SLAED2(3F) SLAED2(3F)
SLAED2 - merge the two sets of eigenvalues together into a single sorted
set
SUBROUTINE SLAED2( K, N, D, Q, LDQ, INDXQ, RHO, CUTPNT, Z, DLAMDA, Q2,
LDQ2, INDXC, W, INDXP, INDX, COLTYP, INFO )
INTEGER CUTPNT, INFO, K, LDQ, LDQ2, N
REAL RHO
INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ), INDXQ(
* )
REAL D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( LDQ2, * ), W( *
), Z( * )
SLAED2 merges the two sets of eigenvalues together into a single sorted
set. Then it tries to deflate the size of the problem. There are two
ways in which deflation can occur: when two or more eigenvalues are
close together or if there is a tiny entry in the Z vector. For each
such occurrence the order of the related secular equation problem is
reduced by one.
K (output) INTEGER
The number of non-deflated eigenvalues, and the order of the
related secular equation. 0 <= K <=N.
N (input) INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
D (input/output) REAL array, dimension (N)
On entry, D contains the eigenvalues of the two submatrices to be
combined. On exit, D contains the trailing (N-K) updated
eigenvalues (those which were deflated) sorted into increasing
order.
Q (input/output) REAL array, dimension (LDQ, N)
On entry, Q contains the eigenvectors of two submatrices in the
two square blocks with corners at (1,1), (CUTPNT,CUTPNT) and
(CUTPNT+1, CUTPNT+1), (N,N). On exit, Q contains the trailing
(N-K) updated eigenvectors (those which were deflated) in its last
N-K columns.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
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SLAED2(3F) SLAED2(3F)
INDXQ (input/output) INTEGER array, dimension (N)
The permutation which separately sorts the two sub-problems in D
into ascending order. Note that elements in the second half of
this permutation must first have CUTPNT added to their values.
Destroyed on exit.
RHO (input/output) REAL
On entry, the off-diagonal element associated with the rank-1 cut
which originally split the two submatrices which are now being
recombined. On exit, RHO has been modified to the value required
by SLAED3.
CUTPNT (input) INTEGER The location of the last eigenvalue in the
leading sub-matrix. min(1,N) <= CUTPNT <= N.
Z (input) REAL array, dimension (N)
On entry, Z contains the updating vector (the last row of the
first sub-eigenvector matrix and the first row of the second subeigenvector
matrix). On exit, the contents of Z have been
destroyed by the updating process.
DLAMDA (output) REAL array, dimension (N) A copy of the first K
eigenvalues which will be used by SLAED3 to form the secular
equation.
Q2 (output) REAL array, dimension (LDQ2, N)
A copy of the first K eigenvectors which will be used by SLAED3 in
a matrix multiply (SGEMM) to solve for the new eigenvectors. Q2
is arranged into three blocks. The first block contains non-zero
elements only at and above CUTPNT, the second contains non-zero
elements only below CUTPNT, and the third is dense.
LDQ2 (input) INTEGER
The leading dimension of the array Q2. LDQ2 >= max(1,N).
INDXC (output) INTEGER array, dimension (N)
The permutation used to arrange the columns of the deflated Q
matrix into three groups: the first group contains non-zero
elements only at and above CUTPNT, the second contains non-zero
elements only below CUTPNT, and the third is dense.
W (output) REAL array, dimension (N)
The first k values of the final deflation-altered z-vector which
will be passed to SLAED3.
INDXP (workspace) INTEGER array, dimension (N)
The permutation used to place deflated values of D at the end of
the array. INDXP(1:K) points to the nondeflated D-values
and INDXP(K+1:N) points to the deflated eigenvalues.
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SLAED2(3F) SLAED2(3F)
INDX (workspace) INTEGER array, dimension (N)
The permutation used to sort the contents of D into ascending
order.
COLTYP (workspace/output) INTEGER array, dimension (N) During
execution, a label which will indicate which of the following
types a column in the Q2 matrix is:
1 : non-zero in the upper half only;
2 : non-zero in the lower half only;
3 : dense;
4 : deflated. On exit, COLTYP(i) is the number of columns of type
i, for i=1 to 4 only.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
SLAED2(3F) SLAED2(3F)
SLAED2 - merge the two sets of eigenvalues together into a single sorted
set
SUBROUTINE SLAED2( K, N, D, Q, LDQ, INDXQ, RHO, CUTPNT, Z, DLAMDA, Q2,
LDQ2, INDXC, W, INDXP, INDX, COLTYP, INFO )
INTEGER CUTPNT, INFO, K, LDQ, LDQ2, N
REAL RHO
INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ), INDXQ(
* )
REAL D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( LDQ2, * ), W( *
), Z( * )
SLAED2 merges the two sets of eigenvalues together into a single sorted
set. Then it tries to deflate the size of the problem. There are two
ways in which deflation can occur: when two or more eigenvalues are
close together or if there is a tiny entry in the Z vector. For each
such occurrence the order of the related secular equation problem is
reduced by one.
K (output) INTEGER
The number of non-deflated eigenvalues, and the order of the
related secular equation. 0 <= K <=N.
N (input) INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
D (input/output) REAL array, dimension (N)
On entry, D contains the eigenvalues of the two submatrices to be
combined. On exit, D contains the trailing (N-K) updated
eigenvalues (those which were deflated) sorted into increasing
order.
Q (input/output) REAL array, dimension (LDQ, N)
On entry, Q contains the eigenvectors of two submatrices in the
two square blocks with corners at (1,1), (CUTPNT,CUTPNT) and
(CUTPNT+1, CUTPNT+1), (N,N). On exit, Q contains the trailing
(N-K) updated eigenvectors (those which were deflated) in its last
N-K columns.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
Page 1
SLAED2(3F) SLAED2(3F)
INDXQ (input/output) INTEGER array, dimension (N)
The permutation which separately sorts the two sub-problems in D
into ascending order. Note that elements in the second half of
this permutation must first have CUTPNT added to their values.
Destroyed on exit.
RHO (input/output) REAL
On entry, the off-diagonal element associated with the rank-1 cut
which originally split the two submatrices which are now being
recombined. On exit, RHO has been modified to the value required
by SLAED3.
CUTPNT (input) INTEGER The location of the last eigenvalue in the
leading sub-matrix. min(1,N) <= CUTPNT <= N.
Z (input) REAL array, dimension (N)
On entry, Z contains the updating vector (the last row of the
first sub-eigenvector matrix and the first row of the second subeigenvector
matrix). On exit, the contents of Z have been
destroyed by the updating process.
DLAMDA (output) REAL array, dimension (N) A copy of the first K
eigenvalues which will be used by SLAED3 to form the secular
equation.
Q2 (output) REAL array, dimension (LDQ2, N)
A copy of the first K eigenvectors which will be used by SLAED3 in
a matrix multiply (SGEMM) to solve for the new eigenvectors. Q2
is arranged into three blocks. The first block contains non-zero
elements only at and above CUTPNT, the second contains non-zero
elements only below CUTPNT, and the third is dense.
LDQ2 (input) INTEGER
The leading dimension of the array Q2. LDQ2 >= max(1,N).
INDXC (output) INTEGER array, dimension (N)
The permutation used to arrange the columns of the deflated Q
matrix into three groups: the first group contains non-zero
elements only at and above CUTPNT, the second contains non-zero
elements only below CUTPNT, and the third is dense.
W (output) REAL array, dimension (N)
The first k values of the final deflation-altered z-vector which
will be passed to SLAED3.
INDXP (workspace) INTEGER array, dimension (N)
The permutation used to place deflated values of D at the end of
the array. INDXP(1:K) points to the nondeflated D-values
and INDXP(K+1:N) points to the deflated eigenvalues.
Page 2
SLAED2(3F) SLAED2(3F)
INDX (workspace) INTEGER array, dimension (N)
The permutation used to sort the contents of D into ascending
order.
COLTYP (workspace/output) INTEGER array, dimension (N) During
execution, a label which will indicate which of the following
types a column in the Q2 matrix is:
1 : non-zero in the upper half only;
2 : non-zero in the lower half only;
3 : dense;
4 : deflated. On exit, COLTYP(i) is the number of columns of type
i, for i=1 to 4 only.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
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