LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
slarot.f
Go to the documentation of this file.
00001 *> \brief \b SLAROT
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *  Definition:
00009 *  ===========
00010 *
00011 *       SUBROUTINE SLAROT( LROWS, LLEFT, LRIGHT, NL, C, S, A, LDA, XLEFT,
00012 *                          XRIGHT )
00013 * 
00014 *       .. Scalar Arguments ..
00015 *       LOGICAL            LLEFT, LRIGHT, LROWS
00016 *       INTEGER            LDA, NL
00017 *       REAL               C, S, XLEFT, XRIGHT
00018 *       ..
00019 *       .. Array Arguments ..
00020 *       REAL               A( * )
00021 *       ..
00022 *  
00023 *
00024 *> \par Purpose:
00025 *  =============
00026 *>
00027 *> \verbatim
00028 *>
00029 *>    SLAROT applies a (Givens) rotation to two adjacent rows or
00030 *>    columns, where one element of the first and/or last column/row
00031 *>    for use on matrices stored in some format other than GE, so
00032 *>    that elements of the matrix may be used or modified for which
00033 *>    no array element is provided.
00034 *>
00035 *>    One example is a symmetric matrix in SB format (bandwidth=4), for
00036 *>    which UPLO='L':  Two adjacent rows will have the format:
00037 *>
00038 *>    row j:     C> C> C> C> C> .  .  .  .
00039 *>    row j+1:      C> C> C> C> C> .  .  .  .
00040 *>
00041 *>    '*' indicates elements for which storage is provided,
00042 *>    '.' indicates elements for which no storage is provided, but
00043 *>    are not necessarily zero; their values are determined by
00044 *>    symmetry.  ' ' indicates elements which are necessarily zero,
00045 *>     and have no storage provided.
00046 *>
00047 *>    Those columns which have two '*'s can be handled by SROT.
00048 *>    Those columns which have no '*'s can be ignored, since as long
00049 *>    as the Givens rotations are carefully applied to preserve
00050 *>    symmetry, their values are determined.
00051 *>    Those columns which have one '*' have to be handled separately,
00052 *>    by using separate variables "p" and "q":
00053 *>
00054 *>    row j:     C> C> C> C> C> p  .  .  .
00055 *>    row j+1:   q  C> C> C> C> C> .  .  .  .
00056 *>
00057 *>    The element p would have to be set correctly, then that column
00058 *>    is rotated, setting p to its new value.  The next call to
00059 *>    SLAROT would rotate columns j and j+1, using p, and restore
00060 *>    symmetry.  The element q would start out being zero, and be
00061 *>    made non-zero by the rotation.  Later, rotations would presumably
00062 *>    be chosen to zero q out.
00063 *>
00064 *>    Typical Calling Sequences: rotating the i-th and (i+1)-st rows.
00065 *>    ------- ------- ---------
00066 *>
00067 *>      General dense matrix:
00068 *>
00069 *>              CALL SLAROT(.TRUE.,.FALSE.,.FALSE., N, C,S,
00070 *>                      A(i,1),LDA, DUMMY, DUMMY)
00071 *>
00072 *>      General banded matrix in GB format:
00073 *>
00074 *>              j = MAX(1, i-KL )
00075 *>              NL = MIN( N, i+KU+1 ) + 1-j
00076 *>              CALL SLAROT( .TRUE., i-KL.GE.1, i+KU.LT.N, NL, C,S,
00077 *>                      A(KU+i+1-j,j),LDA-1, XLEFT, XRIGHT )
00078 *>
00079 *>              [ note that i+1-j is just MIN(i,KL+1) ]
00080 *>
00081 *>      Symmetric banded matrix in SY format, bandwidth K,
00082 *>      lower triangle only:
00083 *>
00084 *>              j = MAX(1, i-K )
00085 *>              NL = MIN( K+1, i ) + 1
00086 *>              CALL SLAROT( .TRUE., i-K.GE.1, .TRUE., NL, C,S,
00087 *>                      A(i,j), LDA, XLEFT, XRIGHT )
00088 *>
00089 *>      Same, but upper triangle only:
00090 *>
00091 *>              NL = MIN( K+1, N-i ) + 1
00092 *>              CALL SLAROT( .TRUE., .TRUE., i+K.LT.N, NL, C,S,
00093 *>                      A(i,i), LDA, XLEFT, XRIGHT )
00094 *>
00095 *>      Symmetric banded matrix in SB format, bandwidth K,
00096 *>      lower triangle only:
00097 *>
00098 *>              [ same as for SY, except:]
00099 *>                  . . . .
00100 *>                      A(i+1-j,j), LDA-1, XLEFT, XRIGHT )
00101 *>
00102 *>              [ note that i+1-j is just MIN(i,K+1) ]
00103 *>
00104 *>      Same, but upper triangle only:
00105 *>                   . . .
00106 *>                      A(K+1,i), LDA-1, XLEFT, XRIGHT )
00107 *>
00108 *>      Rotating columns is just the transpose of rotating rows, except
00109 *>      for GB and SB: (rotating columns i and i+1)
00110 *>
00111 *>      GB:
00112 *>              j = MAX(1, i-KU )
00113 *>              NL = MIN( N, i+KL+1 ) + 1-j
00114 *>              CALL SLAROT( .TRUE., i-KU.GE.1, i+KL.LT.N, NL, C,S,
00115 *>                      A(KU+j+1-i,i),LDA-1, XTOP, XBOTTM )
00116 *>
00117 *>              [note that KU+j+1-i is just MAX(1,KU+2-i)]
00118 *>
00119 *>      SB: (upper triangle)
00120 *>
00121 *>                   . . . . . .
00122 *>                      A(K+j+1-i,i),LDA-1, XTOP, XBOTTM )
00123 *>
00124 *>      SB: (lower triangle)
00125 *>
00126 *>                   . . . . . .
00127 *>                      A(1,i),LDA-1, XTOP, XBOTTM )
00128 *> \endverbatim
00129 *
00130 *  Arguments:
00131 *  ==========
00132 *
00133 *> \verbatim
00134 *>  LROWS  - LOGICAL
00135 *>           If .TRUE., then SLAROT will rotate two rows.  If .FALSE.,
00136 *>           then it will rotate two columns.
00137 *>           Not modified.
00138 *>
00139 *>  LLEFT  - LOGICAL
00140 *>           If .TRUE., then XLEFT will be used instead of the
00141 *>           corresponding element of A for the first element in the
00142 *>           second row (if LROWS=.FALSE.) or column (if LROWS=.TRUE.)
00143 *>           If .FALSE., then the corresponding element of A will be
00144 *>           used.
00145 *>           Not modified.
00146 *>
00147 *>  LRIGHT - LOGICAL
00148 *>           If .TRUE., then XRIGHT will be used instead of the
00149 *>           corresponding element of A for the last element in the
00150 *>           first row (if LROWS=.FALSE.) or column (if LROWS=.TRUE.) If
00151 *>           .FALSE., then the corresponding element of A will be used.
00152 *>           Not modified.
00153 *>
00154 *>  NL     - INTEGER
00155 *>           The length of the rows (if LROWS=.TRUE.) or columns (if
00156 *>           LROWS=.FALSE.) to be rotated.  If XLEFT and/or XRIGHT are
00157 *>           used, the columns/rows they are in should be included in
00158 *>           NL, e.g., if LLEFT = LRIGHT = .TRUE., then NL must be at
00159 *>           least 2.  The number of rows/columns to be rotated
00160 *>           exclusive of those involving XLEFT and/or XRIGHT may
00161 *>           not be negative, i.e., NL minus how many of LLEFT and
00162 *>           LRIGHT are .TRUE. must be at least zero; if not, XERBLA
00163 *>           will be called.
00164 *>           Not modified.
00165 *>
00166 *>  C, S   - REAL
00167 *>           Specify the Givens rotation to be applied.  If LROWS is
00168 *>           true, then the matrix ( c  s )
00169 *>                                 (-s  c )  is applied from the left;
00170 *>           if false, then the transpose thereof is applied from the
00171 *>           right.  For a Givens rotation, C**2 + S**2 should be 1,
00172 *>           but this is not checked.
00173 *>           Not modified.
00174 *>
00175 *>  A      - REAL array.
00176 *>           The array containing the rows/columns to be rotated.  The
00177 *>           first element of A should be the upper left element to
00178 *>           be rotated.
00179 *>           Read and modified.
00180 *>
00181 *>  LDA    - INTEGER
00182 *>           The "effective" leading dimension of A.  If A contains
00183 *>           a matrix stored in GE or SY format, then this is just
00184 *>           the leading dimension of A as dimensioned in the calling
00185 *>           routine.  If A contains a matrix stored in band (GB or SB)
00186 *>           format, then this should be *one less* than the leading
00187 *>           dimension used in the calling routine.  Thus, if
00188 *>           A were dimensioned A(LDA,*) in SLAROT, then A(1,j) would
00189 *>           be the j-th element in the first of the two rows
00190 *>           to be rotated, and A(2,j) would be the j-th in the second,
00191 *>           regardless of how the array may be stored in the calling
00192 *>           routine.  [A cannot, however, actually be dimensioned thus,
00193 *>           since for band format, the row number may exceed LDA, which
00194 *>           is not legal FORTRAN.]
00195 *>           If LROWS=.TRUE., then LDA must be at least 1, otherwise
00196 *>           it must be at least NL minus the number of .TRUE. values
00197 *>           in XLEFT and XRIGHT.
00198 *>           Not modified.
00199 *>
00200 *>  XLEFT  - REAL
00201 *>           If LLEFT is .TRUE., then XLEFT will be used and modified
00202 *>           instead of A(2,1) (if LROWS=.TRUE.) or A(1,2)
00203 *>           (if LROWS=.FALSE.).
00204 *>           Read and modified.
00205 *>
00206 *>  XRIGHT - REAL
00207 *>           If LRIGHT is .TRUE., then XRIGHT will be used and modified
00208 *>           instead of A(1,NL) (if LROWS=.TRUE.) or A(NL,1)
00209 *>           (if LROWS=.FALSE.).
00210 *>           Read and modified.
00211 *> \endverbatim
00212 *
00213 *  Authors:
00214 *  ========
00215 *
00216 *> \author Univ. of Tennessee 
00217 *> \author Univ. of California Berkeley 
00218 *> \author Univ. of Colorado Denver 
00219 *> \author NAG Ltd. 
00220 *
00221 *> \date November 2011
00222 *
00223 *> \ingroup real_matgen
00224 *
00225 *  =====================================================================
00226       SUBROUTINE SLAROT( LROWS, LLEFT, LRIGHT, NL, C, S, A, LDA, XLEFT,
00227      $                   XRIGHT )
00228 *
00229 *  -- LAPACK auxiliary routine (version 3.4.0) --
00230 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00231 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00232 *     November 2011
00233 *
00234 *     .. Scalar Arguments ..
00235       LOGICAL            LLEFT, LRIGHT, LROWS
00236       INTEGER            LDA, NL
00237       REAL               C, S, XLEFT, XRIGHT
00238 *     ..
00239 *     .. Array Arguments ..
00240       REAL               A( * )
00241 *     ..
00242 *
00243 *  =====================================================================
00244 *
00245 *     .. Local Scalars ..
00246       INTEGER            IINC, INEXT, IX, IY, IYT, NT
00247 *     ..
00248 *     .. Local Arrays ..
00249       REAL               XT( 2 ), YT( 2 )
00250 *     ..
00251 *     .. External Subroutines ..
00252       EXTERNAL           SROT, XERBLA
00253 *     ..
00254 *     .. Executable Statements ..
00255 *
00256 *     Set up indices, arrays for ends
00257 *
00258       IF( LROWS ) THEN
00259          IINC = LDA
00260          INEXT = 1
00261       ELSE
00262          IINC = 1
00263          INEXT = LDA
00264       END IF
00265 *
00266       IF( LLEFT ) THEN
00267          NT = 1
00268          IX = 1 + IINC
00269          IY = 2 + LDA
00270          XT( 1 ) = A( 1 )
00271          YT( 1 ) = XLEFT
00272       ELSE
00273          NT = 0
00274          IX = 1
00275          IY = 1 + INEXT
00276       END IF
00277 *
00278       IF( LRIGHT ) THEN
00279          IYT = 1 + INEXT + ( NL-1 )*IINC
00280          NT = NT + 1
00281          XT( NT ) = XRIGHT
00282          YT( NT ) = A( IYT )
00283       END IF
00284 *
00285 *     Check for errors
00286 *
00287       IF( NL.LT.NT ) THEN
00288          CALL XERBLA( 'SLAROT', 4 )
00289          RETURN
00290       END IF
00291       IF( LDA.LE.0 .OR. ( .NOT.LROWS .AND. LDA.LT.NL-NT ) ) THEN
00292          CALL XERBLA( 'SLAROT', 8 )
00293          RETURN
00294       END IF
00295 *
00296 *     Rotate
00297 *
00298       CALL SROT( NL-NT, A( IX ), IINC, A( IY ), IINC, C, S )
00299       CALL SROT( NT, XT, 1, YT, 1, C, S )
00300 *
00301 *     Stuff values back into XLEFT, XRIGHT, etc.
00302 *
00303       IF( LLEFT ) THEN
00304          A( 1 ) = XT( 1 )
00305          XLEFT = YT( 1 )
00306       END IF
00307 *
00308       IF( LRIGHT ) THEN
00309          XRIGHT = XT( NT )
00310          A( IYT ) = YT( NT )
00311       END IF
00312 *
00313       RETURN
00314 *
00315 *     End of SLAROT
00316 *
00317       END
 All Files Functions