MATLAB Function Reference |

Rank 1 update to Cholesky factorization

**Syntax**

**Description**

```
R1 = cholupdate(R,x)
```

where `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 factorization.

```
[R1,p] = cholupdate(R,x,'-')
```

will not return an error message. If `p`

is `0`

, `R1`

is the Cholesky factor of `A - x*x`

'. If `p`

is greater than `0`

, `R1`

is the Cholesky factor of the original `A`

. If `p`

is `1`

, `cholupdate`

failed because the downdated matrix is not positive definite. If `p`

is `2`

, `cholupdate`

failed because the upper triangle of `R`

was not a valid Cholesky factor.

**Remarks**

`cholupdate `

works only for full matrices.

**Example**

A = pascal(4) A = 1 1 1 1 1 2 3 4 1 3 6 10 1 4 10 20 R = chol(A) R = 1 1 1 1 0 1 2 3 0 0 1 3 0 0 0 1 x = [0 0 0 1]';

This is called a rank one update to `A`

since `rank(x*x')`

is `1`

:

Instead of computing the Cholesky factor with `R1 = chol(A + x*x')`

, we can use `cholupdate`

:

Next destroy the positive definiteness (and actually make the matrix singular) by subtracting `1`

from the last element of `A`

. The downdated matrix is:

R1 = chol(A-x*x') ??? Error using ==> chol Matrix must be positive definite. R1 = cholupdate(R,x,'-') ??? Error using ==> cholupdate Downdated matrix must be positive definite.

However, subtracting `0.5`

from the last element of `A`

produces a positive definite matrix, and we can use `cholupdate`

to compute its Cholesky factor:

x = [0 0 0 1/sqrt(2)]'; R1 = cholupdate(R,x,'-') R1 = 1.0000 1.0000 1.0000 1.0000 0 1.0000 2.0000 3.0000 0 0 1.0000 3.0000 0 0 0 0.7071

**Algorithm **

`cholupdate`

uses the algorithms from the LINPACK subroutines `ZCHUD`

and `ZCHDD`

. `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.

**See Also**

**References **

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

cholinc | circshift |

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