Purpose
To compute the eigenvalues of a selected 2-by-2 diagonal block pair of an upper quasi-triangular pencil, to reduce the selected block pair to the standard form and to split it in the case of real eigenvalues, by constructing orthogonal matrices UT and VT. The transformations UT and VT are applied to the pair (A,E) by computing (UT'*A*VT, UT'*E*VT ), to the matrices U and V, by computing U*UT and V*VT.Specification
SUBROUTINE MB03QW( N, L, A, LDA, E, LDE, U, LDU, V, LDV,
$ ALPHAR, ALPHAI, BETA, INFO )
C .. Scalar Arguments ..
INTEGER INFO, L, LDA, LDE, LDU, LDV, N
DOUBLE PRECISION ALPHAI(*), ALPHAR(*), BETA(*)
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), E(LDE,*), U(LDU,*), V(LDV,*)
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the matrices A, E, UT and VT. N >= 2.
L (input) INTEGER
Specifies the position of the block. 1 <= L < N.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the upper quasi-triangular matrix A whose
selected 2-by-2 diagonal block is to be processed.
On exit, the leading N-by-N part of this array contains
the upper quasi-triangular matrix A after its selected
block has been split and/or put in the LAPACK standard
form.
LDA INTEGER
The leading dimension of the array A. LDA >= N.
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading N-by-N part of this array must
contain the upper triangular matrix E whose selected
2-by-2 diagonal block is to be processed.
On exit, the leading N-by-N part of this array contains
the transformed upper triangular matrix E (in the LAPACK
standard form).
LDE INTEGER
The leading dimension of the array E. LDE >= N.
U (input/output) DOUBLE PRECISION array, dimension (LDU,N)
On entry, the leading N-by-N part of this array must
contain a transformation matrix U.
On exit, the leading N-by-N part of this array contains
U*UT, where UT is the transformation matrix used to
split and/or standardize the selected block pair.
LDU INTEGER
The leading dimension of the array U. LDU >= N.
V (input/output) DOUBLE PRECISION array, dimension (LDV,N)
On entry, the leading N-by-N part of this array must
contain a transformation matrix V.
On exit, the leading N-by-N part of this array contains
V*VT, where VT is the transformation matrix used to
split and/or standardize the selected block pair.
LDV INTEGER
The leading dimension of the array V. LDV >= N.
ALPHAR (output) DOUBLE PRECISION array, dimension (2)
ALPHAI (output) DOUBLE PRECISION array, dimension (2)
BETA (output) DOUBLE PRECISION array, dimension (2)
(ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
block pair of the pencil (A,B), k=1,2, i = sqrt(-1).
Note that BETA(k) may be zero.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Let A1 = ( A(L,L) A(L,L+1) ), E1 = ( E(L,L) E(L,L+1) ),
( A(L+1,L) A(L+1,L+1) ) ( E(L+1,L) E(L+1,L+1) )
be the specified 2-by-2 diagonal block pair of the pencil (A,B).
If the eigenvalues of (A1,E1) are complex, then they are computed
and stored in ER and EI, where the real part is stored in ER and
the positive imaginary part in EI. The 2-by-2 block pair is
reduced if necessary to the standard form, such that
A(L,L) = A(L+1,L+1), and A(L,L+1) and A(L+1,L) have oposite signs.
If the eigenvalues are real, the 2-by-2 block pair is reduced to
upper triangular form such that ABS(A(L,L)) >= ABS(A(L+1,L+1)).
In both cases, an orthogonal rotation U1' is constructed such that
U1'*A1*U1 has the appropriate form. Let UT be an extension of U1
to an N-by-N orthogonal matrix, using identity submatrices. Then A
and E are replaced by UT'*A*UT and UT'*E*UT, and the contents of
the arrays U and V are U * UT and V * VT, respectively.
Further Comments
NoneExample
Program Text
NoneProgram Data
NoneProgram Results
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