SLAROT 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:     C> C> C> C> C> .  .  .  .


    row j+1:      C> C> C> C> C> .  .  .  .


    '*' 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:     C> C> C> C> C> p  .  .  .


    row j+1:   q  C> C> C> C> C> .  .  .  .


    The element p would have to be set correctly, then that column


    is rotated, setting p to its new value.  The next call to


    SLAROT 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 SLAROT(.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 SLAROT( .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 SLAROT( .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 SLAROT( .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 SLAROT( .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 )

  LROWS  - LOGICAL


           If .TRUE., then SLAROT 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   - REAL


           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 thereof is applied from the


           right.  For a Givens rotation, C**2 + S**2 should be 1,


           but this is not checked.


           Not modified.


  A      - REAL 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 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 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 SLAROT, 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  - REAL


           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 - REAL


           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.

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