DLATDF - use the LU factorization of the n-by-n matrix Z computed by DGETC2 and computes a contribution to the reciprocal Dif-estimate by solving Z * x = b for x, and choosing the r.h.s

SUBROUTINE DLATDF(

IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, JPIV )

INTEGER IJOB, LDZ, N

DOUBLE PRECISION RDSCAL, RDSUM

INTEGER IPIV( * ), JPIV( * )

DOUBLE PRECISION RHS( * ), Z( LDZ, * )

DLATDF uses the LU factorization of the n-by-n matrix Z computed by DGETC2 and computes a contribution to the reciprocal Dif-estimate by solving Z * x = b for x, and choosing the r.h.s. b such that the norm of x is as large as possible. On entry RHS = b holds the contribution from earlier solved sub-systems, and on return RHS = x.

The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q, where P and Q are permutation matrices. L is lower triangular with unit diagonal elements and U is upper triangular.

IJOB (input) INTEGER

IJOB = 2: First compute an approximative null-vector e of Z using DGECON, e is normalized and solve for Zx = +-e - f with the sign giving the greater value of 2-norm(x). About 5 times as expensive as Default. IJOB .ne. 2: Local look ahead strategy where all entries of the r.h.s. b is choosen as either +1 or -1 (Default).

N (input) INTEGER

The number of columns of the matrix Z.

Z (input) DOUBLE PRECISION array, dimension (LDZ, N)

On entry, the LU part of the factorization of the n-by-n matrix Z computed by DGETC2: Z = P * L * U * Q

LDZ (input) INTEGER

The leading dimension of the array Z. LDA >= max(1, N).

RHS (input/output) DOUBLE PRECISION array, dimension N.

On entry, RHS contains contributions from other subsystems. On exit, RHS contains the solution of the subsystem with entries acoording to the value of IJOB (see above).

RDSUM (input/output) DOUBLE PRECISION

On entry, the sum of squares of computed contributions to the Dif-estimate under computation by DTGSYL, where the scaling factor RDSCAL (see below) has been factored out. On exit, the corresponding sum of squares updated with the contributions from the current sub-system. If TRANS = 'T' RDSUM is not touched. NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.

RDSCAL (input/output) DOUBLE PRECISION

On entry, scaling factor used to prevent overflow in RDSUM. On exit, RDSCAL is updated w.r.t. the current contributions in RDSUM. If TRANS = 'T', RDSCAL is not touched. NOTE: RDSCAL only makes sense when DTGSY2 is called by DTGSYL.

IPIV (input) INTEGER array, dimension (N).

The pivot indices; for 1 <= i <= N, row i of the matrix has been interchanged with row IPIV(i).

JPIV (input) INTEGER array, dimension (N).

The pivot indices; for 1 <= j <= N, column j of the matrix has been interchanged with column JPIV(j).

Based on contributions by

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

This routine is a further developed implementation of algorithm BSOLVE in [1] using complete pivoting in the LU factorization.

[1] Bo Kagstrom and Lars Westin,

Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.

[2] Peter Poromaa,

On Efficient and Robust Estimators for the Separation between two Regular Matrix Pairs with Applications in Condition Estimation. Report IMINF-95.05, Departement of Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.