SLAED3(3F) SLAED3(3F)
SLAED3  find the roots of the secular equation, as defined by the values
in D, W, and RHO, between KSTART and KSTOP
SUBROUTINE SLAED3( K, KSTART, KSTOP, N, D, Q, LDQ, RHO, CUTPNT, DLAMDA,
Q2, LDQ2, INDXC, CTOT, W, S, LDS, INFO )
INTEGER CUTPNT, INFO, K, KSTART, KSTOP, LDQ, LDQ2, LDS, N
REAL RHO
INTEGER CTOT( * ), INDXC( * )
REAL D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( LDQ2, * ), S(
LDS, * ), W( * )
SLAED3 finds the roots of the secular equation, as defined by the values
in D, W, and RHO, between KSTART and KSTOP. It makes the appropriate
calls to SLAED4 and then updates the eigenvectors by multiplying the
matrix of eigenvectors of the pair of eigensystems being combined by the
matrix of eigenvectors of the KbyK system which is solved here.
This code makes very mild assumptions about floating point arithmetic. It
will work on machines with a guard digit in add/subtract, or on those
binary machines without guard digits which subtract like the Cray XMP,
Cray YMP, Cray C90, or Cray2. It could conceivably fail on
hexadecimal or decimal machines without guard digits, but we know of
none.
K (input) INTEGER
The number of terms in the rational function to be solved by
SLAED4. K >= 0.
KSTART (input) INTEGER
KSTOP (input) INTEGER The updated eigenvalues Lambda(I), KSTART
<= I <= KSTOP are to be computed. 1 <= KSTART <= KSTOP <= K.
N (input) INTEGER
The number of rows and columns in the Q matrix. N >= K
(deflation may result in N>K).
D (output) REAL array, dimension (N)
D(I) contains the updated eigenvalues for KSTART <= I <= KSTOP.
Q (output) REAL array, dimension (LDQ,N)
Initially the first K columns are used as workspace. On output
the columns KSTART to KSTOP contain the updated eigenvectors.
Page 1
SLAED3(3F) SLAED3(3F)
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
RHO (input) REAL
The value of the parameter in the rank one update equation. RHO
>= 0 required.
CUTPNT (input) INTEGER
The location of the last eigenvalue in the leading submatrix.
min(1,N) <= CUTPNT <= N.
DLAMDA (input/output) REAL array, dimension (K)
The first K elements of this array contain the old roots of the
deflated updating problem. These are the poles of the secular
equation. May be changed on output by having lowest order bit set
to zero on Cray XMP, Cray YMP, Cray2, or Cray C90, as
described above.
Q2 (input) REAL array, dimension (LDQ2, N)
The first K columns of this matrix contain the nondeflated
eigenvectors for the split problem.
LDQ2 (input) INTEGER
The leading dimension of the array Q2. LDQ2 >= max(1,N).
INDXC (input) INTEGER array, dimension (N)
The permutation used to arrange the columns of the deflated Q
matrix into three groups: the first group contains nonzero
elements only at and above CUTPNT, the second contains nonzero
elements only below CUTPNT, and the third is dense. The rows of
the eigenvectors found by SLAED4 must be likewise permuted before
the matrix multiply can take place.
CTOT (input) INTEGER array, dimension (4)
A count of the total number of the various types of columns in Q,
as described in INDXC. The fourth column type is any column
which has been deflated.
W (input/output) REAL array, dimension (K)
The first K elements of this array contain the components of the
deflationadjusted updating vector. Destroyed on output.
S (workspace) REAL array, dimension (LDS, K)
Will contain the eigenvectors of the repaired matrix which will
be multiplied by the previously accumulated eigenvectors to
update the system.
LDS (input) INTEGER
The leading dimension of S. LDS >= max(1,K).
Page 2
SLAED3(3F) SLAED3(3F)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = i, the ith argument had an illegal value.
> 0: if INFO = 1, an eigenvalue did not converge
SLAED3(3F) SLAED3(3F)
SLAED3  find the roots of the secular equation, as defined by the values
in D, W, and RHO, between KSTART and KSTOP
SUBROUTINE SLAED3( K, KSTART, KSTOP, N, D, Q, LDQ, RHO, CUTPNT, DLAMDA,
Q2, LDQ2, INDXC, CTOT, W, S, LDS, INFO )
INTEGER CUTPNT, INFO, K, KSTART, KSTOP, LDQ, LDQ2, LDS, N
REAL RHO
INTEGER CTOT( * ), INDXC( * )
REAL D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( LDQ2, * ), S(
LDS, * ), W( * )
SLAED3 finds the roots of the secular equation, as defined by the values
in D, W, and RHO, between KSTART and KSTOP. It makes the appropriate
calls to SLAED4 and then updates the eigenvectors by multiplying the
matrix of eigenvectors of the pair of eigensystems being combined by the
matrix of eigenvectors of the KbyK system which is solved here.
This code makes very mild assumptions about floating point arithmetic. It
will work on machines with a guard digit in add/subtract, or on those
binary machines without guard digits which subtract like the Cray XMP,
Cray YMP, Cray C90, or Cray2. It could conceivably fail on
hexadecimal or decimal machines without guard digits, but we know of
none.
K (input) INTEGER
The number of terms in the rational function to be solved by
SLAED4. K >= 0.
KSTART (input) INTEGER
KSTOP (input) INTEGER The updated eigenvalues Lambda(I), KSTART
<= I <= KSTOP are to be computed. 1 <= KSTART <= KSTOP <= K.
N (input) INTEGER
The number of rows and columns in the Q matrix. N >= K
(deflation may result in N>K).
D (output) REAL array, dimension (N)
D(I) contains the updated eigenvalues for KSTART <= I <= KSTOP.
Q (output) REAL array, dimension (LDQ,N)
Initially the first K columns are used as workspace. On output
the columns KSTART to KSTOP contain the updated eigenvectors.
Page 1
SLAED3(3F) SLAED3(3F)
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
RHO (input) REAL
The value of the parameter in the rank one update equation. RHO
>= 0 required.
CUTPNT (input) INTEGER
The location of the last eigenvalue in the leading submatrix.
min(1,N) <= CUTPNT <= N.
DLAMDA (input/output) REAL array, dimension (K)
The first K elements of this array contain the old roots of the
deflated updating problem. These are the poles of the secular
equation. May be changed on output by having lowest order bit set
to zero on Cray XMP, Cray YMP, Cray2, or Cray C90, as
described above.
Q2 (input) REAL array, dimension (LDQ2, N)
The first K columns of this matrix contain the nondeflated
eigenvectors for the split problem.
LDQ2 (input) INTEGER
The leading dimension of the array Q2. LDQ2 >= max(1,N).
INDXC (input) INTEGER array, dimension (N)
The permutation used to arrange the columns of the deflated Q
matrix into three groups: the first group contains nonzero
elements only at and above CUTPNT, the second contains nonzero
elements only below CUTPNT, and the third is dense. The rows of
the eigenvectors found by SLAED4 must be likewise permuted before
the matrix multiply can take place.
CTOT (input) INTEGER array, dimension (4)
A count of the total number of the various types of columns in Q,
as described in INDXC. The fourth column type is any column
which has been deflated.
W (input/output) REAL array, dimension (K)
The first K elements of this array contain the components of the
deflationadjusted updating vector. Destroyed on output.
S (workspace) REAL array, dimension (LDS, K)
Will contain the eigenvectors of the repaired matrix which will
be multiplied by the previously accumulated eigenvectors to
update the system.
LDS (input) INTEGER
The leading dimension of S. LDS >= max(1,K).
Page 2
SLAED3(3F) SLAED3(3F)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = i, the ith argument had an illegal value.
> 0: if INFO = 1, an eigenvalue did not converge
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