Linear Algebra and Its Applications · Problem 3.3.31

Orthogonality — Gilbert Strang

Mathematics · recursive-least-squares

Problem

If you know the average

\hat{x}_9

of 9 numbers

b_1, \ldots, b_9

, how can you quickly find the average

\hat{x}_{10}

with one more number

b_{10}

? The idea of recursive least squares is to avoid adding 10 numbers. What coefficient of

\hat{x}_9

correctly gives

\hat{x}_{10}

\hat{x}_{10} = \frac{1}{10}\hat{b}_{10} + \_\_\_\_\, \hat{x}_9 = \frac{1}{10}(b_1 + \cdots + b_{10}).