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This example shows how to use the `fixed.forgettingFactor`

and `fixed.forgettingFactorInverse`

functions.

The growth in the QR decomposition can be seen by looking at the magnitude of the first element $$R(1,1)$$of the upper-triangular factor $$R$$, which is equal to the Euclidean norm of the first column of matrix $$A$$,

$$|R(1,1)|=||A(:,1)|{|}_{2}$$ .

To see this, create matrix $$A$$ as a column of ones of length $$n$$ and compute $$R$$ of the economy-size QR decomposition.

n = 1e4; A = ones(n,1);

Then $$|R(1,1)|=||A(:,1)|{|}_{2}=\sqrt{\sum _{i=1}^{n}{1}^{2}}=\sqrt{n}$$.

R = fixed.qlessQR(A)

R = 100.0000

norm(A)

ans = 100

sqrt(n)

ans = 100

The diagonal elements of the upper-triangular factor $$R$$ of the QR decomposition may be positive, negative, or zero, but `fixed.qlessQR`

and `fixed.qrAB`

always return the diagonal elements of $$R$$ as non-negative.

In a real-time application, such as when data is streaming continuously from a radar array, you can update the QR decomposition with an exponential forgetting factor $$\alpha $$ where $$0<\alpha <1$$. Use the `fixed.forgettingFactor`

function to compute a forgetting factor $$\alpha $$ that acts as if the matrix were being integrated over $$m$$ rows to maintain a gain of about $$\sqrt{m}$$. The relationship between $$\alpha $$ and $$m$$ is $$\alpha ={e}^{-1/(2m)}$$.

m = 16; alpha = fixed.forgettingFactor(m); R_alpha = fixed.qlessQR(A,alpha)

R_alpha = 3.9377

sqrt(m)

ans = 4

If you are working with a system and have been given a forgetting factor $$\alpha $$, and want to know the effective number of rows $$m$$ that you are integrating over, then you can use the `fixed.forgettingFactorInverse`

function. The relationship between $$m$$ and $$\alpha $$ is $$m=\frac{-1}{2\mathrm{log}(\alpha )}$$.

fixed.forgettingFactorInverse(alpha)

ans = 16

`fixed.forgettingFactor`

|`fixed.forgettingFactorInverse`

|`fixed.qlessQR`

|`fixed.qlessQRMatrixSolve`

|`fixed.qlessQRUpdate`