Algorithm 1 R code that applies sequential Kalman filter (KF) update with known observation constants (measurement vector) and rank deficient state covariance matrix.
Input	x is (ks + ka)-by-1 design based initial estimate of state vector
	V is (ks + ka)-by-(ks + ka) design based initial estimate of covariance matrix
	y is ka-by-1 conformable vector of auxiliary census constants
	ks is number of study variables
	ka is number of auxiliary variablesk = ks + ka
1	w < − matrix(0,k,1)           # initialize with k-by-1 zero vector
2	I.wh < − diag(k)             # initialize with k-by-k identity matrix
3	for ( i in ka:1 ) {         # i = { ka, (ka − 1), ..., 1}
4	k < − ks + i
5	if ( V[k,k] < tol ) next # tol near zero (e.g., tol = 0.0001)
6	c = 2*abs( y[i] − x[k])/V[k,k]
7	if ( c > 1 ) { # divergence constraint from Equation (A17)
8	V[1:k,k] < −V[1:k,k]*c
9	V[k,1:k] < −V[k,1:k]*c
10	} # end of divergence “if statement”
11	w[1:k] < −V[1:k,k]/V[k,k]
12	I.wh[1:k,k] < −(-w[1:k] ) # subtract vector w from k-th column
13	x[1:k] < − ( I.wh[1:k,1:k] %*% x[1:k] ) + ( w[1:k]*y[i] )
14	V[1:k,1:k] < −I.wh[1:k,1:k] %*% V[1:k,1:k] %*% t( I.wh[1:k,1:k] )
15	}                            # end of i-th iteration
Definitions	[a:b,c] is the matrix partition, rows a to b, column c
	<− is the replacement operator
	abs(z) is the absolute value of scalar z
	t(A) is the transpose of matrix A
	%*% is the matrix multiplication operator
	w is the k-by-1 vector of computed optimal Kalman weights
	I.wh is the k-by-k matrix of complementary Kalman weights
Output	x is (ks + ka)-by-1 state vector after Kalman updates with ka auxiliary variables
	V is (ks + ka)-by-(ks + ka) covariance matrix with ka auxiliary variables