Matrix computations &
Numerical linear algebra

David Gleich

Purdue University

Fall 2012

Course number CS-51500

Tuesday and Thursday, 10:30-11:45am

Location Lawson 1106

Quiz 20

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Consider the first step of the Arnoldi iteration. We know $\vq_1$ with $\normof{\vq_1} = 1$. Given that:

$$\mA \vq_1 = H_{1,1} \vq_1 + H_{2,1} \vq_2$$

where $\vq_2$ is orthogonal to $\vq_1$ with $\normof{\vq_2} = 1$, how can you find $H_{1,1}, H_{2,1}$ and $\vq_2$?

Solution

Let $\vv = \mA \vq_1 = H_{1,1} \vq_1 + H_{2,1} \vq_2$. Then $H_{1,1} = \vq_1^T \vv$. Also, $\vv - H_{1,1} \vq_1 = H_{2,1} \vq_2$ so $H_{2,1} = \normof{\vv - H_{1,1} \vq_1}$ and $\vq_2 = (\vv - H_{1,1} \vq_1)/H_{2,1}$ (assuming that $H_{2,1}$ is non-zero.)