chol (MATLAB Functions)

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

For full matrices X, chol uses the LAPACK routines listed in the following table.

Real
Complex

X double
DPOTRF
ZPOTRF

X single
SPOTRF
CPOTRF

	Real	Complex
`X` `double`	DPOTRF	ZPOTRF
`X` `single`	SPOTRF	CPOTRF

References

[1] Anderson, E., Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK User's Guide (http://www.netlib.org/lapack/lug/ lapack_lug.html), Third Edition, SIAM, Philadelphia, 1999.

See Also

cholinc, cholupdate

checkout cholinc