functioncholeskyDownDate2

Information

Syntax

Ldd = Matrices.choleskyDownDate(L, v);
Ldd = Matrices.choleskyDownDate(L, v, true);

Description

This function computes the rank-1-downdated Cholesky factorization Ldd, with

Add = Ldd*Ldd^T = A - v*v^T = L*L^T - v*v^T

from the input L, i.e. the left (lower) Cholesky factor of the original matrix A. The algorithm is taken from [1].

Matrix Ldd is calculated by

[v, Ldd]^T = H *[0, L]^T

with orthogonal Matrix H such that

v*v^T + Ldd*Ldd^T = [v, Ldd] * [v, Ldd]^T = [0, L]*H^T *H*[0, L]^T = [0, L]*[0, L]^T = L*L^T = A,

i.e., by orthogonal transformation

H = H_1*...*H_n.

The matrices H_i are Givens matrices computed such that

H_1*H_2*...*H_n*[z, a^T]^T = [1, 0, ..., 0]^T,

with a is the solution of

L*a = v

and

z = ||a||.

The following sequence illustrate the principle of calculating the H_i, starting with H_n

|z|       |z|       |z|       |z|
|a|  H_3  |a|  H_2  |a|  H_1  |0|
|a| ----> |a| --->  |0| --->  |0|
|a|       |0|       |0|       |0|

Note, that the z and a are different in each column. It is shown in [1] that this algorithm results in the modified Cholesky factor Ldd.

With the boolean input "upper" the user specifies whether the matrix L is lower or upper triangular matrix (left or right Cholesky factor). If "upper==true", the output Ldd is also upper triangular. Default is "upper==false".

References

 [1] Dongarra J. J., Bunch J. R., Moler G. B., Stewart G.W. (1987):

LINPACK Users' Guide. Society for Industrial Mathematics.
 

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Revisions