CLAROT
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
November 2006
Purpose
CLAROT applies a (Givens) rotation to two adjacent rows or
columns, where one element of the first and/or last column/row
for use on matrices stored in some format other than GE, so
that elements of the matrix may be used or modified for which
no array element is provided.
One example is a symmetric matrix in SB format (bandwidth=4), for
which UPLO='L': Two adjacent rows will have the format:
row j: * * * * * . . . .
row j+1: * * * * * . . . .
'*' indicates elements for which storage is provided,
'.' indicates elements for which no storage is provided, but
are not necessarily zero; their values are determined by
symmetry. ' ' indicates elements which are necessarily zero,
and have no storage provided.
Those columns which have two '*'s can be handled by SROT.
Those columns which have no '*'s can be ignored, since as long
as the Givens rotations are carefully applied to preserve
symmetry, their values are determined.
Those columns which have one '*' have to be handled separately,
by using separate variables "p" and "q":
row j: * * * * * p . . .
row j+1: q * * * * * . . . .
The element p would have to be set correctly, then that column
is rotated, setting p to its new value. The next call to
CLAROT would rotate columns j and j+1, using p, and restore
symmetry. The element q would start out being zero, and be
made non-zero by the rotation. Later, rotations would presumably
be chosen to zero q out.
Typical Calling Sequences: rotating the i-th and (i+1)-st rows.
General dense matrix:
CALL CLAROT(.TRUE.,.FALSE.,.FALSE., N, C,S,
A(i,1),LDA, DUMMY, DUMMY)
General banded matrix in GB format:
j = MAX(1, i-KL )
NL = MIN( N, i+KU+1 ) + 1-j
CALL CLAROT( .TRUE., i-KL.GE.1, i+KU.LT.N, NL, C,S,
A(KU+i+1-j,j),LDA-1, XLEFT, XRIGHT )
[ note that i+1-j is just MIN(i,KL+1) ]
Symmetric banded matrix in SY format, bandwidth K,
lower triangle only:
j = MAX(1, i-K )
NL = MIN( K+1, i ) + 1
CALL CLAROT( .TRUE., i-K.GE.1, .TRUE., NL, C,S,
A(i,j), LDA, XLEFT, XRIGHT )
Same, but upper triangle only:
NL = MIN( K+1, N-i ) + 1
CALL CLAROT( .TRUE., .TRUE., i+K.LT.N, NL, C,S,
A(i,i), LDA, XLEFT, XRIGHT )
Symmetric banded matrix in SB format, bandwidth K,
lower triangle only:
[ same as for SY, except:]
. . . .
A(i+1-j,j), LDA-1, XLEFT, XRIGHT )
[ note that i+1-j is just MIN(i,K+1) ]
Same, but upper triangle only:
. . .
A(K+1,i), LDA-1, XLEFT, XRIGHT )
Rotating columns is just the transpose of rotating rows, except
for GB and SB: (rotating columns i and i+1)
GB:
j = MAX(1, i-KU )
NL = MIN( N, i+KL+1 ) + 1-j
CALL CLAROT( .TRUE., i-KU.GE.1, i+KL.LT.N, NL, C,S,
A(KU+j+1-i,i),LDA-1, XTOP, XBOTTM )
[note that KU+j+1-i is just MAX(1,KU+2-i)]
SB: (upper triangle)
. . . . . .
A(K+j+1-i,i),LDA-1, XTOP, XBOTTM )
SB: (lower triangle)
. . . . . .
A(1,i),LDA-1, XTOP, XBOTTM )
columns, where one element of the first and/or last column/row
for use on matrices stored in some format other than GE, so
that elements of the matrix may be used or modified for which
no array element is provided.
One example is a symmetric matrix in SB format (bandwidth=4), for
which UPLO='L': Two adjacent rows will have the format:
row j: * * * * * . . . .
row j+1: * * * * * . . . .
'*' indicates elements for which storage is provided,
'.' indicates elements for which no storage is provided, but
are not necessarily zero; their values are determined by
symmetry. ' ' indicates elements which are necessarily zero,
and have no storage provided.
Those columns which have two '*'s can be handled by SROT.
Those columns which have no '*'s can be ignored, since as long
as the Givens rotations are carefully applied to preserve
symmetry, their values are determined.
Those columns which have one '*' have to be handled separately,
by using separate variables "p" and "q":
row j: * * * * * p . . .
row j+1: q * * * * * . . . .
The element p would have to be set correctly, then that column
is rotated, setting p to its new value. The next call to
CLAROT would rotate columns j and j+1, using p, and restore
symmetry. The element q would start out being zero, and be
made non-zero by the rotation. Later, rotations would presumably
be chosen to zero q out.
Typical Calling Sequences: rotating the i-th and (i+1)-st rows.
General dense matrix:
CALL CLAROT(.TRUE.,.FALSE.,.FALSE., N, C,S,
A(i,1),LDA, DUMMY, DUMMY)
General banded matrix in GB format:
j = MAX(1, i-KL )
NL = MIN( N, i+KU+1 ) + 1-j
CALL CLAROT( .TRUE., i-KL.GE.1, i+KU.LT.N, NL, C,S,
A(KU+i+1-j,j),LDA-1, XLEFT, XRIGHT )
[ note that i+1-j is just MIN(i,KL+1) ]
Symmetric banded matrix in SY format, bandwidth K,
lower triangle only:
j = MAX(1, i-K )
NL = MIN( K+1, i ) + 1
CALL CLAROT( .TRUE., i-K.GE.1, .TRUE., NL, C,S,
A(i,j), LDA, XLEFT, XRIGHT )
Same, but upper triangle only:
NL = MIN( K+1, N-i ) + 1
CALL CLAROT( .TRUE., .TRUE., i+K.LT.N, NL, C,S,
A(i,i), LDA, XLEFT, XRIGHT )
Symmetric banded matrix in SB format, bandwidth K,
lower triangle only:
[ same as for SY, except:]
. . . .
A(i+1-j,j), LDA-1, XLEFT, XRIGHT )
[ note that i+1-j is just MIN(i,K+1) ]
Same, but upper triangle only:
. . .
A(K+1,i), LDA-1, XLEFT, XRIGHT )
Rotating columns is just the transpose of rotating rows, except
for GB and SB: (rotating columns i and i+1)
GB:
j = MAX(1, i-KU )
NL = MIN( N, i+KL+1 ) + 1-j
CALL CLAROT( .TRUE., i-KU.GE.1, i+KL.LT.N, NL, C,S,
A(KU+j+1-i,i),LDA-1, XTOP, XBOTTM )
[note that KU+j+1-i is just MAX(1,KU+2-i)]
SB: (upper triangle)
. . . . . .
A(K+j+1-i,i),LDA-1, XTOP, XBOTTM )
SB: (lower triangle)
. . . . . .
A(1,i),LDA-1, XTOP, XBOTTM )
Arguments
| LROWS |
LOGICAL
If .TRUE., then CLAROT will rotate two rows. If .FALSE.,
then it will rotate two columns. Not modified. |
| LLEFT |
LOGICAL
If .TRUE., then XLEFT will be used instead of the
corresponding element of A for the first element in the second row (if LROWS=.FALSE.) or column (if LROWS=.TRUE.) If .FALSE., then the corresponding element of A will be used. Not modified. |
| LRIGHT |
LOGICAL
If .TRUE., then XRIGHT will be used instead of the
corresponding element of A for the last element in the first row (if LROWS=.FALSE.) or column (if LROWS=.TRUE.) If .FALSE., then the corresponding element of A will be used. Not modified. |
| NL |
INTEGER
The length of the rows (if LROWS=.TRUE.) or columns (if
LROWS=.FALSE.) to be rotated. If XLEFT and/or XRIGHT are used, the columns/rows they are in should be included in NL, e.g., if LLEFT = LRIGHT = .TRUE., then NL must be at least 2. The number of rows/columns to be rotated exclusive of those involving XLEFT and/or XRIGHT may not be negative, i.e., NL minus how many of LLEFT and LRIGHT are .TRUE. must be at least zero; if not, XERBLA will be called. Not modified. C, S - COMPLEX Specify the Givens rotation to be applied. If LROWS is true, then the matrix ( c s ) ( _ _ ) (-s c ) is applied from the left; if false, then the transpose (not conjugated) thereof is applied from the right. Note that in contrast to the output of CROTG or to most versions of CROT, both C and S are complex. For a Givens rotation, |C|**2 + |S|**2 should be 1, but this is not checked. Not modified. |
| A |
COMPLEX array.
The array containing the rows/columns to be rotated. The
first element of A should be the upper left element to be rotated. Read and modified. |
| LDA |
INTEGER
The "effective" leading dimension of A. If A contains
a matrix stored in GE, HE, or SY format, then this is just the leading dimension of A as dimensioned in the calling routine. If A contains a matrix stored in band (GB, HB, or SB) format, then this should be *one less* than the leading dimension used in the calling routine. Thus, if A were dimensioned A(LDA,*) in CLAROT, then A(1,j) would be the j-th element in the first of the two rows to be rotated, and A(2,j) would be the j-th in the second, regardless of how the array may be stored in the calling routine. [A cannot, however, actually be dimensioned thus, since for band format, the row number may exceed LDA, which is not legal FORTRAN.] If LROWS=.TRUE., then LDA must be at least 1, otherwise it must be at least NL minus the number of .TRUE. values in XLEFT and XRIGHT. Not modified. |
| XLEFT |
COMPLEX
If LLEFT is .TRUE., then XLEFT will be used and modified
instead of A(2,1) (if LROWS=.TRUE.) or A(1,2) (if LROWS=.FALSE.). Read and modified. |
| XRIGHT |
COMPLEX
If LRIGHT is .TRUE., then XRIGHT will be used and modified
instead of A(1,NL) (if LROWS=.TRUE.) or A(NL,1) (if LROWS=.FALSE.). Read and modified. |