cholesky-like algorithm

["John Pitchard" <johnpitchard@googlemail.com>]

 

Dear all,

 

I would like to solve the following set of equations:

 

x(1) =b(11)* f(1)

x(2)=b(21)*f(1)+b(22)*f(2)

x(3)=b(31)*f(1)+b(32)*f(2) +b(33)*f(3)

x(4)=b(41)*f(1)+b(42)*f(2) +b(43)*f(3)+b(44)*f(4)

x(5)=b(51)*f(1)+b(52)*f(2) +b(53)*f(3)+b(54)*f(5)+b(55)*f(5)

x(6)=b(61)*f(1)+b(62)*f(2) +b(63)*f(3)+b(64)*f(5)+b(65)*f(6))+b(66)*f(6)

....

....

etc

where the number in brackets is the subscript.

 

The vectors x(1), x(2) and x(3) are known beforehand.

 

I have 2 correlation matrices: the first with the actual correlations

between the 3 variables x(1), x(2) and x(3). The second matrix has the

target correlations between x(1), x(2),... etc

 

The actual matrix is put in the target because these

are the true correlations between the variables, i.e.

target.correlation.matrix[1:3,1:3] <- true.correlation.matrix

However, the resulting matrix is not positive-definite.

 

Is it still possible to solve the system of equations where f(4), f(5),... are random vectors sampled from a standard normal distribution?

I essentially want to calculate the b's first followed by the x's. Is there an agorithm to do this?

 

I would be grateful for some help.

Many thanks,

John