cholupdate

Rank 1 update to Cholesky factorization

Syntax

*

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

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 factoriza- tion.

[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:

*

A + x*x'
ans =

*

1 1 1 1
1 2 3 4
1 3 6 10
1 4 10 21

Instead of computing the Cholesky factor with R1 = chol(A + x*x'), we can use cholupdate:

*

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 1.4142

Next destroy the positive definiteness (and actually make the matrix singular) by subtracting 1 from the last element of A. The downdated matrix is:

*

A - x*x'
ans =

1 1 1 1
1 2 3 4
1 3 6 10
1 4 10 19

Compare chol with cholupdate:

*

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.

circshift

Shift array circularly

Syntax

*

B = circshift(A,shiftsize)

Description

B = circshift(A,shiftsize) circularly shifts the values in the array, A, by shiftsize elements. shiftsize is a vector of integer scalars where the n-th element specifies the shift amount for the n-th dimension of array A. If an element in shiftsize is positive, the values of A are shifted down (or to the right). If it is negative, the values of A are shifted up (or to the left). If it is 0, the values in that dimension are not shifted.

Example

Circularly shift first dimension values down by 1.

*

A = [ 1 2 3;4 5 6; 7 8 9]
A =
1 2 3
4 5 6
7 8 9

B = circshift(A,1)
B =
7 8 9
1 2 3
4 5 6

Circularly shift first dimension values down by 1 and second dimension values to the left by 1.

*

B = circshift(A,[1 -1]);
B =
8 9 7
2 3 1
5 6 4

cla

Clear current axes

Syntax

*

cla
cla reset

Description

cla deletes from the current axes all graphics objects whose handles are not hidden (i.e., their HandleVisibility property is set to on).

cla reset deletes from the current axes all graphics objects regardless of the setting of their HandleVisibility property and resets all axes properties, except Position and Units, to their default values.

Remarks

The cla command behaves the same way when issued on the command line as it does in callback routines - it does not recognize the HandleVisibility setting of callback. This means that when issued from within a callback routine, cla deletes only those objects whose HandleVisibility property is set to on.