Calculates a Cholesky-factor of the inverse covariance matrix (ICM) of a multivariable autoregressive process.
C = cholicm(Af,Ab,Sf,Sb,N)
The Inverse Covariance Matrix S of a multivariable autoregressive noise process according to  is calculated. A Cholesky factor C is returned such that C'C = S
The noise model contains a causal and an auticausal part, both of which describe the actual noise v(k). If e(k) is a Gaussian white innovation, the model is given by:
Right arrow denote the causal (Forward) components while left arrows denote the anti-causal (Backward) ones.
The multivariable noise model can be obtained using the destmar function
Af is an l x ld matrix containing the causal part of the noise process. Af = [Af1,Af2,...,Afd]
Ab is an l x ld matrix containing the anti-causal part of the noise process. Ab = [Ab1,Ab2,...,Abd]
Sf is an l x l matrix describing the covariance E[ef ef']
Sb is an l x l matrix describing the covariance E[eb eb']
N is the number of samples
C is a Cholesky factor of the ICM. This matrix is stored in LAPACK/BLAS band-storage; its size is (d + 1)l x N, and the bottom row contains the diagonal of C. The row above contains a zero, and then the first superdiagonal of C. The row above contains two zeros, and then the second superdiagonal, etc. The top row contains ((d + 1)l - 1) zeros, and then the ((d + 1)l - 1) th superdiagonal.
A covariance matrix of a stationary process is always positive definite. However, it is very well possible to specify filter coefficients Af, Ab and covariances Sf and Sb such that the theoretical ICM calculated per  is not positive definite. In such cases, no Cholesky factor can be calculated, and an identity matrix will be returned along with a warning message. The filter should be checked and adjusted in these cases.
The upper-triangular block-band of a sparse banded inverse covariance matrix according to  is filled. A direct sparse Cholesky factorization is subsequently performed using MATLAB's internal chol function.
 B. Davis, Parameter Estimation in Nonlinear Dynamical Systems with Correlated Noise. PhD thesis, Universite Catholique de Louvain-La-Neuve, Belgium, 2001.