chol (MATLAB Function Reference)

chol	Examples See Also

Cholesky factorization

Syntax

R = chol(X)
[R,p] = chol(X)

Description

The chol function uses only the diagonal and upper triangle of X. The lower triangular is assumed to be the (complex conjugate) transpose of the upper. That is, X is Hermitian.

R = chol(X),

where X is positive definite produces an upper triangular R so that R'*R = X. If X is not positive definite, an error message is printed.

[R,p] = chol(X),

with two output arguments, never produces an error message. If X is positive definite, then p is 0 and R is the same as above. If X is not positive definite, then p is a positive integer and R is an upper triangular matrix of order q = p-1 so that R'*R = X(1:q,1:q).

Examples

The binomial coefficients arranged in a symmetric array create an interesting positive definite matrix.

n = 5;
X = pascal(n)
X =
    1    1    1    1    1
    1    2    3    4    5
    1    3    6   10   15
    1    4   10   20   35
    1    5   15   35   70

It is interesting because its Cholesky factor consists of the same coefficients, arranged in an upper triangular matrix.

R = chol(X)
R =
    1    1    1    1    1
    0    1    2    3    4
    0    0    1    3    6
    0    0    0    1    4
    0    0    0    0    1

Destroy the positive definiteness (and actually make the matrix singular) by subtracting 1 from the last element.

X(n,n) = X(n,n)-1
X =
    1    1    1    1    1
    1    2    3    4    5
    1    3    6   10   15
    1    4   10   20   35
    1    5   15   35   69

Now an attempt to find the Cholesky factorization fails.

Algorithm

chol uses the algorithm from the LINPACK subroutine ZPOFA. For a detailed description of the use of the Cholesky decomposition, see Chapter 8 of the LINPACK Users' Guide.

References

[1] Dongarra, J.J., J.R. Bunch, C.B. Moler, and G.W. Stewart, LINPACK Users' Guide, SIAM, Philadelphia, 1979.