zpbstf.f

          SUBROUTINE ZPBSTF( UPLO, N, KD, AB, LDAB, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          UPLO
          INTEGER            INFO, KD, LDAB, N
    *     ..
    *     .. Array Arguments ..
          COMPLEX*16         AB( LDAB, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  ZPBSTF computes a split Cholesky factorization of a complex
    *  Hermitian positive definite band matrix A.
    *
    *  This routine is designed to be used in conjunction with ZHBGST.
    *
    *  The factorization has the form  A = S**H*S  where S is a band matrix
    *  of the same bandwidth as A and the following structure:
    *
    *    S = ( U    )
    *        ( M  L )
    *
    *  where U is upper triangular of order m = (n+kd)/2, and L is lower
    *  triangular of order n-m.
    *
    *  Arguments
    *  =========
    *
    *  UPLO    (input) CHARACTER*1
    *          = 'U':  Upper triangle of A is stored;
    *          = 'L':  Lower triangle of A is stored.
    *
    *  N       (input) INTEGER
    *          The order of the matrix A.  N >= 0.
    *
    *  KD      (input) INTEGER
    *          The number of superdiagonals of the matrix A if UPLO = 'U',
    *          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
    *
    *  AB      (input/output) COMPLEX*16 array, dimension (LDAB,N)
    *          On entry, the upper or lower triangle of the Hermitian band
    *          matrix A, stored in the first kd+1 rows of the array.  The
    *          j-th column of A is stored in the j-th column of the array AB
    *          as follows:
    *          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
    *
    *          On exit, if INFO = 0, the factor S from the split Cholesky
    *          factorization A = S**H*S. See Further Details.
    *
    *  LDAB    (input) INTEGER
    *          The leading dimension of the array AB.  LDAB >= KD+1.
    *
    *  INFO    (output) INTEGER
    *          = 0: successful exit
    *          < 0: if INFO = -i, the i-th argument had an illegal value
    *          > 0: if INFO = i, the factorization could not be completed,
    *               because the updated element a(i,i) was negative; the
    *               matrix A is not positive definite.
    *
    *  Further Details
    *  ===============
    *
    *  The band storage scheme is illustrated by the following example, when
    *  N = 7, KD = 2:
    *
    *  S = ( s11  s12  s13                     )
    *      (      s22  s23  s24                )
    *      (           s33  s34                )
    *      (                s44                )
    *      (           s53  s54  s55           )
    *      (                s64  s65  s66      )
    *      (                     s75  s76  s77 )
    *
    *  If UPLO = 'U', the array AB holds:
    *
    *  on entry:                          on exit:
    *
    *   *    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53' s64' s75'
    *   *   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54' s65' s76'
    *  a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77
    *
    *  If UPLO = 'L', the array AB holds:
    *
    *  on entry:                          on exit:
    *
    *  a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77
    *  a21  a32  a43  a54  a65  a76   *   s12' s23' s34' s54  s65  s76   *
    *  a31  a42  a53  a64  a64   *    *   s13' s24' s53  s64  s75   *    *
    *
    *  Array elements marked * are not used by the routine; s12' denotes
    *  conjg(s12); the diagonal elements of S are real.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ONE, ZERO
          PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            UPPER
          INTEGER            J, KLD, KM, M
          DOUBLE PRECISION   AJJ
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          EXTERNAL           LSAME
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           XERBLA, ZDSCAL, ZHER, ZLACGV
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          DBLE, MAX, MIN, SQRT
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters.
    *
          INFO = 0
          UPPER = LSAME( UPLO, 'U' )
          IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
             INFO = -1
          ELSE IF( N.LT.0 ) THEN
             INFO = -2
          ELSE IF( KD.LT.0 ) THEN
             INFO = -3
          ELSE IF( LDAB.LT.KD+1 ) THEN
             INFO = -5
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'ZPBSTF', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.EQ.0 )
         $   RETURN
    *
          KLD = MAX( 1, LDAB-1 )
    *
    *     Set the splitting point m.
    *
          M = ( N+KD ) / 2
    *
          IF( UPPER ) THEN
    *
    *        Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
    *
             DO 10 J = N, M + 1, -1
    *
    *           Compute s(j,j) and test for non-positive-definiteness.
    *
                AJJ = DBLE( AB( KD+1, J ) )
                IF( AJJ.LE.ZERO ) THEN
                   AB( KD+1, J ) = AJJ
                   GO TO 50
                END IF
                AJJ = SQRT( AJJ )
                AB( KD+1, J ) = AJJ
                KM = MIN( J-1, KD )
    *
    *           Compute elements j-km:j-1 of the j-th column and update the
    *           the leading submatrix within the band.
    *
                CALL ZDSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 )
                CALL ZHER( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1,
         $                 AB( KD+1, J-KM ), KLD )
       10    CONTINUE
    *
    *        Factorize the updated submatrix A(1:m,1:m) as U**H*U.
    *
             DO 20 J = 1, M
    *
    *           Compute s(j,j) and test for non-positive-definiteness.
    *
                AJJ = DBLE( AB( KD+1, J ) )
                IF( AJJ.LE.ZERO ) THEN
                   AB( KD+1, J ) = AJJ
                   GO TO 50
                END IF
                AJJ = SQRT( AJJ )
                AB( KD+1, J ) = AJJ
                KM = MIN( KD, M-J )
    *
    *           Compute elements j+1:j+km of the j-th row and update the
    *           trailing submatrix within the band.
    *
                IF( KM.GT.0 ) THEN
                   CALL ZDSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD )
                   CALL ZLACGV( KM, AB( KD, J+1 ), KLD )
                   CALL ZHER( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD,
         $                    AB( KD+1, J+1 ), KLD )
                   CALL ZLACGV( KM, AB( KD, J+1 ), KLD )
                END IF
       20    CONTINUE
          ELSE
    *
    *        Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
    *
             DO 30 J = N, M + 1, -1
    *
    *           Compute s(j,j) and test for non-positive-definiteness.
    *
                AJJ = DBLE( AB( 1, J ) )
                IF( AJJ.LE.ZERO ) THEN
                   AB( 1, J ) = AJJ
                   GO TO 50
                END IF
                AJJ = SQRT( AJJ )
                AB( 1, J ) = AJJ
                KM = MIN( J-1, KD )
    *
    *           Compute elements j-km:j-1 of the j-th row and update the
    *           trailing submatrix within the band.
    *
                CALL ZDSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD )
                CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD )
                CALL ZHER( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD,
         $                 AB( 1, J-KM ), KLD )
                CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD )
       30    CONTINUE
    *
    *        Factorize the updated submatrix A(1:m,1:m) as U**H*U.
    *
             DO 40 J = 1, M
    *
    *           Compute s(j,j) and test for non-positive-definiteness.
    *
                AJJ = DBLE( AB( 1, J ) )
                IF( AJJ.LE.ZERO ) THEN
                   AB( 1, J ) = AJJ
                   GO TO 50
                END IF
                AJJ = SQRT( AJJ )
                AB( 1, J ) = AJJ
                KM = MIN( KD, M-J )
    *
    *           Compute elements j+1:j+km of the j-th column and update the
    *           trailing submatrix within the band.
    *
                IF( KM.GT.0 ) THEN
                   CALL ZDSCAL( KM, ONE / AJJ, AB( 2, J ), 1 )
                   CALL ZHER( 'Lower', KM, -ONE, AB( 2, J ), 1,
         $                    AB( 1, J+1 ), KLD )
                END IF
       40    CONTINUE
          END IF
          RETURN
    *
       50 CONTINUE
          INFO = J
          RETURN
    *
    *     End of ZPBSTF
    *
          END