Language Reference: INVUPDT Function :: SAS/IML(R) 9.2 User's Guide

Language Reference

INVUPDT Function

updates a matrix inverse

INVUPDT( matrix, vector<, scalar>)

The inputs to the INVUPDT function are as follows:

matrix: is an nonsingular matrix. In most applications matrix is symmetric positive definite.
vector: is an or vector.
scalar: is a numeric scalar.

The Sherman-Morrison-Woodbury formula is

$(a + u{v}^')^{-1} = a^{-1} - a^{-1} u (i + v^'a^{-1}u)^{-1} v^'a^{-1}$

where

is an

nonsingular matrix and

and

are

. The formula shows that a rank

update to

corresponds to a rank

update of $a^{-1}$ .

The INVUPDT function is used primarily to update a matrix inverse. The function implements the Sherman-Morrison-Woodbury formula for rank-one updates with

and

, where

is an

vector and

is a scalar.

If $m = a^{-1}$ , then you can call the INVUPDT function as follows:

  
    R=invupdt(M,X,w);

This statement computes the following matrix:

$r = m - w{mx} (i + w{x}^'{mx})^{-1} x^'m$

The matrix

is equivalent to $(a + w{xx}^')^{-1}$ . If

is symmetric positive definite, then so is

If is not specified, then it is given a default value of 1.

A common use of the INVUPDT function is in linear regression. If is a design matrix, $m = (z^'z)^{-1}$ is the associated inverse crossproduct matrix, and 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 by using the following statement:

  
    M2=invupdt(M,v);

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

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

The following program demonstrates adding or removing observations from a linear fit and updating the inverse crossproduct matrix:

  
    X = {0,1,1,1,2,2,3,4,4}; 
    Y = {1,1,2,6,2,3,3,3,4}; 
  
    /* find linear fit */ 
    Z = j(nrow(X),1,1) || X; /* design matrix */ 
    M = inv(Z`*Z); 
  
    b = M*Z`*Y;        /* LS estimate */ 
    resid = Y - Z*b;   /* residuals */ 
    print "Original Fit", b resid; 
  
    /* residual for observation (1,6) seems too large. 
       Take obs number 4 out of data set and refit. */ 
    v = z[4,]; 
    M = invupdt( M, v, -1 ); /* update inverse crossprod */ 
  
    keepObs = (1:3) || (5:nrow(X)); 
    Z = Z[keepObs,]; 
    Y = Y[keepObs,]; 
    b = M*Z`*Y;        /* new LS estimate */ 
    print "After deleting observation 4", b; 
  
    /* Add a new obs (x,y) = (0,2) and refit. */ 
    obs = {0 2}; 
    v = 1 || obs[1];    /* new row in design matrix */ 
    M = invupdt( M, v ); 
  
    Z = Z // v; 
    Y = Y // obs[2]; 
    b = M*Z`*Y;        /* new LS estimate */ 
    print "After adding observation (0,2)", b;

The output is as follows:

  
                          Original Fit 
  
                               B     RESID 
  
                       2.0277778 -1.027778 
                           0.375 -1.402778 
                                 -0.402778 
                                 3.5972222 
                                 -0.777778 
                                 0.2222222 
                                 -0.152778 
                                 -0.527778 
                                 0.4722222 
  
  
                  After deleting observation 4 
  
                                B 
  
                                    1 
                            0.6470588 
  
  
                 After adding observation (0,2) 
  
                                B 
  
                                  1.3 
                            0.5470588

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