Tianyi May 5, 2016 at 2:22 pm #

    I am really confused about the first question. I am not sure exactly what the question is asking, and I am not sure how exactly I should approach it. Can I have some hint?

    • yuliy May 5, 2016 at 2:56 pm #

      You should find polynomials p_1=1; p2=x+a21 p1, p3=x^2+a31 p1 +a32 p2,… which are orthogonal with respect to the given scalar product.

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