week 12

\(\def\Real{\mathbb{R}}
\def\Comp{\mathbb{C}}
\def\Rat{\mathbb{Q}}
\def\Field{\mathbb{F}}
\def\Fun{\mathbf{Fun}}
\def\e{\mathbf{e}}
\def\f{\mathbf{f}}
\def\bv{\mathbf{v}}
\)

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
    \[
    p(x)=ax^3+bx^2+cx+d
    \]
    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
    \[
    Q=\left(
    \begin{array}{cc}
    3&-1\\
    -1&3\\
    \end{array}
    \right)
    \]
    Run two iterations of the conjugate gradient method for the system
    \[
    Qx=b,
    \]
    where \(b=(1,0)’\).

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