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    1:       SUBROUTINE DPBSTF( UPLO, N, KD, AB, LDAB, INFO )
    2: *
    3: *  -- LAPACK routine (version 3.2) --
    4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    6: *     November 2006
    7: *
    8: *     .. Scalar Arguments ..
    9:       CHARACTER          UPLO
   10:       INTEGER            INFO, KD, LDAB, N
   11: *     ..
   12: *     .. Array Arguments ..
   13:       DOUBLE PRECISION   AB( LDAB, * )
   14: *     ..
   15: *
   16: *  Purpose
   17: *  =======
   18: *
   19: *  DPBSTF computes a split Cholesky factorization of a real
   20: *  symmetric positive definite band matrix A.
   21: *
   22: *  This routine is designed to be used in conjunction with DSBGST.
   23: *
   24: *  The factorization has the form  A = S**T*S  where S is a band matrix
   25: *  of the same bandwidth as A and the following structure:
   26: *
   27: *    S = ( U    )
   28: *        ( M  L )
   29: *
   30: *  where U is upper triangular of order m = (n+kd)/2, and L is lower
   31: *  triangular of order n-m.
   32: *
   33: *  Arguments
   34: *  =========
   35: *
   36: *  UPLO    (input) CHARACTER*1
   37: *          = 'U':  Upper triangle of A is stored;
   38: *          = 'L':  Lower triangle of A is stored.
   39: *
   40: *  N       (input) INTEGER
   41: *          The order of the matrix A.  N >= 0.
   42: *
   43: *  KD      (input) INTEGER
   44: *          The number of superdiagonals of the matrix A if UPLO = 'U',
   45: *          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
   46: *
   47: *  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
   48: *          On entry, the upper or lower triangle of the symmetric band
   49: *          matrix A, stored in the first kd+1 rows of the array.  The
   50: *          j-th column of A is stored in the j-th column of the array AB
   51: *          as follows:
   52: *          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
   53: *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
   54: *
   55: *          On exit, if INFO = 0, the factor S from the split Cholesky
   56: *          factorization A = S**T*S. See Further Details.
   57: *
   58: *  LDAB    (input) INTEGER
   59: *          The leading dimension of the array AB.  LDAB >= KD+1.
   60: *
   61: *  INFO    (output) INTEGER
   62: *          = 0: successful exit
   63: *          < 0: if INFO = -i, the i-th argument had an illegal value
   64: *          > 0: if INFO = i, the factorization could not be completed,
   65: *               because the updated element a(i,i) was negative; the
   66: *               matrix A is not positive definite.
   67: *
   68: *  Further Details
   69: *  ===============
   70: *
   71: *  The band storage scheme is illustrated by the following example, when
   72: *  N = 7, KD = 2:
   73: *
   74: *  S = ( s11  s12  s13                     )
   75: *      (      s22  s23  s24                )
   76: *      (           s33  s34                )
   77: *      (                s44                )
   78: *      (           s53  s54  s55           )
   79: *      (                s64  s65  s66      )
   80: *      (                     s75  s76  s77 )
   81: *
   82: *  If UPLO = 'U', the array AB holds:
   83: *
   84: *  on entry:                          on exit:
   85: *
   86: *   *    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53  s64  s75
   87: *   *   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54  s65  s76
   88: *  a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77
   89: *
   90: *  If UPLO = 'L', the array AB holds:
   91: *
   92: *  on entry:                          on exit:
   93: *
   94: *  a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77
   95: *  a21  a32  a43  a54  a65  a76   *   s12  s23  s34  s54  s65  s76   *
   96: *  a31  a42  a53  a64  a64   *    *   s13  s24  s53  s64  s75   *    *
   97: *
   98: *  Array elements marked * are not used by the routine.
   99: *
  100: *  =====================================================================
  101: *
  102: *     .. Parameters ..
  103:       DOUBLE PRECISION   ONE, ZERO
  104:       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
  105: *     ..
  106: *     .. Local Scalars ..
  107:       LOGICAL            UPPER
  108:       INTEGER            J, KLD, KM, M
  109:       DOUBLE PRECISION   AJJ
  110: *     ..
  111: *     .. External Functions ..
  112:       LOGICAL            LSAME
  113:       EXTERNAL           LSAME
  114: *     ..
  115: *     .. External Subroutines ..
  116:       EXTERNAL           DSCAL, DSYR, XERBLA
  117: *     ..
  118: *     .. Intrinsic Functions ..
  119:       INTRINSIC          MAX, MIN, SQRT
  120: *     ..
  121: *     .. Executable Statements ..
  122: *
  123: *     Test the input parameters.
  124: *
  125:       INFO = 0
  126:       UPPER = LSAME( UPLO, 'U' )
  127:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
  128:          INFO = -1
  129:       ELSE IF( N.LT.0 ) THEN
  130:          INFO = -2
  131:       ELSE IF( KD.LT.0 ) THEN
  132:          INFO = -3
  133:       ELSE IF( LDAB.LT.KD+1 ) THEN
  134:          INFO = -5
  135:       END IF
  136:       IF( INFO.NE.0 ) THEN
  137:          CALL XERBLA( 'DPBSTF', -INFO )
  138:          RETURN
  139:       END IF
  140: *
  141: *     Quick return if possible
  142: *
  143:       IF( N.EQ.0 )
  144:      $   RETURN
  145: *
  146:       KLD = MAX( 1, LDAB-1 )
  147: *
  148: *     Set the splitting point m.
  149: *
  150:       M = ( N+KD ) / 2
  151: *
  152:       IF( UPPER ) THEN
  153: *
  154: *        Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
  155: *
  156:          DO 10 J = N, M + 1, -1
  157: *
  158: *           Compute s(j,j) and test for non-positive-definiteness.
  159: *
  160:             AJJ = AB( KD+1, J )
  161:             IF( AJJ.LE.ZERO )
  162:      $         GO TO 50
  163:             AJJ = SQRT( AJJ )
  164:             AB( KD+1, J ) = AJJ
  165:             KM = MIN( J-1, KD )
  166: *
  167: *           Compute elements j-km:j-1 of the j-th column and update the
  168: *           the leading submatrix within the band.
  169: *
  170:             CALL DSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 )
  171:             CALL DSYR( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1,
  172:      $                 AB( KD+1, J-KM ), KLD )
  173:    10    CONTINUE
  174: *
  175: *        Factorize the updated submatrix A(1:m,1:m) as U**T*U.
  176: *
  177:          DO 20 J = 1, M
  178: *
  179: *           Compute s(j,j) and test for non-positive-definiteness.
  180: *
  181:             AJJ = AB( KD+1, J )
  182:             IF( AJJ.LE.ZERO )
  183:      $         GO TO 50
  184:             AJJ = SQRT( AJJ )
  185:             AB( KD+1, J ) = AJJ
  186:             KM = MIN( KD, M-J )
  187: *
  188: *           Compute elements j+1:j+km of the j-th row and update the
  189: *           trailing submatrix within the band.
  190: *
  191:             IF( KM.GT.0 ) THEN
  192:                CALL DSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD )
  193:                CALL DSYR( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD,
  194:      $                    AB( KD+1, J+1 ), KLD )
  195:             END IF
  196:    20    CONTINUE
  197:       ELSE
  198: *
  199: *        Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
  200: *
  201:          DO 30 J = N, M + 1, -1
  202: *
  203: *           Compute s(j,j) and test for non-positive-definiteness.
  204: *
  205:             AJJ = AB( 1, J )
  206:             IF( AJJ.LE.ZERO )
  207:      $         GO TO 50
  208:             AJJ = SQRT( AJJ )
  209:             AB( 1, J ) = AJJ
  210:             KM = MIN( J-1, KD )
  211: *
  212: *           Compute elements j-km:j-1 of the j-th row and update the
  213: *           trailing submatrix within the band.
  214: *
  215:             CALL DSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD )
  216:             CALL DSYR( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD,
  217:      $                 AB( 1, J-KM ), KLD )
  218:    30    CONTINUE
  219: *
  220: *        Factorize the updated submatrix A(1:m,1:m) as U**T*U.
  221: *
  222:          DO 40 J = 1, M
  223: *
  224: *           Compute s(j,j) and test for non-positive-definiteness.
  225: *
  226:             AJJ = AB( 1, J )
  227:             IF( AJJ.LE.ZERO )
  228:      $         GO TO 50
  229:             AJJ = SQRT( AJJ )
  230:             AB( 1, J ) = AJJ
  231:             KM = MIN( KD, M-J )
  232: *
  233: *           Compute elements j+1:j+km of the j-th column and update the
  234: *           trailing submatrix within the band.
  235: *
  236:             IF( KM.GT.0 ) THEN
  237:                CALL DSCAL( KM, ONE / AJJ, AB( 2, J ), 1 )
  238:                CALL DSYR( 'Lower', KM, -ONE, AB( 2, J ), 1,
  239:      $                    AB( 1, J+1 ), KLD )
  240:             END IF
  241:    40    CONTINUE
  242:       END IF
  243:       RETURN
  244: *
  245:    50 CONTINUE
  246:       INFO = J
  247:       RETURN
  248: *
  249: *     End of DPBSTF
  250: *
  251:       END