# 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 $$b=(1,0)’$$.

### 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.