|MATLAB Function Reference|
Rank 1 update to Cholesky factorization
R1 = cholupdate(R,x)
R = chol(A) is the original Cholesky factorization of
A, returns the upper triangular Cholesky factor of
A + x*x', where
x is a column vector of appropriate length.
cholupdate uses only the diagonal and upper triangle of
R. The lower triangle of
R is ignored.
R1 = cholupdate(R,x,'+')
is the same as
R1 = cholupdate(R,x).
R1 = cholupdate(R,x,'-')
returns the Cholesky factor of
A - x*x'. An error message reports when R is not a valid Cholesky factor or when the downdated matrix is not positive definite and so does not have a Cholesky factoriza- tion.
[R1,p] = cholupdate(R,x,'-')
will not return an error message. If
R1 is the Cholesky factor of
A - x*x'. If
p is greater than
R1 is the Cholesky factor of the original
cholupdate failed because the downdated matrix is not positive definite. If
cholupdate failed because the upper triangle of
R was not a valid Cholesky factor.
cholupdate works only for full matrices.
This is called a rank one update to
Instead of computing the Cholesky factor with
R1 = chol(A + x*x'), we can use
Next destroy the positive definiteness (and actually make the matrix singular) by subtracting
1 from the last element of
A. The downdated matrix is:
0.5 from the last element of
A produces a positive definite matrix, and we can use
cholupdate to compute its Cholesky factor:
cholupdate uses the algorithms from the LINPACK subroutines
cholupdate is useful since computing the new Cholesky factor from scratch is an algorithm, while simply updating the existing factor in this way is an algorithm.
 Dongarra, J.J., J.R. Bunch, C.B. Moler, and G.W. Stewart, LINPACK Users' Guide, SIAM, Philadelphia, 1979.