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

Required Arguments

R — N by N upper triangular matrix containing the upper triangular factor to be updated.   (Input)
Only the upper triangle of R is referenced.

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

RNEW — N by N upper triangular matrix containing the updated triangular factor of
RT R + XXT.   (Output)
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).

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

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 LUPCH (R, X, RNEW [,…])

Specific:                             The specific interface names are S_LUPCH and D_LUPCH.

FORTRAN 77 Interface

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

Double:                              The double precision name is DLUPCH.

Description

The routine LUPCH is based on the LINPACK routine SCHUD; 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, LUPCH computes the factorization

In the program

is called RNEW.

LUPCH determines an orthogonal matrix U as the product GN…G1 of Givens rotations, such that

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

is obtained.

Each Givens rotation, Gi, is chosen to zero out an element in xT. The matrix
Gi is (N + 1) × (N + 1) and has the form

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

Example

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

 

      USE IMSL_LIBRARIES
!                                 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 = (  1.0  -3.0   2.0)

!                                     ( -3.0  10.0  -5.0)

!                                     (  2.0  -5.0   6.0)

!

      DATA A/1.0, -3.0, 2.0, -3.0, 10.0, -5.0, 2.0, -5.0, 6.0/

!

!                                 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)

!                                 Update the factorization

      CALL LUPCH (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   1.000  -3.000   2.000
2           1.000   1.000
3                   1.000

          Z

     1        2        3

1860.0    433.0   -254.0

      FACNEW

     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

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