LDNCH

Downdates the RT R Cholesky factorization of a real symmetric positive definite matrix after a rank-one matrix is removed.

Required Arguments

R — N by N upper triangular matrix containing the upper triangular factor to be downdated. (Input)

Only the upper triangle of R is referenced.

Only the upper triangle of R is referenced.

X — Vector of length N determining the rank-one matrix to be subtracted from the factorization RT R. (Input)

RNEW — N by N upper triangular matrix containing the downdated triangular factor of RT R − X XT. (Output)

Only the upper triangle of RNEW is referenced. If R is not needed, R and RNEW can share the same storage locations.

Only the upper triangle of RNEW is referenced. If R is not needed, R and RNEW can share the same storage locations.

Optional Arguments

N — Order of the matrix. (Input)

Default: N = size (R,2).

Default: N = size (R,2).

LDR — Leading dimension of R exactly as specified in the dimension statement of the calling program. (Input)

Default: LDR = size (R,1).

Default: LDR = size (R,1).

LDRNEW — Leading dimension of RNEW exactly as specified in the dimension statement of the calling program. (Input)

Default: LDRNEW = size (RNEW,1).

Default: LDRNEW = size (RNEW,1).

CS — Vector of length N containing the cosines of the rotations. (Output)

SN — Vector of length N containing the sines of the rotations. (Output)

FORTRAN 90 Interface

Generic: CALL LDNCH (R, X, RNEW [, …])

Specific: The specific interface names are S_LDNCH and D_LDNCH.

FORTRAN 77 Interface

Single: CALL LDNCH (N, R, LDR, X, RNEW, LDRNEW, CS, SN)

Double: The double precision name is DLDNCH.

Description

The routine LDNCH is based on the LINPACK routine SCHDD; see Dongarra et al. (1979).

The Cholesky factorization of a matrix is A = RT R, where R is an upper triangular matrix. Given this factorization, LDNCH computes the factorization

In the program

is called RNEW. This is not always possible, since A - xxT may not be positive definite.

LDNCH determines an orthogonal matrix U as the product GN…G1of Givens rotations, such that

By multiplying this equation by its transpose and noting that UT U = I, the desired result

is obtained.

Let a be the solution of the linear system RT a = x and let

The Givens rotations, Gi, are chosen such that

The Gi are (N + 1) × (N + 1) matrices of the form

where Ik is the identity matrix of order k; and ci = cosθi = CS(I), si = sinθi = SN(I) for some θi.

The Givens rotations are then used to form

The matrix

is upper triangular and

because

Comments

Informational error

Type | Code | Description |
---|---|---|

4 | 1 | RTR − X XT is not positive definite. R cannot be downdated. |

Example

A linear system Az = b is solved using the Cholesky factorization of A. This factorization is then downdated, and the system (A − xxT)z = b is solved using this downdated factorization.

USE LDNCH_INT

USE LFTDS_INT

USE LFSDS_INT

USE WRRRN_INT

! Declare variables

INTEGER LDA, LDFACT, N

PARAMETER (LDA=3, LDFACT=3, N=3)

REAL A(LDA,LDA), FACT(LDFACT,LDFACT), FACNEW(LDFACT,LDFACT), &

X(N), B(N), CS(N), SN(N), Z(N)

!

! Set values for A

! A = ( 10.0 3.0 5.0)

! ( 3.0 14.0 -3.0)

! ( 5.0 -3.0 7.0)

!

DATA A/10.0, 3.0, 5.0, 3.0, 14.0, -3.0, 5.0, -3.0, 7.0/

!

! Set values for X and B

DATA X/3.0, 2.0, 1.0/

DATA B/53.0, 20.0, 31.0/

! Factor the matrix A

CALL LFTDS (A, FACT)

! Solve the original system

CALL LFSDS (FACT, B, Z)

! Print the results

CALL WRRRN (’FACT’, FACT, ITRING=1)

CALL WRRRN (’Z’, Z, 1, N, 1)

! Downdate the factorization

CALL LDNCH (FACT, X, FACNEW)

! Solve the updated system

CALL LFSDS (FACNEW, B, Z)

! Print the results

CALL WRRRN (’FACNEW’, FACNEW, ITRING=1)

CALL WRRRN (’Z’, Z, 1, N, 1)

!

END

Output

FACT

1 2 3

1 3.162 0.949 1.581

2 3.619 -1.243

3 1.719

Z

1 2 3

4.000 1.000 2.000

FACNEW

1 2 3

1 1.000 -3.000 2.000

2 1.000 1.000

3 1.000

Z

1 2 3

1859.9 433.0 -254.0