sgbbrd.f

          SUBROUTINE SGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
         $                   LDQ, PT, LDPT, C, LDC, WORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          VECT
          INTEGER            INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC
    *     ..
    *     .. Array Arguments ..
          REAL               AB( LDAB, * ), C( LDC, * ), D( * ), E( * ),
         $                   PT( LDPT, * ), Q( LDQ, * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SGBBRD reduces a real general m-by-n band matrix A to upper
    *  bidiagonal form B by an orthogonal transformation: Q' * A * P = B.
    *
    *  The routine computes B, and optionally forms Q or P', or computes
    *  Q'*C for a given matrix C.
    *
    *  Arguments
    *  =========
    *
    *  VECT    (input) CHARACTER*1
    *          Specifies whether or not the matrices Q and P' are to be
    *          formed.
    *          = 'N': do not form Q or P';
    *          = 'Q': form Q only;
    *          = 'P': form P' only;
    *          = 'B': form both.
    *
    *  M       (input) INTEGER
    *          The number of rows of the matrix A.  M >= 0.
    *
    *  N       (input) INTEGER
    *          The number of columns of the matrix A.  N >= 0.
    *
    *  NCC     (input) INTEGER
    *          The number of columns of the matrix C.  NCC >= 0.
    *
    *  KL      (input) INTEGER
    *          The number of subdiagonals of the matrix A. KL >= 0.
    *
    *  KU      (input) INTEGER
    *          The number of superdiagonals of the matrix A. KU >= 0.
    *
    *  AB      (input/output) REAL array, dimension (LDAB,N)
    *          On entry, the m-by-n band matrix A, stored in rows 1 to
    *          KL+KU+1. The j-th column of A is stored in the j-th column of
    *          the array AB as follows:
    *          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
    *          On exit, A is overwritten by values generated during the
    *          reduction.
    *
    *  LDAB    (input) INTEGER
    *          The leading dimension of the array A. LDAB >= KL+KU+1.
    *
    *  D       (output) REAL array, dimension (min(M,N))
    *          The diagonal elements of the bidiagonal matrix B.
    *
    *  E       (output) REAL array, dimension (min(M,N)-1)
    *          The superdiagonal elements of the bidiagonal matrix B.
    *
    *  Q       (output) REAL array, dimension (LDQ,M)
    *          If VECT = 'Q' or 'B', the m-by-m orthogonal matrix Q.
    *          If VECT = 'N' or 'P', the array Q is not referenced.
    *
    *  LDQ     (input) INTEGER
    *          The leading dimension of the array Q.
    *          LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
    *
    *  PT      (output) REAL array, dimension (LDPT,N)
    *          If VECT = 'P' or 'B', the n-by-n orthogonal matrix P'.
    *          If VECT = 'N' or 'Q', the array PT is not referenced.
    *
    *  LDPT    (input) INTEGER
    *          The leading dimension of the array PT.
    *          LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
    *
    *  C       (input/output) REAL array, dimension (LDC,NCC)
    *          On entry, an m-by-ncc matrix C.
    *          On exit, C is overwritten by Q'*C.
    *          C is not referenced if NCC = 0.
    *
    *  LDC     (input) INTEGER
    *          The leading dimension of the array C.
    *          LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
    *
    *  WORK    (workspace) REAL array, dimension (2*max(M,N))
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit.
    *          < 0:  if INFO = -i, the i-th argument had an illegal value.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          REAL               ZERO, ONE
          PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            WANTB, WANTC, WANTPT, WANTQ
          INTEGER            I, INCA, J, J1, J2, KB, KB1, KK, KLM, KLU1,
         $                   KUN, L, MINMN, ML, ML0, MN, MU, MU0, NR, NRT
          REAL               RA, RB, RC, RS
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           SLARGV, SLARTG, SLARTV, SLASET, SROT, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, MIN
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          EXTERNAL           LSAME
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters
    *
          WANTB = LSAME( VECT, 'B' )
          WANTQ = LSAME( VECT, 'Q' ) .OR. WANTB
          WANTPT = LSAME( VECT, 'P' ) .OR. WANTB
          WANTC = NCC.GT.0
          KLU1 = KL + KU + 1
          INFO = 0
          IF( .NOT.WANTQ .AND. .NOT.WANTPT .AND. .NOT.LSAME( VECT, 'N' ) )
         $     THEN
             INFO = -1
          ELSE IF( M.LT.0 ) THEN
             INFO = -2
          ELSE IF( N.LT.0 ) THEN
             INFO = -3
          ELSE IF( NCC.LT.0 ) THEN
             INFO = -4
          ELSE IF( KL.LT.0 ) THEN
             INFO = -5
          ELSE IF( KU.LT.0 ) THEN
             INFO = -6
          ELSE IF( LDAB.LT.KLU1 ) THEN
             INFO = -8
          ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. LDQ.LT.MAX( 1, M ) ) THEN
             INFO = -12
          ELSE IF( LDPT.LT.1 .OR. WANTPT .AND. LDPT.LT.MAX( 1, N ) ) THEN
             INFO = -14
          ELSE IF( LDC.LT.1 .OR. WANTC .AND. LDC.LT.MAX( 1, M ) ) THEN
             INFO = -16
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'SGBBRD', -INFO )
             RETURN
          END IF
    *
    *     Initialize Q and P' to the unit matrix, if needed
    *
          IF( WANTQ )
         $   CALL SLASET( 'Full', M, M, ZERO, ONE, Q, LDQ )
          IF( WANTPT )
         $   CALL SLASET( 'Full', N, N, ZERO, ONE, PT, LDPT )
    *
    *     Quick return if possible.
    *
          IF( M.EQ.0 .OR. N.EQ.0 )
         $   RETURN
    *
          MINMN = MIN( M, N )
    *
          IF( KL+KU.GT.1 ) THEN
    *
    *        Reduce to upper bidiagonal form if KU > 0; if KU = 0, reduce
    *        first to lower bidiagonal form and then transform to upper
    *        bidiagonal
    *
             IF( KU.GT.0 ) THEN
                ML0 = 1
                MU0 = 2
             ELSE
                ML0 = 2
                MU0 = 1
             END IF
    *
    *        Wherever possible, plane rotations are generated and applied in
    *        vector operations of length NR over the index set J1:J2:KLU1.
    *
    *        The sines of the plane rotations are stored in WORK(1:max(m,n))
    *        and the cosines in WORK(max(m,n)+1:2*max(m,n)).
    *
             MN = MAX( M, N )
             KLM = MIN( M-1, KL )
             KUN = MIN( N-1, KU )
             KB = KLM + KUN
             KB1 = KB + 1
             INCA = KB1*LDAB
             NR = 0
             J1 = KLM + 2
             J2 = 1 - KUN
    *
             DO 90 I = 1, MINMN
    *
    *           Reduce i-th column and i-th row of matrix to bidiagonal form
    *
                ML = KLM + 1
                MU = KUN + 1
                DO 80 KK = 1, KB
                   J1 = J1 + KB
                   J2 = J2 + KB
    *
    *              generate plane rotations to annihilate nonzero elements
    *              which have been created below the band
    *
                   IF( NR.GT.0 )
         $            CALL SLARGV( NR, AB( KLU1, J1-KLM-1 ), INCA,
         $                         WORK( J1 ), KB1, WORK( MN+J1 ), KB1 )
    *
    *              apply plane rotations from the left
    *
                   DO 10 L = 1, KB
                      IF( J2-KLM+L-1.GT.N ) THEN
                         NRT = NR - 1
                      ELSE
                         NRT = NR
                      END IF
                      IF( NRT.GT.0 )
         $               CALL SLARTV( NRT, AB( KLU1-L, J1-KLM+L-1 ), INCA,
         $                            AB( KLU1-L+1, J1-KLM+L-1 ), INCA,
         $                            WORK( MN+J1 ), WORK( J1 ), KB1 )
       10          CONTINUE
    *
                   IF( ML.GT.ML0 ) THEN
                      IF( ML.LE.M-I+1 ) THEN
    *
    *                    generate plane rotation to annihilate a(i+ml-1,i)
    *                    within the band, and apply rotation from the left
    *
                         CALL SLARTG( AB( KU+ML-1, I ), AB( KU+ML, I ),
         $                            WORK( MN+I+ML-1 ), WORK( I+ML-1 ),
         $                            RA )
                         AB( KU+ML-1, I ) = RA
                         IF( I.LT.N )
         $                  CALL SROT( MIN( KU+ML-2, N-I ),
         $                             AB( KU+ML-2, I+1 ), LDAB-1,
         $                             AB( KU+ML-1, I+1 ), LDAB-1,
         $                             WORK( MN+I+ML-1 ), WORK( I+ML-1 ) )
                      END IF
                      NR = NR + 1
                      J1 = J1 - KB1
                   END IF
    *
                   IF( WANTQ ) THEN
    *
    *                 accumulate product of plane rotations in Q
    *
                      DO 20 J = J1, J2, KB1
                         CALL SROT( M, Q( 1, J-1 ), 1, Q( 1, J ), 1,
         $                          WORK( MN+J ), WORK( J ) )
       20             CONTINUE
                   END IF
    *
                   IF( WANTC ) THEN
    *
    *                 apply plane rotations to C
    *
                      DO 30 J = J1, J2, KB1
                         CALL SROT( NCC, C( J-1, 1 ), LDC, C( J, 1 ), LDC,
         $                          WORK( MN+J ), WORK( J ) )
       30             CONTINUE
                   END IF
    *
                   IF( J2+KUN.GT.N ) THEN
    *
    *                 adjust J2 to keep within the bounds of the matrix
    *
                      NR = NR - 1
                      J2 = J2 - KB1
                   END IF
    *
                   DO 40 J = J1, J2, KB1
    *
    *                 create nonzero element a(j-1,j+ku) above the band
    *                 and store it in WORK(n+1:2*n)
    *
                      WORK( J+KUN ) = WORK( J )*AB( 1, J+KUN )
                      AB( 1, J+KUN ) = WORK( MN+J )*AB( 1, J+KUN )
       40          CONTINUE
    *
    *              generate plane rotations to annihilate nonzero elements
    *              which have been generated above the band
    *
                   IF( NR.GT.0 )
         $            CALL SLARGV( NR, AB( 1, J1+KUN-1 ), INCA,
         $                         WORK( J1+KUN ), KB1, WORK( MN+J1+KUN ),
         $                         KB1 )
    *
    *              apply plane rotations from the right
    *
                   DO 50 L = 1, KB
                      IF( J2+L-1.GT.M ) THEN
                         NRT = NR - 1
                      ELSE
                         NRT = NR
                      END IF
                      IF( NRT.GT.0 )
         $               CALL SLARTV( NRT, AB( L+1, J1+KUN-1 ), INCA,
         $                            AB( L, J1+KUN ), INCA,
         $                            WORK( MN+J1+KUN ), WORK( J1+KUN ),
         $                            KB1 )
       50          CONTINUE
    *
                   IF( ML.EQ.ML0 .AND. MU.GT.MU0 ) THEN
                      IF( MU.LE.N-I+1 ) THEN
    *
    *                    generate plane rotation to annihilate a(i,i+mu-1)
    *                    within the band, and apply rotation from the right
    *
                         CALL SLARTG( AB( KU-MU+3, I+MU-2 ),
         $                            AB( KU-MU+2, I+MU-1 ),
         $                            WORK( MN+I+MU-1 ), WORK( I+MU-1 ),
         $                            RA )
                         AB( KU-MU+3, I+MU-2 ) = RA
                         CALL SROT( MIN( KL+MU-2, M-I ),
         $                          AB( KU-MU+4, I+MU-2 ), 1,
         $                          AB( KU-MU+3, I+MU-1 ), 1,
         $                          WORK( MN+I+MU-1 ), WORK( I+MU-1 ) )
                      END IF
                      NR = NR + 1
                      J1 = J1 - KB1
                   END IF
    *
                   IF( WANTPT ) THEN
    *
    *                 accumulate product of plane rotations in P'
    *
                      DO 60 J = J1, J2, KB1
                         CALL SROT( N, PT( J+KUN-1, 1 ), LDPT,
         $                          PT( J+KUN, 1 ), LDPT, WORK( MN+J+KUN ),
         $                          WORK( J+KUN ) )
       60             CONTINUE
                   END IF
    *
                   IF( J2+KB.GT.M ) THEN
    *
    *                 adjust J2 to keep within the bounds of the matrix
    *
                      NR = NR - 1
                      J2 = J2 - KB1
                   END IF
    *
                   DO 70 J = J1, J2, KB1
    *
    *                 create nonzero element a(j+kl+ku,j+ku-1) below the
    *                 band and store it in WORK(1:n)
    *
                      WORK( J+KB ) = WORK( J+KUN )*AB( KLU1, J+KUN )
                      AB( KLU1, J+KUN ) = WORK( MN+J+KUN )*AB( KLU1, J+KUN )
       70          CONTINUE
    *
                   IF( ML.GT.ML0 ) THEN
                      ML = ML - 1
                   ELSE
                      MU = MU - 1
                   END IF
       80       CONTINUE
       90    CONTINUE
          END IF
    *
          IF( KU.EQ.0 .AND. KL.GT.0 ) THEN
    *
    *        A has been reduced to lower bidiagonal form
    *
    *        Transform lower bidiagonal form to upper bidiagonal by applying
    *        plane rotations from the left, storing diagonal elements in D
    *        and off-diagonal elements in E
    *
             DO 100 I = 1, MIN( M-1, N )
                CALL SLARTG( AB( 1, I ), AB( 2, I ), RC, RS, RA )
                D( I ) = RA
                IF( I.LT.N ) THEN
                   E( I ) = RS*AB( 1, I+1 )
                   AB( 1, I+1 ) = RC*AB( 1, I+1 )
                END IF
                IF( WANTQ )
         $         CALL SROT( M, Q( 1, I ), 1, Q( 1, I+1 ), 1, RC, RS )
                IF( WANTC )
         $         CALL SROT( NCC, C( I, 1 ), LDC, C( I+1, 1 ), LDC, RC,
         $                    RS )
      100    CONTINUE
             IF( M.LE.N )
         $      D( M ) = AB( 1, M )
          ELSE IF( KU.GT.0 ) THEN
    *
    *        A has been reduced to upper bidiagonal form
    *
             IF( M.LT.N ) THEN
    *
    *           Annihilate a(m,m+1) by applying plane rotations from the
    *           right, storing diagonal elements in D and off-diagonal
    *           elements in E
    *
                RB = AB( KU, M+1 )
                DO 110 I = M, 1, -1
                   CALL SLARTG( AB( KU+1, I ), RB, RC, RS, RA )
                   D( I ) = RA
                   IF( I.GT.1 ) THEN
                      RB = -RS*AB( KU, I )
                      E( I-1 ) = RC*AB( KU, I )
                   END IF
                   IF( WANTPT )
         $            CALL SROT( N, PT( I, 1 ), LDPT, PT( M+1, 1 ), LDPT,
         $                       RC, RS )
      110       CONTINUE
             ELSE
    *
    *           Copy off-diagonal elements to E and diagonal elements to D
    *
                DO 120 I = 1, MINMN - 1
                   E( I ) = AB( KU, I+1 )
      120       CONTINUE
                DO 130 I = 1, MINMN
                   D( I ) = AB( KU+1, I )
      130       CONTINUE
             END IF
          ELSE
    *
    *        A is diagonal. Set elements of E to zero and copy diagonal
    *        elements to D.
    *
             DO 140 I = 1, MINMN - 1
                E( I ) = ZERO
      140    CONTINUE
             DO 150 I = 1, MINMN
                D( I ) = AB( 1, I )
      150    CONTINUE
          END IF
          RETURN
    *
    *     End of SGBBRD
    *
          END