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Factorization of Structured Matrices >> Factorization of Structured Matrices > ldl_blocktoep

ldl_blocktoep

LDL factorization of a strongly regular hermitian block Toeplitz matrix with positive-definite first block

Calling Sequence

[L,d] = ldl_blocktoep(BC)

Parameters

BC

M*r x r matrix

L

lower triangular M*r x M*r matrix

d

M*r x 1 vector, entries are +1 or -1

Description

The routine computes the LDL factorization of a hermitian but possibly indefinite block Toeplitz matrix T=[T(0) T(1)' T(2)' ... T(M-1)'; T(1) T(0) T(1)' ... T(M-2)'; ... ;T(M-1) T(M-2) T(M-3) ... T(0)] with first block column BC=[T(0);T(1);...;T(M-1)]. Here, the blocks T(0),...,T(M-1) are r x r matrices. We assume that the first block T(0)=T(0)'=BC(1:r,1:r) is hermitian and positive-definite and that the matrix T is strongly regular (i.e., all leading minors are non-zero). The algorithm computes a lower triangular matrix L and a vector d with entries +1 / -1 such that T=L*diag(d)*L'.

Examples

BC = (1:5)'+%i*(0:4)';        // first column with BC(1,1)=BC(1,1)'>0
T = toeplitz(BC,BC');        // corresponding Toeplitz matrix
[L,d] = ldl_blocktoep(BC);    // perform LDL factorization
disp(clean(T-L*diag(d)*L'));    // show residual; should be zero

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