INVUPDT Function

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INVUPDT Function

updates a matrix inverse

INVUPDT( matrix, vector<, scalar>)

The inputs to the INVUPDT function are as follows:

matrix: is an n ×n positive definite matrix.
vector: is an n ×1 or 1 ×n vector.
scalar: is a numeric scalar.

The INVUPDT function updates a matrix inverse. For example, let

   r=invupdt(a,x,w);

where R is an n ×w matrix; A is an n ×n positive definite matrix; X is an n ×1 or 1 ×n vector; and w is an optional scalar (if not specified, w has default value 1).

The INVUPDT function computes the matrix expression

R = A - wAX (1 + w X'AX)^-1 X'A^-1X'A

or, in matrix language,

   r=a-w*a*x*inv(1+w*x`*a*x)*x`*a;

The INVUPDT function is used primarily to update a matrix inverse because the function has the property

INVUPDT(B^-1,X,w) = (B + wXX')^-1 .

If Z is a design matrix and X is a new observation to be used in estimating the parameters of a linear model, then the inverse crossproducts matrix that includes the new observation can be updated from the old inverse by

   c2=invupdt(c,x);

where C = INV(Z'Z). Note that

   c2=inv((z//x)`*(z//x));

If w is 1, the function adds an observation to the inverse; if w is -1, the function removes an observation from the inverse. If weighting is used, w is the weight.

To perform the computation, the INVUPDT function uses about 2n² multiplications and additions, where n is the row dimension of the positive definite argument matrix.

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