\(\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 \(b=(1,0)’\).

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?

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.