BSTREXC(3) ScaLAPACK routine of NEC Numeric Library Collection BSTREXC(3)
NAME
BSTREXC - reorders the real Schur factorization of a real matrix A =
Q*T*Q**T, so that the diagonal block of T with row index IFST is moved
to row ILST
SYNOPSIS
SUBROUTINE BSTREXC( N, T, LDT, IFST, ILST, NITRAF, ITRAF, NDTRAF,
DTRAF, WORK, INFO )
INTEGER IFST, ILST, INFO, LDT, N, NDTRAF, NITRAF
INTEGER ITRAF( * )
REAL DTRAF( * ), T( LDT, * ), WORK( * )
PURPOSE
BSTREXC reorders the real Schur factorization of a real matrix A =
Q*T*Q**T, so that the diagonal block of T with row index IFST is moved
to row ILST.
The real Schur form T is reordered by an orthogonal similarity trans-
formation Z**T*T*Z. In contrast to the LAPACK routine DTREXC, the
orthogonal matrix Z is not explicitly constructed but represented by
paramaters contained in the arrays ITRAF and DTRAF, see further
details.
T must be in Schur canonical form (as returned by DHSEQR), that is,
block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
2-by-2 diagonal block has its diagonal elements equal and its off-diag-
onal elements of opposite sign.
ARGUMENTS
N (input) INTEGER
The order of the matrix T. N >= 0.
T (input/output) REAL array, dimension (LDT,N)
On entry, the upper quasi-triangular matrix T, in Schur Schur
canonical form.
On exit, the reordered upper quasi-triangular matrix, again in
Schur canonical form.
LDT (input) INTEGER
The leading dimension of the array T. LDT >= max(1,N).
IFST (input/output) INTEGER
ILST (input/output) INTEGER
Specify the reordering of the diagonal blocks of T.
The block with row index IFST is moved to row ILST, by a
sequence of transpositions between adjacent blocks.
On exit, if IFST pointed on entry to the second row of a 2-by-2
block, it is changed to point to the first row; ILST always
points to the first row of the block in its final position
(which may differ from its input value by +1 or -1).
1 <= IFST <= N; 1 <= ILST <= N.
NITRAF (input/output) INTEGER
On entry, length of the array ITRAF.
As a minimum requirement, NITRAF >= max(1,|ILST-IFST|).
If there are 2-by-2 blocks in T then NITRAF must be larger; a
safe choice is NITRAF >= max(1,2*|ILST-IFST|).
On exit, actual length of the array ITRAF.
ITRAF (output) INTEGER array, length NITRAF
List of parameters for representing the transformation matrix
Z, see further details.
NDTRAF (input/output) INTEGER
On entry, length of the array DTRAF.
As a minimum requirement, NDTRAF >= max(1,2*|ILST-IFST|).
If there are 2-by-2 blocks in T then NDTRAF must be larger; a
safe choice is NDTRAF >= max(1,5*|ILST-IFST|).
On exit, actual length of the array DTRAF.
DTRAF (output) REAL array, length NDTRAF
List of parameters for representing the transformation matrix
Z, see further details.
WORK (workspace) REAL array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1: two adjacent blocks were too close to swap (the problem
is very ill-conditioned); T may have been partially
reordered, and ILST points to the first row of the
current position of the block being moved.
= 2: the 2 by 2 block to be reordered split into two 1 by 1
blocks and the second block failed to swap with an
adjacent block. ILST points to the first row of the
current position of the whole block being moved.
FURTHER DETAILS
The orthogonal transformation matrix Z is a product of NITRAF elemen-
tary orthogonal transformations. The parameters defining these trans-
formations are stored in the arrays ITRAF and DTRAF as follows:
Consider the i-th transformation acting on rows/columns POS, POS+1, ...
If this transformation is
(1) a Givens rotation with cosine C and sine S then
ITRAF(i) = POS,
DTRAF(j) = C, DTRAF(j+1) = S;
(2) a Householder reflector H = I - tau * v * v' with
v = [ 1; v2; v3 ] then
ITRAF(i) = N + POS,
DTRAF(j) = tau, DTRAF(j+1) = v2, DTRAF(j+2) = v3;
(3) a Householder reflector H = I - tau * v * v' with
v = [ v1; v2; 1 ] then
ITRAF(i) = 2*N + POS,
DTRAF(j) = v1, DTRAF(j+1) = v2, DTRAF(j+2) = tau;
Note that the parameters in DTRAF are stored consecutively.
ScaLAPACK routine 31 October 2017 BSTREXC(3)